/ BROADBAND MICROWAVE MEASUREMENTS OF TWO DIMENSIONAL QUANTUM MATTER by Wei Liu A dissertation submitted to The Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy. Baltimore, Maryland January, 2013 © Wei Liu 2013 All rights reserved UMI Number: 3572735 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Di!ss0?t&iori P iiblist’Mlg UMI 3572735 Published by ProQuest LLC 2013. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 Abstract Employing broadband microwave spectroscopy, we study a field tuned 2D superconductor-metal quantum phase transition in a low-disorder InOx film. We measure the complex conductance as a function of temperature, frequency and magnetic field. The zero field transition temperature Tc = T k tb is determined as the temperature where the phase stiffness starts to acquire frequency dependence. The thermal phase tran sition is consistent with the Kosterlitz-Thouless-Berezinsky formalism. The AC data demonstrate the critical slowing down close to ity of a vortex plasma model above T k tb T k tb and in general the applicabil • For finite field measurements, we find the strong suppression in the superfluid stiffness above the nominal quantum critical point Bcross where different isotherms of resistance as a function of magnetic field cross each other. The critical slowing down of the fluctuation rate near a continuous quantum phase transition supports a possible scenario that the quantum critical point is at a much lower field B sm above which the film transits into an anomalous metallic state with superconducting correlations. A phase diagram is established that includes the magnetic field dependence of the superconducting transition temperatures, phase ABSTRACT stiffness at the lowest and highest accessed frequencies, and the fluctuation rates at the base temperature. Primary Reader: N. Peter Armitage Secondary Reader: Predrag Nikolic Acknowledgments My interests in science were inspired actually by anecdotes of scientists I read when I was a kid; how they would do something ridiculous cause they were so focused on their research and thinking. From then on, for some reason, I was always imagining being one of those nerds. So now here I am. After 13 years of majoring in physics and doing research in physics, I am much closer to the type of people I always want to be. Of course, there is no end in the pursuing. I have grown a lot in the past few years in Hopkins and this in turn makes my stay and PhD life the most valuable years in my life. W hat I learned here is not just knowledge, but the way of thinking and solving problems. So many people here I would like to acknowledge for their help, support, love and companionship. I could not have carried on in the journey of pursuing my PhD without them. First I would like to acknowledge my advisor Prof. N. Peter Armitage. He has set a great example to me about how to be a great scientist by his passion and consistent pursuit in physics. I appreciate all his support and trust along the way. He always ACKNOWLEDGMENTS has the patience for my questions even though most of time my questions may be simple and naive and may not be of physics nature. There have been countless times that I call from the lab about any possible situations happening in the lab and Peter always has time to discuss. I also feel grateful for all the travel opportunities that I have throughout these years to present my work to colleagues outside the university. I feel really lucky to have Peter as my PhD advisor. Natalia Drichko has been a great mentor in life for me. She always has time when I want to talk to her. Thank you for being so generous of your time. My lab mate Luke Bilbro, who joined the lab half a year earlier, has been a great accompany for more than five years of working together in the lab. I would like to thank him for his support and friendship throughout almost my whole graduate school. Rolando Valdes Aguilar has taught me so much about physics, ways of thinking and countless useful techniques and skills in writing and giving presentation. I thank him for his inspiring and useful conversations. I also thank all the rest lab members: Andreas, Chris, LiDong, Mohammad, Yuval, Grace, Liang, Nick and Ji, thank you all for the great time together and friendship. I am glad to have this great opportunity to work with you all. I acknowledge my long time collaborator Prof. Sambandamurthy at University of Buffalo for his efforts and supports in many ways. I enjoyed all the conversations and discussions with him. I also thank Yufeng from Prof. Ruoff’s group at UT Austin for the high quality graphene samples. ACKNOWLEDGMENTS I would like to acknowledge all the professors who taught me during my graduate school. I have learned so much from all the lectures and valuable discussions. I also have been fortunate to meet all my dear friends through the department, JHU Taekwondo club and the school. I thank my family for always being supportive, especially my husband Brian. Without their love and understanding, this thesis work would have been impossible. D edication Dedicated to my family Contents A bstract ii Acknowledgments iv List of Figures xi 1 Introduction 1 1.1 Ginzburg-Landau th e o r y .......................................................................... 2 Enhanced conductivity above Tcby amplitude fluctuations . . 5 1.1.1 1.2 Kosterlitz-Thouless-Berezinsky phase tr a n s itio n .................................. 8 1.2.1 2D XY m o d e l................................................................................ 9 1.2.2 Superfluid stiffness and the universal j u m p ................................ 14 1.2.3 Nonlinear IV characteristic........................................................... 16 1.3 2D superconductor-insulator quantum phase tr a n s itio n ....................... 17 1.4 AC response of a su p erco n d u cto r.......................................................... 23 1.5 Thesis overview ......................................................................................... 26 viii CONTENTS 2 Broadband Corbino microwave spectrom eter 29 2.1 Experimental setup overview................................................................... 31 2.2 Data a n a ly s is ............................................................................................ 39 2.3 C alibrations............................................................................................... 42 2.3.1 Room temperature c a lib ra tio n s.................................................. 43 2.3.2 Low temperature c a lib ra tio n s..................................................... 52 2.3.2.1 Effects of the superconducting c a b l e ......................... 53 2.3.2.2 Error coefficients........................................................... 60 2.3.3 2.4 3 4 5 Microwave measurements in a perpendicular magnetic field. . Review of the Corbino spectrometers from other g ro u p s...................... 62 67 AC studies o f th e zero field phase transition 69 3.1 Superconducting fluctuations................................................................... 69 3.2 Dynamics of KTB phase tr a n s itio n ....................................................... 73 3.3 Sample Details ......................................................................................... 76 3.4 R esults........................................................................................................ 78 3.5 Conclusion.................................................................................................. 89 2D field tuned superconductor-m etal quantum phase transition 91 4.1 Dynamics of 2D quantum phase tr a n s itio n .......................................... 93 4.2 Conclusion.................................................................................................. 107 Summary 108 ix CONTENTS A Transmission line m odel 111 A.0.1 Impedance m a tc h in g .................................................................... B 113 Experim ental procedures 116 B.l Measurements at zero fie ld ...................................................................... 116 B.1.1 Operating sequence....................................................................... 118 Measurements at finite field s................................................................... 125 B.2.1 Operating sequence....................................................................... 126 B.2 C Reflection coefflcients for zero-field m easurem ents 130 D M agnetic field distributions 134 E Microwave conductance o f InO* at finite fields 140 F AC conductance o f CVD grown graphene 148 F .l Electrodynamics of single layer graphene...................................................................................................... 149 F.2 Conclusion................................................................................................... 158 Bibliography 160 V ita 178 x List of Figures 1.1 1.2 1.3 GL free energy density function for a > 0 and a < 0....................... The spin configuration of a vortex....................................................... A proposed phase diagram for the dirty boson m o d e l ................... 2.1 2.2 2.3 Scheme of the overall experimental s e t u p ......................................... 31 Pictures of the Corbino p r o b e ........................................................... 33 Scheme and a picture of the Au pattern for the sampleprepared for microwave m easurem ents..................................................................... 34 Scheme plot of the sample s ta g e ........................................................ 35 Pictures of the setup in our l a b ........................................................ 36 Magnitude of S ll for different open su b stra te s............................... 46 Comparison of calibrated impedance for 40 nm NiCr on Siusing dif ferent open stan d ard s 47 Effects of the position of the set s c r e w ............................................ 48 Impedance of a 10 nm A1 film with different spring configurations . . 50 Impedance of InOj at room temperature showing the importance of substrate correction.............................................................................. 50 Magnitude of the substrate im p e d a n c e ............................................ 52 Four temperature scans for the bulk copper at 7 GHz with a 20 cm superconducting cable in the transmission l i n e ................................ 55 Magnitude and phase of S™ of Si as a function of temperature at 7 GHz with a 10 cm superconducting cable in the transmission line . . 56 Magnitude and phase of S'™ of InOx as a function of temperature at 7 G H z ....................................................................................................... 59 Temperature dependence of the magnitude and phase of the three error terms at 9 G H z .................................................................................... 61 The hysteresis of the magnet in our sy stem ...................................... 64 Magnitude and phase of S™ of a Si standard as a function of field at 300 mK for 4 fixed frequencies............................................................ 65 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 xi 4 13 19 LIST OF FIGURES 2.18 Ratio of real conductance of InOx measured at 3.5 Tesla but calibrated by standards measured at 2 Tesla and 3.5 T e s l a .................................. 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 4.1 4.2 4.3 4.4 4.5 4.6 A .l B.l 66 Rq as a function of temperature for one of the InOx films................... Measured R q as a function of temperature along with two fitted effec tive normal sheet resistances.................................................................... The AFM image, TEM diffraction pattern and TEM image of a co deposited insulating InOx f i lm ................................................................. Sheet resistance of a granular InOx film ................................................ Temperature and frequency dependence of the complex conductance at zero f i e l d ................................................................................................ The phase stiffness at different frequencies as a function of temperature The phase stiffness on a linear scale near Tc ....................................... The phase and magnitude of the complex c o n d u c ta n c e .................... The phase and magnitude of the scaling function .............................. Comparison of the scaling forms with the 2D AL formalism.............. Different fittings to the temperature dependence of the characteristic fluctuation frequen cy................................................................................ 71 72 77 77 79 81 83 84 85 86 90 Co-measured sheet resistance as a function of temperature in different fields.............................................................................................................. 95 Rq measured in a separate dilution fridge e n v iro n m en t..................... 98 Frequency dependence of the real and imaginary conductance at dif ferent f i e l d s ................................................................................................ 99 Frequency dependence of the phase stiffness at 300 mK at different fields 100 Experimentally measured real and imaginary conductance and fitted data to a Lorentzian lineshape model....................................................... 103 Fluctuation rates at different magnetic fields and a proposed phase d ia g ra m ...................................................................................................... 105 A schematic plot of the transmission line m o d e l................................. Ill Examples of temperature profiles for the standardized zero field mea surements...................................................................................................... 117 C .l C.2 C.3 Magnitude and phase of S'™ of one Nb film at 7 G H z ........................ Magnitude and phase of S™ of one NiCr film at 7 G H z ..................... Magnitude and phase of S™ of a glass substrate at 7 G H z.................. 131 132 133 D .l D.2 D.3 D.4 The design of our cryostat from Janis company ................................. Contour plot of B z .................................................................................. Contour plot of B r .................................................................................... Contour plot of the magnitude of the magnetic field............................. 136 137 138 139 LIST OF FIGURES E .l E.2 E.3 E.4 E.5 E.6 E.T Complex Complex Complex Complex Complex Complex Complex conductance conductance conductance conductance conductance conductance conductance and and and and and and and phase phase phase phase phase phase phase stiffness stiffness stiffness stiffness stiffness stiffness stiffness F .l F.2 at at at at at at at B B B B B B B = = = = = = = 0 T e s l a ........ 1T e s l a .................... 2T e s l a .................... 3.5 T e s l a ............... 4 T e s l a ........ 5T e s l a .................... 6T e s l a .................... 141 142 143 144 145 146 147 Raman spectra and DC values of graphene on S i ......................... 151 Calibrated impedance and conductance of graphene in the microwave ra n g e ................................................................................................... 153 F.3 Ratio of the real and imaginary conductance of graphene in the mi crowave r a n g e 155 F.4 Complex normalized conductance of graphene in terahertz range . . . 157 xiii Chapter 1 Introduction Superconductivity was first discovered by Heike Kamerlingh Onnes in 1911. The three hallmarks of superconductivity are zero electrical resistance, Meissner effect (perfect diamagnetism) and flux quantization. One of the early phenomenological descriptions of superconductivity that remains useful nowadays was proposed by the Londons [1]. The two famous London equations 47rA2 C^ t V x Js = c2 a3dt -B - E. (1.1) (1.2) govern the microscopic electronic and magnetic fields. J s is the current density. A is the penetration length which is defined as 47rA2/c2 = m /( n se2) where n8 is the number density of superconducting electrons. These equations were followed by the phenomenological mean-field Ginzburg and Landau (GL) theory [2], and then the fundamental microscopic BCS theory [3]. 1 CHAPTER 1. INTRODUCTION The well accepted microscopic BCS theory, was developed by Bardeen, Cooper and Schrieffer to explain the superconducting state. According to this theory, even a weak attractive interaction between electrons causes an instability of the Fermi-sea to the formation of bound pairs of electrons with equal and opposite momentum and spin. These bound pairs of electrons, so called Cooper pairs, comprise the superconducting charge carriers. Therefore, superconductivity can be understood as an effect caused by a condensation of Cooper pairs into a macroscopic quantum state. In conventional superconductors, the attraction is caused by the electron phonon interaction. One of the key predictions of the BCS theory is the existent of an energy gap A, of order kBTc. A minimum energy Eg — 2A is required to break a Cooper pair and create two quasi-particle excitations. In the weak coupling limit, near the critical temperature Tc, Eg depends on temperature as Eg = 3.52ksTcy / l —(T /T c). This expression is independent of materials and has been confirmed in numerous experiments. 1.1 Ginzburg-Landau theory Despite the triumphal success of BCS theory, the full microscopic theory becomes very difficult in some situations, such as systems with the presence of inhomogeneous ns. Another powerful tool that has been widely used to describe superconductivity is the Ginzburg-Landau (GL) theory, which was postulated as a phenomenological model. GL theory is also very useful in describing thermal and quantum supercon 2 CHAPTER 1. INTRODUCTION ducting fluctuations, which are the main focuses of this thesis. In this section, I will mainly discuss the GL mean field theory and the Aslamazov-Larkin conductiv ity. The treatment given here loosely follows the more explicit treatments given in references [4,5]. In the GL theory, a pseudowavefunction tp(r) = A eltp is introduced as a complex order parameter, which describes the center-of-mass motion of the electron pairs. |^ (r)|2 is the local number density of superconducting electrons ns(r), which goes to zero at Tc when there are no thermal fluctuations present. Assuming ip is small and varies slowly in space, the free energy of a superconductor consists of three main parts Fs = Fn + Condensation energy + Kinetic energy + Field energy Fn is the free energy in the normal phase. Therefore, the free energy density can be written as (1.3) If ip — 0, fs = + Here, because of the pairing, e* = 2e and m* = 2m e, where m e is the electron mass, a and j3 (fi > 0) are both material dependent parameters. In the absence of fields and gradients, equation 1.3 becomes (1.4) The minimum of f s - f n occurs at |-0|2 = 0, f s - /„ = 0 when a > 0, and \ip\2 — | = |^oo|2, fs — fn = —|g = when a < 0 as shown in 1.1. Thus, or must change sign at the GL transition temperature Tc. Following Ginzburg and Landau, 3 CHAPTER 1. INTRODUCTION fs - fn fs - fn A A a > 0 a< 0 Figure 1.1: GL free energy density function for a > 0 and a < 0. we take a = ao(t — 1), where t = T /T c, a 0 > 0 and j3 independent of temperature. One important quantity in the GL theory is the coherence length £. Let’s first examine the GL differential equation which can be obtained by taking d f s/dip = 0 since a choice of ip should minimize the free energy. Without fields, the GL differential equation becomes aip + 0\ip\2tp - — — V 2ip = 0. 2m * (1.5) Assuming ip — ipoo+Sip and keeping the leading terms in Sip, one can rewrite equation 1.5 as fj2 aSip + 3/31^00\28ip - - — V 2Sip = 0. 2m * ( 1.6) Substituting |-0oo|2 = — we obtain (1.7) 2m ' 4 CHAPTER 1. INTRODUCTION The solution to this equation in one dimension is of the form 6xp ~ « T>- s i - where <18> The expression of Sip shows that a perturbation of ip by a small Sip will decay in a length of order £(T). Therefore f(T ) is called the GL coherence length, the charac teristic length for the variation of -0(r). £(T) diverges as T —>Tc. 1.1.1 Enhanced conductivity above Tc by am pli tude fluctuations Thermodynamic fluctuations in 3D clean superconductors are generally small due to the long characteristic coherence length and large superfluid density. These super conductors usually feature a very sharp transition between the normal and supercon ducting states. However, in dirty 3D superconductors, the coherence length is reduced by the mean free path. As a result, thermal fluctuations round the sharp edges of the superconducting transition and bring down the transition temperature. Moreover, they make small but measurable contributions to superconducting properties above Te. We start by first estimating the size of thermal fluctuations of xp near the free energy minimum. This can be obtained from J^jjr) and one has (Sip)2 = ksT /\a\. To better understand the effects of thermal fluctuations, we start with the free energy in the momentum space. xp(r) can be expanded as xp(r) = ]T]k p \e ik r. At 5 CHAPTER 1. INTRODUCTION T > Tc, the free energy density in equation 1.3 in the zero-field case can be rewritten as ' k x We drop the f l^]4 term in the expression above because for the GL mean field theory |^ (r)|2 oc ns is small in the fluctuation regime above Tc except in the narrow critical region near Tc. The thermodynamic average of |^k|2 can be calculated as 12 I l^k|2exp( - f / k BT)d?ip______ k BT f exp(—f / k BT)d?ip < l ^ k| > | a | ( l + **£*) (L10) £ again is the coherence length defined in equation 1.8. The spatial correlation func tion of -0(r) and ^(r') is g( R = r - r') = < < | ^ | 2 > exp(ik • R). >= (1.11) k Plugging in the expression for < 1^12 > and integrating over k, we get g{R) = wtvrexpj im-n) 2irn This shows that (112) R r) is exponentially correlated over the length scale £(T) in the fluc tuation regime above Tc. One can interpret the thermal fluctuations as evanescent short-lived droplets of superconductivity of size ~ £(T) [5]. The presence of these superconducting droplets above Tc must affect the conductivity measured in experi ments. This excess conductivity, which is also called paraconductivity, is in general small, but can be studied especially in low dimensional superconducting samples. To calculate the enhanced conductivity above Tc due to the thermal fluctuations, one has to include the lifetime of the fluctuations. This is because the contribution CHAPTER 1. INTRODUCTION of a fluctuation to the paraconductivity is proportional to the time during which the fluctuation-induced superconducting pairs exist to be accelerated by the external field. This can be modeled by the time dependent Ginzburg-Landau equation as ah dip 8kB(T - Tc) at ( . l2 h2 2 ^ V 2) i/>. (a + m 2- (1.13) This equation is equivalent to assuming the deviation of tp from its equilibrium value to relax exponentially in time. The relaxation time for k = 0 mode is T“ (r) - s M ^ j ' <U4> The linearized TDGL equation can be written as -T O IT J = (1 - f 2V V . (1.15) Thus, the high-energy modes with k > 0 decays more rapidly with a relaxation rate i Tk = i± * £ . (1.16) TGL In the absence of such fluctuations, the normal DC conductivity is on= ne2T/m where r is the mean scattering time of the normal electrons. Wemightexpect that the fluctuation conductivity can be written as _ _ 2e2 af luc ~ m * 2 s k < f^ k l2 > 2 Tk ■ ( ) Inserting < \ip\n\2 > and Tk from expression 1.10 and 1.16 and integrating over the k space, one finds the two-dimensional Aslamazov-Larkin fluctuation conductivity AL _ 20 e2 T 16h d T - T c' 7 / i I q\ ^ ^ CHAPTER 1. INTRODUCTION Another contribution (Maki term) to the fluctuation conductivity comes from the enhanced normal electron conductivity induced by superconducting fluctuations. In the presence of pair-breaking effects, the contribution from the Maki term is strongly suppressed. This pair-breaking parameter should be large for 2D amorphous super conducting films [6,7] and one can safely drop this additional term for these materials. 1.2 K osterlitz-Thouless-Berezinsky phase transition The GL mean field phase transition described in section 1.1 takes place when the magnitude of the order parameter first becomes non-zero. However, two dimensional systems show transition temperatures that can be substantially below the GL tran sition temperature. Unlike 3D superconductors in the clean limit, superconducting fluctuations are largely enhanced in 2D films due to reduced superfluid density, re duced dimensionality and reduced coherence length in the presence of disorder for films such as InOx. In this case, the transition temperature is controlled mainly by fluctuations in the phase of the order parameter induced by the presence of thermally activated free vortices. However, if one writes down the Hamiltonian for a two dimensional system with continuous symmetries and sufficiently short range interactions, the Mermin-Wagner theorem states that the system can not show spontaneous symmetry breaking. It 8 CHAPTER 1. INTRODUCTION means that at any finite temperature it is impossible to have G(r -» oo) = lim (e-M?) fM #)) ± 0. r— >oo (1.19) This expressions means that a true long-range order is impossible. To explain the phase transition in two dimensions, Berezinsky (1971) [8] and Kosterlitz and Thouless (1973) [9] demonstrated that if one includes topological defects (vortices and anti-vortices), there is a transition at T k tb separating power law correlated and ex ponentially correlated states. In this picture, there are thermally activated vortices above Tk t b and G(r) decays exponentially in r; For T < T Kt b , vortices can only exist in bound pairs with opposite vorticity and G(r) decays in power law in distance. In this section, I will give a short overview of the 2-dimensional Kosterlitz-ThoulessBerezinsky (KTB) phase transition. Readers can refer to reference [9-15] for detailed calculations and discussions as well as renormalization group (RG) equations for the KTB transition and also the analogy of KTB physics to 2D Coulomb gas physics. 1.2.1 2D X Y m odel The classical 2D XY model can be used to describe phase transitions in superfluid 4He, 3He films and superconducting films (note that the 2D XY model can only be applied to superconducting films if the in plane magnetic penetration length is bigger than the sample dimension [16]. Vortices and antivortices are thus held together by logarithmical confining potentials). In this model, the system consists of spins 9 CHAPTER 1. INTRODUCTION of unit length arranged on a 2D square lattice. These spins can be represented as conventional vectors S* = S cos 0t\ + S sin <9j. Oi is the angle of the spin on site i with respect to some arbitrary polar direction in the 2D vector space. Using |5 |2 = 1, the Hamiltonian for the system can be written as: H = - J ] T Si • ^ = - J ] T cos(0i - Oj). <i,j> <i,j> (1.20) Here, J is the coupling constant (spin stiffness) between the nearest neighbor spins. J > 0 for ferromagnetic interactions. < i , j > sums over all the nearest neighbors on the 2D square lattice. The Hamiltonian remains the same for a rotation of all the spins by the same angle, thus shows an 0(2) symmetry. However, its ground state with all the spins pointing in the same direction must spontaneously break this 0(2) symmetry. If we assume that the spins are nearly parallel from site to site at low temperatures (T < J), we can have cos(0j —8j) ~ 1 —|(0, —8j)2 (spin-wave approximation). In this approximation and in the continuum limit, equation 1.20 can be expressed as H ~ i J <MV0)2, (1.21) Two dimension is a critical dimension for a lot of phenomena. To demonstrate the particularity of two dimension, we consider a d-dimensional XY model and generalize the Hamiltonian in equation 1.21 to a d-dimensional cubic lattice by extending the 10 CHAPTER 1. INTRODUCTION integral over r to d-dimension. The average of Sx in d dimension is [14] M = < S x > = < cos0(r) > = < cos 0(0) > J D9e~l" ’eie). = (1.22) (1.23) Here J D S = n J l , d 8 „ integrates over all possible configurations. After Fourier transforming the phase fields and some algebra, < S* > reduces to [14] — -s^ jC -^ ) (L24) Sd is related to the function integral of all possible configurations of the phase field. Integral I(L) = dkkd~3 strongly depends on the dimensionality as following [14]: m L2~d d < 2 ln(L/a) d =2 5 W - 2 ^>2 Here, the integration at short distances is cutoff by the vortex core size fo ~ o. while at large distances is cutoff by the finite system size L. < Sx > must be zero in the limit of large systems for d < 2 at any finite temperatures. This indicates that no true ordered phase exists at low temperatures for d < 2, in agreement with the Mermin-Wagner theorem. One can also calculate the two point correlation function in d dimension [14] exp( G(r) = < S(r) • S(0) > ~ < (l)^ r r) d = x d = 2 exp (—const. * T ) d > 2 11 CHAPTER 1. INTRODUCTION where 77 = kBT/2irJ. As showed by G(r -> 0 0 ), the system features an ordered phase at low temperatures for d > 2. For d = 2, G(r) decays algebraically indicating that the system is “critical” at any finite temperature. The spin-wave approximation fails to account for the phase transitions in 2D. This is because this approximation only describes small continuous deviation from the ground state configurations. It rules out the possible existence of vortices and all vortex like excitations. We notice that the Hamiltonian in equation 1.20 is invariant under discrete local transformations 0i —» 0* ± 2ir. To the contrary, this additional symmetry is not conserved in the spin-wave approximation. So the approximation breaks down when we include vortices in the system. A vortex is a topological defect in which the phase winds by ±2ir in going around the defect (Fig. 1.2) where 27m if all paths encircle the center of the vortex 0 if all paths do not encircle the center of the vortex V Here, n does not depend on the choice of the path. The phase transition is topological in nature by including vortex excitations. To estimate the energy cost to introduce a vortex into the lattice, we assume 6(r) = 9(r). Therefore, we get 27rr|W (r)| = 2irn from the previous close path integral. Using equation 1.21, one obtains E (1.25) (1.26) Ec is the core energy. It is obvious that vortices with higher winding numbers are not CHAPTER 1. INTRODUCTION / / y j / S H U ' '• - - X ^ \ W \ U W W W s t t t \ \ \ w / / \ \ / / / Figure 1.2: The spin configuration of a vortex. energetically favorable since the energy of a vortex is quadratic in the charge. For a large enough system, the energy of one vertex diverges logarithmically with the size of the system. To obtain the free energy of a vortex, we estimate the entropy S = ka ln(L2/a 2) from the number of independent places where a vortex of size a can be placed in a system of size L. The free energy is given by F = Ec + ( i r J - 2 k BT ) \n { - ) . (1.27) CL F —Ec changes from positive to negative through T k t b = n J /2 k s from below. Above Tk tb the system canlower the free energy by allowing vortex excitations. In reality it requires less energy to create a bound vortex anti vortex pair (E = 2Ec + Ei ln(R/a) where E\ oc J and R is the distance between the vortex and an tivortex [14]). At T < Tk tb vortices can exist only in bound pairs with opposite 13 CHAPTER 1. INTRODUCTION vorticity. These pairs do not disrupt phase coherence because their net vorticity is zero. The vortex and antivortex in the pair interact via a logarithmic confining po tential. Bound pairs of vortices at short distances screen the interaction potential of pairs with longer separations, thus renormalize the bare spin stiffness J measured on long length scales. KTB phase transition happens in a fashion that the vortex pairs with the largest separation that are bound at vortices begin to proliferate above Tk tb T < T ktb unbind at Tk t b ■ Free and the state with a quasi long range order is destroyed. 1.2.2 Superfluid stiffness and th e universal jum p In this thesis, I always use the superfluid stiffness (or the phase stiffness) Tg = J to describe the effects of thermally activated vortex pairs in the superconductors. We need to find the connection between the 2D XY model and 2D superconductors. In superconductors, each vortex carries one quantized magnetic flux and has a phase circulation 2n around the normal core of the vortex. Similar to spin stiffness, phase stiffness describes the energy scale required to twist the phase of superconducting order parameter 0(r) of superfluid or superconductors. The increase in the free energy due to the twist is given by F(V0) - F(0) = \T e(V6)2. The gradient to the phase field is related to the superfluid velocity us(r) = ^ V 0 . m* is the effective mass for the bosons causing the superfluidity. For 4He films m* is the mass of the helium atom and for a superconductor it is the mass of a Cooper pair. Together with F(V0) —F(0) = 14 CHAPTER 1. INTRODUCTION | Nm*Vg, one concludes that phase stiffness is determined as Tg = N h2/m*. At T > T k tb , the appearance of the first unbound vortex pair causes the phase stiffness to drop discontinuously to zero. This so called “universal jump” is a par ticular feature associated with phase transitions of the KTB variety. Despite the discontinuity in the stiffness the phase transition itself is still continuous. The uni versal jump occurs at Tk tb = f Tg {A T k tb = Tg for Tg defined by expression 3.2). The two point correlation function G(r) starts to decay as G(r) ~ e~r/^ T) with free vortices in the system. £(T) diverges in an unusual fashion as £(T) ~ e-b\T-TKTB\~112 (L28) when one approaches Tt k b from above. This divergence is much faster than any power law. This exponential dependence comes from f 2 ~ l / n F where nF is the vortex density. The vortices are thermally activated and hence nF oc exp{-(3E) with /3 = l / k BT. This is just a simple picture for this fast divergence. A more rigorous way to obtain the stretched exponential dependence on temperature is to use the RG equations. Another feature of KTB physics is that the temperature dependence of the re sistance curve above TTk b can be well described by the Halperin-Nelson form [17]. A rough estimation gives R oc nF ~ £ -2 ~ eib\T-TKTB\~1/2 Experimental confirma tion can be found in reference [12]. Although this agreement is often considered as an indication that the phase transition is indeed induced by phase fluctuations, the author in reference [12] pointed out that “this conclusion may be too rash”. Two 15 CHAPTER 1. INTRODUCTION reasons: (1) the resistance form is for T very close to T k tb (the reduced temperature should be way less than 1 ) when £ still has the unusual dependence on temperature. However, most data are from outside this region. (2) The measured resistance curves have little structure and the fitting form have too many free parameters. Therefore, it is important for us to look at the AC response of 2D superconductors instead of utilizing only DC probes to bear on this problem. 1.2.3 N onlinear IV characteristic In additional to the universal jump in phase stiffness and also the universal func tional form of the resistance, KTB physics shows another jump in the nonlinear IV characteristic. In the KTB model, in principal there are no free vortices present be low T k tb , and hence no flux-flow resistance. However, each bound pair has a small probability of being “ionized” provided a finite current j is imposed across the super conducting film. Therefore there is always some exceedingly weak dissipation in a 2D superconducting film. The critical current is actually zero in 2D. The theoretic pre diction for the flux-flow resistance generated below IV characteristic of the form V ~ J “ , where a = 1 Tk + tb is equivalent to a nonlinear 2 7Arg . V is linear in J for small enough current when the system is in the normal state. At T — T K t b >a = 3. Thus one expects a universal jump i n a = l - » a = 3 when T Tk tb from above [18-21]. In 2D, the potential for the opposite charges depends logarithmically on distance. As a result, the KTB physics can be mapped to a 2D Coulomb gas model. The 16 CHAPTER 1. INTRODUCTION two models share the same grand canonical partition function and they fall into the same universality class. The polarization of bound vortices by a current is equivalent to the electric polarization in a medium. In the case of a Coulomb gas system, the dielectric constant e is renormalized by the screening from dipoles of different intra pair separations. An increase in dielectric constant indicates the bound pairs become more polarizable, corresponding to a decrease in the phase stiffness. At T k t b , free charges present in the system and drive 1 /e (or Tg in the case of superconductors) discontinuously to zero. 1.3 2D superconductor-insulator quantum phase transition One motivation of our project is to understand how superconductivity is destroyed across the quantum phase transition boundary. For the 2D thermal superconducting transition, as we detail in the previous section, the appearance of the first thermal activated vortices kills the superconducting state. It has been shown experimentally that certain 2D thin films undergo a quantum phase transition from a superconductor to an insulator with infinite resistance at zero temperature. This quantum phase transition is driven by a non-thermal parameter in the Hamiltonian, such as the applied magnetic field, the film thickness, the disorder level, the pressure and so on. Most of the measurements in this field are performed at sub-Kelvin temperature so 17 CHAPTER 1. INTRODUCTION that the thermal energy K b T is sufficiently small. In most experiments, one measures the resistance per square R q through transport. R q is obtained by setting the length and width of the film equal to each other (to form a square). By doing so, /?□ should not depend on the dimension of a film (except thickness). One important theory model for a 2D superconductor-insulator transition (SIT) is the “dirty boson” point of view which was proposed by Fisher [22]. A phase diagram related to this model as a function of magnetic field, temperature and disorder is displayed in 1.3. In this picture, the 2D quantum phase transition is a transition directly from a true superconductor to a true insulator (defined at zero temperature and a true insulator is only well defined at absolute 0 K with a diverging resistance). A metallic state can only exist at the quantum critical point (QCP) with a universal resistance of order h /4e2 ~ 6.4 Kfi. According to the scaling theory, the resistance curve at the transition point should have little dependence on temperature at low temperatures and also the magnetoresistance curves for a field tuned SIT should have a well-defined crossing point, which is in general interpreted as the location of QCP. The dirty boson model describes a 2D SIT caused by the quantum fluctuations purely in the order parameter’s phase. It suggests that superconductivity in 2D is destroyed by phase fluctuations instead of by suppression of the magnitude of the order parameter. If this is true, Cooper pairs can also exist in the insulating phase, but they cannot move around freely. In 2D SIT, a duality transformation of 18 CHAPTER 1. INTRODUCTION ELECTRON GLASS-: NORMAL VORTEX LATTICE ^SU PER OONOUCTOI NORMAL Figure 1.3: A proposed phase diagram for the dirty boson model. Reprinted from [22]. CHAPTER 1. INTRODUCTION Cooper pairs and vortices can map superconductors and insulators onto each other [23]: in the superconducting phase, we have Bose condensation of Cooper pairs, but localized vortices; in the insulating phase, we have Bose condensation of vortices, but localized Cooper pairs. The two phases are separated by a quantum critical point at zero temperature. This duality view can also been reasoned from the Heisenberg uncertainty principle between phase and particle number: AnAip > 1. Bosons can stay in either an eigenstate of phase (A<p = 0 indicating a phase coherence) which is a superconductor, or an eigenstate of particle number (An = 0 indicating localized Cooper pairs) which is an insulator. Theoretically one can argue that the 2D SIT is always bosonic. However, the bosonic region can be arbitrary narrow and might not be feasible to probe directly in some experiments. Therefore, this pure bosonic model still remains an open problem in experiments. On the experimental side, while the transport, microwave cavity and STM measurements on highly disordered InOx films [24-26] and the transport on TiN films [27] seem to favor the scenario that pairing exists on both sides of the transition, electron tunneling measurements on ultrathin quench condensed Bi films near the SIT [28] suggest that the superconducting gap becomes very small and approaches zero at the transition point. In this scenario, Cooper pairs are broken at the transition and the insulating state is dominated by fermion physics [29]. The appearance of a Fermion insulator across the QCP is not predicted by the dirty boson model. 20 CHAPTER 1. INTRODUCTION Another experimental observation that cannot be explained by the dirty boson model is the possible existence of a range of metallic states at zero temperature. For many samples, at the crossing point is pretty close to the universal resistance. The notable exceptions from R q are MoGe and other low-disorder films. The transitions in these samples are also sample dependent. Moreover, these samples show signs of low temperature intervening metallic states between the superconducting and insu lating states. One possible explanation by Kapitulnik et al. [30] is that the effects of dissipation come from the coupling of the system to a dissipative heat bath. However, the source of the heat bath is still unknown. Despite all the disagreements on the possible mechanisms how superconductivity is destructed across the transition, one common feature for all the continuous QPT is the diverging correlation lengths £ and the diverging correlation time £T in the vicinity of the QCP [31,32]. Theoretically, a d-dimensional quantum system can be mapped to a (d+l)-dimensional classical system, where the extra dimension is imaginary time with a size set by h/3. For example, the time evolution of a ID Josephson junction array (ID XY model) is equivalent to a configuration of a 1+1D classical XY model. The coupling constant in the ID system corresponds to the temperature of the 2D classical system. The ordered and disordered phases in the classical model represent the superconducting and insulating phases in the quantum system. This analogy enables us to generalize the critical behaviors near the critical point of a classical transition. At T = 0 for the quantum system, £ and £r diverge as S = X — X c 21 0 CHAPTER 1. INTRODUCTION in the following fashion e ~ i«r (1.29) (1.30) 8 describes how far the system is away from the critical point X c (X = B if the transition is tuned by applying a magnetic field). These diverging behaviors and the associated scaling analysis represent universal behaviors of physical quantities and should be insensitive to microscopic details of the quantum system. Our experiments look at the dynamics of the 2D SIT. We explicitly measure the frequency dependence of the complex conductance and impedance of the supercon ductor across the transition. As we will demonstrate in Chapter 3 that our broadband microwave probe is sensitive to the temporal correlations of the superconducting fluc tuations. There, we show the critical slowing down (corresponding to the diverging correlation length) as the system is approaching the thermal transition point, where resistance goes to zero at the same temperature. This shows that the AC approach is a suitable tool to study the quantum phase transition since one can determine the true location of the QCP. This critical slowing down behavior for a continuous QPT is the theoretical background for data presented in Chapter 4. There, we show data that support a superconductor metal transition and the critical magnetic field might be way below the phenomenological crossing point of the iso-thermal magnetoresistance curves at low temperatures. The understanding of a 2D QPT in a superconduct ing film should help us to understand the dynamics of different phases of strongly 22 CHAPTER 1. INTRODUCTION correlated electronic systems, including but not limited to quantum hall effects and superconducting fluctuations in high temperature superconductors. 1.4 AC response of a superconductor Since this thesis work is about properties of 2 D superconductors at finite frequen cies, we would like to understand the measurements performed in our experiments. To this end, we need to know the optical conductivity of a superconductor, espe cially the low frequency response of a superconductor, when the probing frequency is much smaller than the superconducting BCS gap of our sample. Technically a Matthis-Bardeen approach should be used to treat the mean field AC conductivity of a superconductor. However, I will mostly use a simpler two fluid approach, which works quite well for states near Tc despite the simplicity of the model. This will be demonstrated in the later chapters. We employ the Drude model to describe the gen eral electrodynamics properties and the optical response of metals. The conduction electrons are modeled as a gas of particles with no Coulomb repulsion. The central assumption of this model is the existence of a mean free time (also called the relax ation time), which is the average time that elapses between collisions of electrons. One can write down the equation of motion of an electron of mass m driven by an electric field E with mean free time r as CHAPTER 1. INTRODUCTION Here, v is the average velocity of the electrons. For a steady state, one has d v/d t = 0 due to the competition between the scattering and the acceleration by E. The current density for n conduction electrons per unit volume is J = —new = ne2r /m E = <ToE, and one produces the Ohms law for a metal. We assume that the applied ac field is of the form E (t) — ’Eoe~lut. The solution to equation 1.31 should be of the same form v = v(ui)rrlwt. Equation 1.31 can be rewritten as (—— r dt — —eE /m (1.32) Using J = —nev = cr(u;)E, one obtains a complex frequency dependent conductivity o{u) = o0-— - — = oy(uj) + ia2(u) = (To 1 + l“ T . 1 - iujt 1 + ar r 1 (1.33) This frequency dependence of the complex conductivity shows that ax is frequency independent with a DC value ao well below the scattering rate to decrease at this 7 7 = 1 /r. o\ starts and falls off with u~2 at higher frequencies. a2 peaks at 7 ; <r2 oc u for low frequencies and a oc u ~ l for high frequencies. When ujt < 1, one has cr0 « <Ti(u) 3> (t2 ( ( j ) . This is the low frequency limit of equation 1.33, the “so called” Hagen-Rubens regime. For the InO* film, the scattering rate of its normal state is way above our accessible frequency range and the Hagen-Rubens limit applies. We can attribute the frequency dependence of the calibrated data for the normal state InOx to the response of its substrate (see Chapter 2 for details). 24 CHAPTER 1. INTRODUCTION In a zero-dissipation limit r —> oc, equation 1.33 becomes TTT iP O (1-34) TIC'^ a2(u) = — THU (1.35) The / sum rule of the conductivity is given by Io ax(u)du = !<r0 = y • (1-36) w2 u p is the plasma frequency and - f is the spectral weight which is essential the area un der the conductivity spectrum. This sum rule demonstrates that the spectral weight should be conserved and is a constant for all temperatures. The AC response of a superconductor can be obtained by considering a two-fluid model where n = ns(T)+ nn(T). ns{T) denotes the superconducting part and nn(T) is the normal fraction with scattering rate rn. The BCS theory gives nn(T) ~ e_A/fcijT at low temperatures. We take the dissipationless limit of the Drude form for the superconducting electrons and assume urn -C 1 for normal electrons, and the complex conductivity is given by cr\ u3) = irnse2 nne2rn & \u ) 4 2m m ~n— -------------------------------- ,nse2 mu ^ 1 ----------------- • ( 1 37 - ) Under the condition hu <C 2A and T < T C, the AC response of a superconductor can be given by simply the zero-dissipation limit of the Drude form. In this case, a x is zero everywhere except at u = 0. According to equation 1.36, the coefficient of the 8 function in ax should be the spectral weight. a2 a \ / u with a prefactor that is 25 CHAPTER 1. INTRODUCTION proportional to the superfiuid density ns. We will refer back to these behaviors of o\ and <72 in later chapters when we discuss our experimental data. At higher frequencies and temperatures, the AC response of a superconductor can be evaluated from the Mattis-Bardeen expression on the basis of the BCS theory. Both thermal energy and the energy of the radiation can create pairs of quasiparticles and contribute to dispassion. One can refer to [5] for a thorough discussion. 1.5 Thesis overview In chapter 2, I give an introduction to the broadband Corbino microwave spec troscopy. In that chapter, I will discuss in length about the design of the spectrometer in our group and also about the calibration procedures, both at room temperatures and low temperatures. There also is a section dedicated to describe the characteriza tion of the Corbino spectrometer at finite magnetic fields. In chapter 3 , 1 discuss the results of the AC measurements of thermal fluctuations in an InOx film. There, I show data for the explicit frequency dependency of the complex conductance of the InOx film and the phase stiffness over a range from 0.21 to 15 GHz at temperatures down to 350 mK. Superfluid stiffness acquires frequency dependence at a transition temperature which is close to the universal jump value. Our observations are consistent with Kosterlitz-Thouless-Berezinsky formalism. We explicitly demonstrate the critical slowing down of the characteristic fluctuation rate 26 CHAPTER 1. INTRODUCTION on the approach to the superconducting state and show in general the applicability of a vortex plasma model above TKTg. Chapter 4 focuses on the microwave measurement on a low disordered InO* film at finite magnetic fields. Data are presented for a field tuned quantum phase transition between the superconducting and the resistive states in the frequency range of 0.05 to 16 GHz. The relevant frequency scale of superconducting fluctuations approaches zero at a field B sm fax below the field Bcross where different isotherms of resistance as a function of magnetic field cross each other. The phase stiffness at the lowest fre quency vanishes from the superconducting side at B w B sm, while the high frequency limit extrapolates to zero near B cross- Our data are consistent with a scenario where B sm is the true quantum critical point for a transition from a superconductor to an anomalous metal, while BcrOSS only signifies a crossover to a regime where supercon ducting correlations make a vanishing contribution to both AC and DC transport measurements in the low-disorder limit. In Chapter 5 , 1 summarize the work presented in this thesis. Appendix A is for readers who are interested in knowing how to calculate sheet impedance from the calibrated reflection coefficients. Appendix B details the exper imental procedures for both zero field and finite field measurements. It is specially targeted at users needing to perform microwave measurements in the lab. Appendix C contains more information about the repeatability of different scans at zero field. Magnetic field distributions and their affects on temperature sensors are covered in 27 CHAPTER 1. INTRODUCTION Appendix D. Appendix E is complementary for the data presented in chapter 4. It contains almost all the data we have for the magnetic field measurements on the low disordered InO* film. I report a study of the complex AC impedance of CVD grown graphene in Ap pendix F. There, we measure the explicit frequency dependence of the complex impedance and conductance of a single-layer graphene over the microwave and tera hertz range of frequencies using microwave Corbino and time domain terahertz spec trometers. We demonstrate how one may resolve a number of technical difficulties in measuring the intrinsic impedance of the graphene layer that this frequency range presents, such as distinguishing contributions to the impedance from the substrate. From our microwave measurements, the AC impedance has little dependence on tem perature and frequency down to liquid helium temperatures. The small contribution to the imaginary impedance comes from either a remaining residual contribution from the substrate or a small deviation of the conductance from the Drude form. 28 Chapter 2 Broadband Corbino microwave spectrometer In the field of condensed m atter physics, spectroscopy is used to investigate the response of various materials to the electromagnetic radiation as a function of fre quency. It employs reflection or transmission measurements to determine the physical and electrical properties of those materials. In general, the characteristic energy scale of materials under study extends over several orders of magnitude. For some mate rials, like normal metals, semiconductors and insulators, the characteristic energies are typically of the order of eV. However, in some systems, especially for supercon ductors, the energy gaps go down to sub-millimeter and even microwave frequencies. At sub-millimeter frequency ranges, the radiation can still be guided through free space and conventional optical components like mirrors and lenses can be used; how- 29 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER ever, microwave radiations have to be guided by coaxial cables or waveguides and their interaction with the sample has to be performed in a well-defined manner. This is because the wavelength of a microwave (typically 0.001-0.3 m) is comparable to the dimensions of the experimental components and samples. Broadband microwave measurements are traditionally very difficult and have to be performed using different techniques. For the amorphous superconducting InOx thin film in our project, we can estimate its superconducting gap from Tc, which is about 170 GHz for a T c « 2.4 K sample. This falls into the microwave frequency range and thus requires us to probe the sam ple using microwave radiation. Most of the microwave measurements in the literature have been carried out by using a microwave cavity. Although a microwave cavity is able to measure both the real and imaginary conductivity with good sensitivity and high accuracy, it only can examine a limited number of frequencies. The introduction of the broadband Corbino spectroscopy to study strongly correlated system is a sig nificant advance in this field. This technique can give us true spectroscopic response of the sample under study as well as provide phase information of the conductance without referring to the Kramer-Kronig relation. At Johns Hopkins University, we successfully incorporated the Corbino system into a He3 cryostat and are able to obtain reliable data down to 300 mK as well as with a perpendicular magnetic field up to 8 Tesla. The purpose of our study is to investigate the thermal and quantum fluctuations in disordered superconductors and the intrinsic 30 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER electronic behaviors in the insulating side of the quantum phase transition. In this chapter, we mainly focus on the experimental setup details. We demonstrate that we overcome the challenges presented especially to this kind of broadband spectrometer and are able to perform our experiment repeatedly and obtain reliable results. 2.1 Experimental setup overview Lockin Amplifier Model: SR830 -j . Network Analyzer J / Bias Tee Model: Affleat N5230A Copper cablo Glass seal adapter y Stainless steel cable Glass seal adaptor, heat sunk 4.2 K Glass seal adapter _ heat sunk 300 mK cable Charcoar pump Copper cable Heat sunk stage for outer conductor of -th e coaxial cables He3 stage Magnet Sample Component Figure 2.1: Schematic of the overall experimental setup showing all the coaxial cables sections and connections in the Corbino microwave spectrometer. The schematic drawing of the overall experimental setup in our group is shown in 31 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER Fig. 2.1. A microwave radiation is generated by a vector network analyzer (Agilent model N5230A), and guided by coaxial cables. The network analyzer can generate microwave signals from 10 MHz up to 40 GHz with 1 Hz resolution and is equipped with two 2.4 mm male connectors. As showed in the scheme plot, 4 sections of semi-rigid coaxial cables are used in our transmission line setup. The overall length of the whole transmission line is roughly about 1.4 meters. We use a number of different coaxial cables with different properties at different points in the system. For the copper coaxial cables (Micro-Coax Company, part number UT-85C-TP-LL) that are used outside the cryostat connecting the network analyzer and also used to connect the superconducting cable and the Corbino probe, the outer conductor is tin plated copper while the inner conductor is made of silver plated copper for better conductivity. Inside the cryostat, the upper longest section in our transmission line is the stainless steel coaxial cable (Micro-Coax Company, part number UT-085-SS). Its outer conductor is 304 stainless steel and the inner conductor is silver plated copper covered steel (SPCW). We added a NbTi superconducting coaxial cable (Keycom Company, part number NbTiNbTi085A) into the system to better isolate the heat load from the room temperature connections, especially from the inner conductor. The transmission line is terminated by a customized Corbino probe, which was made from a 2.4 mm Rosenberger in series adapter (part number: 09K121-K00S3). As one can see from the left picture in Fig. 2 .2 , we took the thread away from one end of the adapter and carefully machine away the extra materials so that the surface 32 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER Figure 2.2: Picture of a Corbino probe showing the flat surface of the probe and the 4 pin configuration of the inner conductor. However, This is not the Corbino probe we used in our experiment as one can see that its surface is a little rough. Although we do not think it will affect the results but we use another Corbino probe with much smoother surface. Due to calibration issues, we cannot disconnect the one in use to take a picture. of the out conductor is flat within 0 .0 0 1 ” deviation. Samples usually have a donut shaped pattern as demonstrated in Fig. 2.3. By using the Corbino disk geometry, the currents in the film flow in the radial directions and only produce magnetic fields that are parallel to the surface of the films. The edge effects of the thin film are effectively eliminated compared to other experimental setups with a square or rectangular geometry. The sample is tightly pressed against the surface of the Corbino probe to make a direct electric contact between the outer Au pad of the film and the outer conductor of the probe. For this particular type of adapters, the inner connector is not flush with the outer conductor (roughly about 0.005” lower). To bridge the gap between the center Au pad of the sample and the inner conductor of the Corbino probe, a small conical shaped center pin (see Fig. 2.4) 33 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER is plugged into the inner conductor. Au c o n ta c t 4- S u b s tra te Figure 2.3: Au pattern of the sample prepared for microwave measurements and the wire connections for 4 probe measurements. On the right we show a picture of one of the samples we used in the experiment. One reason to choose the Rosenberger adapter is because of the special design of its inner conductor, which has 4 fingers as displayed in the right picture of Fig. 2.2. These fingers hold the center pin in place and also provide some springy force necessary to keep the pin in the correct configurations when a sample is attached. The interaction between the sample and the Corbino probe is also fixed by a spring behind the sample. The spring force is carefully set to be the same for all the samples and calibration standards by adjusting the length of the set screw. The same reference plane is assured for each sample by using a caliper to set the set screw in place. This improves the overall quality of our data dramatically. A particular experimental challenge in performing these experiments in a He3 34 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER Corbino Probe _ ^ C en ter Pm Sam ple ' C opper housing \ S et screw ' C opper su p p o rt Figure 2.4: Scheme plot of the sample stage. cryostat was to isolate the heat load from the top of the coaxial line which always stays at room temperature. We thermal anchor the outer conductor of the transmission line at two locations: the top flange of the inner vacuum chamber (IVC) and the He3 pot. To ensure good thermal contact, we made a copper housing for the adapter between stainless steel cable and NbTi cable and pressed this copper housing tight against the top flange of the IVC. Another copper housing for the adapter between NbTi cable and copper cable was screwed tightly on top of the He3 pot. This guarantees that the section of the superconducting cable is thermal anchored at 4.2 K and at the base temperature of the cryostat. This can be easily viewed in the picture in Fig. 2.5. For the connection between NbTi and copper cables, we also used copper wires and the thermal grease around the cables, connectors and adapters to make sure that 35 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER the outer conductors of the cables are well thermally anchored along the connection. The main reason for this is because we were concerned that the outer stainless steel conductors of the microwave adapters may not provide enough heat transfer. Figure 2.5: Pictures of the set up showing the manner how we heat sink the coaxial cable. On the right we show the front image of the sample stage. It is more difficult to heat sink the inner conductor since the dielectric in the cable is usually made of teflon and in most adapters and connectors is just air. This means that although we can manage to cool the outer conductor to 300 mK, the inner conductor might stay at a much higher temperature due to the poor thermal conductivity of air and teflon. Before we added the superconducting coaxial cable, the base temperature of the set up was only 500 mK and the holding time was less 36 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER than half an hour. To thermally anchor the inner conductor properly, we incorporated three hermetically sealed glass bead adapters (Kawashima Manufacturing Company, part number: KPC185FFHA) as displayed in Fig. 2.1. Two of these special adapters were heat sunk respectively at 4.2 K and the He3 stage that were separated by a roughly 10 cm long superconducting NbTi coaxial cable as showed in Fig. 2.5. We were able to reach 290 mK without microwave radiations and the holding time can be at least a day, which is enough for one cycle of our experimental procedures. The transition temperature of the superconducting cables is around incident microwave power level was chosen to be -27 dBm (~ 2 8 —9 K. The //W and this is the lowest power that we could set the network analyzer to). It turns out that this power level is not heating up the sample too much at the base temperature yet high enough compared with the noise level. All the cables, adapters and connectors (except the section outside the dewar which stays at room temperature for all the measurements) were thermal cycled a couple of times in liquid nitrogen before and after assembly to reduce the effects of thermal contractions in later measurements. After we assembled all the cables, we keep all the connections untouched since undoing any of the connections would ruin the calibration and change the signal. We use a lift and two rails to guide and move the dewar up and down for low temperature experiments. The coaxial cables also get “aged” after many times of cooling. The “aging” of the cables causes oscillations in frequency in the calibrated data. However, the overall effects to the final results of the sample under study are minor as we will 37 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER discuss in more details in the later text. Using a vector network analyzer, we measure the complex reflection coefficients S'™ from the sample that terminates the otherwise open-ended transmission line. S™ is then calibrated by measurements of three standards: open, load, and short. I will give more details about the calibration standards and procedures in the following sections. The useful frequency range for our studies usually lies in between 50 MHz to 16 GHz. Above 16 GHz, microwave reflections are dominated by the resonance in the sample holder stage or their interference with other components in the setup. At low frequencies, usually lower than 45 MHz, microwave data appear to be contaminated by the finite contact resistance (~ 2 Ohms) of the Corbino press fit contact. In addition to AC measurements, we can measure the two point DC resistance simultaneously with a lock-in amplifier (Model SR830) by adding a bias tee (Agilent 11612B Bias Network, 45 MHz to 50 GHz) into the transmission line. The measure ment frequency of the lock-in amplifier for most measurements is 13 Hz. We found that due to this pass frequency of the bias tee, setting the spectrometer to scan at frequencies lower than 45 MHz would introduce a large amount of noise into the DC measurements. We also found out that an accurate determination of the resistance for the load sample is very important. The uncertainty in measuring the DC value of the load sample (either from the uncertainty in determining the contact resistance or from the noise introduced by setting the spectrometer to scan below 45 MHz) will propagate 38 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER to the conductance of the thin films under study. The excitation currents for the lock in amplifier that we used during different measurements were within the range from 100 nA to 200 nA. Since it is a 2 probe DC measurement, to reduce the contact resistant, 100 - 350 nm thick donut shaped gold contact were evaporated on to all the samples except the open standard. An iron shadow mask was used during the gold evaporation and was held on top of the sample with a permanent magnet. The inner Ti and outer r 2 diameters of the donut shaped gold contact were 0.7 and 2.3 mm respectively. We also tried lithography at the beginning but photoresist needs to dried at around 100 C and the heating process would anneal the InOz films under study. So according to our experience, shadow masks do less harm to the samples than lithography. 2.2 D ata analysis The actual reflection coefficient from the sample surface differs from the measured S™ due to the effects of extraneous reflections, damping, and phase shifts in the transmission line. .S^ can be calculated as: Sa = *~*ii ~ ^ D EH + E s i S R - E o Y (2 ( l) } Here, the complex error coefficients E D, Es , and ER represent the effects of di rectivity (signal reaches the detector directly without interacting with the sample), source match (signal coming from the sample is reflected again and adds to the signal 39 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER approaching the sample), and reflection tracking (damping and phase shift) in the transmission lines. This implies that performing three reference measurements on standard samples with known reflection coefficients is needed to determine the three unknown error coefficients. Interested readers can refer to reference [33] for the full inversion. This calibration procedure however gives us another challenge that is the repeata bility of each measurement as the three error coefficients are very temperature depen dent in general and especially in our setup since we included a superconducting cable. We need to make sure that the temperature profile is the same when we perform the microwave measurements on three standards and the sample under study. To this end, we established a particular cool-down procedure. Liquid nitrogen and then liquid helium were introduced into the bath and temperatures were allowed to equilibrate for over 12 hours before starting measurements. A very slow and repeatable scan was performed for each sample from the base temperature up to 10 K in about 9 hours. I will discuss the low temperature calibrations with more details in the next section. The detailed measurement procedures can be found in Appendix B. After we determine E d , E s , and E r for each temperature and frequency, S h can be obtained via equation 2.1. To extract the sample sheet impedance Z eJ }, in principle the standard equation <2-2) may be used. Here Z q = 50 ohms is the characteristic impedance of the cable and 40 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER g = 27r/ ln(r 2 /ri) is the geometric factor where T2 and r\ are the outer and inner radii of the donut shaped sample. However, this Z R J ? will be the impedance of the film under study only when the substrate contribution is negligible. In what follows, Z§ub is the effective substrate impedance from everything that lies behind the film. For a thin film where the sample thickness is much smaller than the skin depth and under the assumption that only TEM waves propagate in the transmission lines, the effective impedance for a thin film of impedance Zs backed by a substrate with characteristic impedance Z§ub [34] is zf S For a sample that has Zs "C Z§ub, = — 5* 1 + #us ! ( 2 .3 ) ' ' ~ 0 and equation 2.3 reduces to Z eJ ^ = Zs- In our case both InO* in the normal state and graphene have a sheet resistance comparable to the Si substrate so Z eJ i = Zs does not hold anymore. In order to obtain the real response of the film, it is necessary to extract Z§ub. To isolate the impedance of the sample under study, we assume that Hagen-Rubens limit holds for InO* in the normal state since our probing frequency is in the microwave range and it is way below the characteristic scattering frequency of the sample (in the range of THz). This implies that Zs in the normal state should be pure real and independent of frequency and could be deduced from the DC resistance exactly. The substrate contribution Z§ub is then extracted from the calibrated InOx data at 5.6 K. With a reasonable assumption that Z§ub is temperature independent at low temperatures, the intrinsic response of the film Zs at any other temperatures can be calculated. 41 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER Complex sheet conductance G = ad is related to sheet impedance as G = \/Z * in the thin film limit (See Appendix A). 2.3 Calibrations Calibrations are very important if one wants to know the actual response from the sample. We need three standards for the three unknown error coefficients and one needs to carefully choose the three calibration standards. A blank high resistivity Si substrate was used as an open standard (Sxx = 1). A 20 nm NiCr film evaporated on Si substrate was used as a load standard. NiCr has a very high scattering rate, so one can assume that the impedance of the NiCr standard is flat in our accessible frequency range. Its S X1 can be evaluated from its simultaneously measured DC resistance R via the relation S X1 = A 20 nm superconducting Nb film (Tc ~ 6 K, which can be clearly seen from the DC resistance of Nb films) sputtered on Si substrate was used as a short standard (Sxl = —1) for thermal fluctuation measurements. A Nb film above Tc is not a good short standard since it is in the normal state and has a finite resistance. Using a superconducting Nb film as a perfect short yields more reliable results than using bulk copper for calibrating superconducting InO* films. This is likely the case because copper is less reflective than the superconducting InOx sample. The small calibration errors from the imperfection in the short standards result in a small error in the phase of the calibrated conductivity thus for highly conductive 42 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER samples giving us some negative real component of conductivity. However, this fact does not affect too much of the calibrated conductivity of InO* once the temperature is above the transition temperature Tc and in the fluctuating regime since the sample is very dissipative and copper turns to be a fairly good short in that case. For measurements in magnetic fields, Nb films cannot be perfect shorts once we apply a perpendicular magnetic field. Therefore we use bulk copper with thick Au film on top as a short standard. This again does not affect our results since InO^ becomes very dissipative with applied magnetic fields. Short only calibration as discussed in reference [33] may not be possible in our setup since the three error terms have very strong temperature dependence especially after the inclusion of the superconducting cable. Different choices of open might affect the high limit of the cutoff of usable frequency ranges. This was again discussed in length in reference [33]. A quick test of different calibration standards shows that the choice of glass, ceramic or Si substrate gives the same conclusion with regards to the experimental data so we did not perform a systematic study of the effects on the cutoff frequency from different open standards. 2.3.1 R oom tem perature calibrations One room temperature calibration was carried out at the end of the first copper coaxial cable using the commercial available calibration standards. Calibrations at the sample stage were also performed. These room temperature calibrations were 43 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER used to characterize the system at the early stage of the building phase. The raw signals are usually very noisy and lossy because we have a very long transmission line system with many connections. The stainless steel cables and superconducting cables are in general very lossy. Room temperature calibrations are very important and we revealed two significant facts for later experiments: the importance of the exact position of the set screw and the substrate correction. The microwave configuration between the sample and the Corbino probe, as dis cussed in the previous section, is defined by the two springy forces provided by the inner conductor of the probe and the spring behind the sample. To ensure we have the same reference plane for all the samples, we should have the same springy force for each configuration. Spring tension determines how hard the sample is pushed against the probe. Different spring tensions may lead to different positions of the center pin, thus change how the microwave radiation interacts with the sample. The main purpose is to maintain the same length of the spring beneath the sample for all the configurations. This approach would also make sure that the center pin will be pushed into the inner conductor at the same depth each time. We pick up a standard for the length of the set screw outside of the sample stage when the sample and the Corbino probe have the best contact, which is usually defined from the minimum point of the simultaneous DC measurement. This at the same time also sets the force in the spring. The length of the set screw changes accordingly for samples with different thickness. Although the reference plane might change once the IVC is in 44 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER high vacuum as discussed later and in low temperature environments, all the refer ence planes should still be the same for all the samples if we have the same starting reference plane and same procedure to pump and cool down the system. The choice of load and short are pretty standard. As pointed out by Ref. [33], the choice of open, however, would affect the usable frequency range since it may change the resonance frequency in the calibrated data. To check response from different open standards, we performed a quick check on different samples with high impedance using a very short copper cable. We attached an unmodified Rosenberger adapter at the end of the copper cable and the whole connection was calibrated by the three commercially available standards. We then replaced the Rosenberger adapter with our Corbino probe. This exchange process changed the signal level some, but the overall change should not be too significant and this room temperature test still gave us some ideas of responses of different possible open standards as shown in Fig. 2.6. Comparing the magnitude of S n of all the open samples, air (no sample is used to terminate the otherwise open transmission line) is a much better open with resonances at much higher frequencies. Highly doped Si, on the other hand, has an impedance far away from a perfect open thus cannot be used as an open standard. Although air is a perfect open, it cannot be used for low temperature calibrations when the commercial standards are not applicable. This is because the center pin would protrude somewhat compared with the situation when there is a sample terminating the transmission line, which changes the reference plane. We can clearly see this from Fig. 2.7 where we 45 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER 1.0 0.8 0.6 co* 0.4 Highly doped Si Ceramic Plastics G lass Air Silicon 0.2 0.0 0 20 10 30 40 Frequency (GHz) Figure 2.6: Magnitude of S n for different open substrates. Magnitude of Sn for highly doped Si is just an illustration for the possible response of a lossy sample. show the calibrated real (Z{) and imaginary (Z 2 ) impedance for 40 nm NiCr film on high resistive Si substrate as a function of frequency at room temperature. Substrate correction is not needed for this sample since its sheet resistance is negligible compared with the one of the Si substrate. Different colors in this plot indicate different sets of calibration standards. Data calibrated with air as open clearly deviates more from the expected value which is derived from the co-measured DC value, especially at high frequencies. The fast increases at around 16 GHz in Z x and Z 2 are caused by the resonance in the sample stage. The huge deviation of data calibrated by air as open from their expected results can be accounted by the strong interaction of the microwave radiation with the setup component and the different position of the 46 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER center pin from all other calibration standards and samples when there is no sample to terminate the line. We also used ceramic and glass to calibrate the sample as displayed by the green and purple curves in Fig. 2.7. From Fig. 2.6, we know that both glass and ceramic develop a resonance with the sample holder at a higher frequency compared with Si. This is also confirmed from the calibrated data of 40 nm NiCr film. The resonance using Si as open formed at about 17 GHz while the first resonance did not show up until 21 GHz for data calibrated by ceramic and glass (not shown in the plot). 250 40 40 40 40 200 nm nm nm nm NiCr using air a s open NiCr using Si a s open NiCr using ceramic a s open NiCr using glass a s open -50 0 2 4 6 8 10 12 14 16 18 Frequency (GHz) Figure 2.7: Real and imaginary impedance of 40 nm NiCr on Si as a function of frequency. 20 nm NiCr film as load and bulk copper as short were used for all the calibrations. Red dashed lines mark the expected impedance for this sample from the co-measured DC value. As discussed above applying the same force in the spring for all the samples is important. A large difference in the reference plane, of course, would give us different 47 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER results of the sample under study. A small difference on the other hand would give some wiggles in frequency in the calibrated data as shown in Fig. 2.8. The difference in experimental setup for the two sets of data is that the set screw was just about 0 .1 mm away from its assumed position for the red curves. As we can see the absolute magnitude and shape of the two sets of data are overall very similar. However, the data in red have small oscillations in frequency in both Z\ and Z<i and this is caused by the small deviation of the reference plane when we set the set screw differently for the 40 nm NiCr film for that test run. 150 ............. _ '3? E sz O 8 £<0 100 •40 nm NiCr with fixed force in the spring ' 40 nm NiCr with set screw about 0.1 mm away from its supposed position 50 0 r ,- r , -50 6 _L _L 8 10 12 14 16 Frequency (GHz) Figure 2.8: Real and imaginary impedance 40 nm NiCr on Si as a function of fre quency. Black curves are the frequency dependence of the calibrated Z\ and Z 2 with the set screw in the supposed position maintaining the same spring force for all the calibration standards and the sample under study. For the red lines, the set screw is in the right position for all the three calibration standards, but it is about 0 .1 mm off from the presumed position for 40 nm NiCr. 48 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER We performed a more systematic study of the position of the set screw as shown in Fig. 2.9. 10 nm A1 film on a Si substrate was calibrated by 20 nm NiCr, bulk copper, and ceramic or glass. Red dashed lines are guide to the eye of the expected value of the impedance. The position of the set screw was chosen by closely watching the DC resistance for 20 nm NiCr film when we tightened the set screw. The 20 nm NiCr has 350 nm thick Au pattern which ensures good electric contact between the sample and the Corbino probe. We took the location when the resistance was the minimum point of reading as the standard position for the set screw. The force of the spring was adjusted to be the same by this standard position for the three calibration standards. For the 10 nm A1 film, we tried different spring configurations as described by the color legend in Fig. 2.9. For different forces in the spring, the calibrated impedance slightly deviates from the expected value, especially at higher frequency. Here we did not distinguish which open standard we used since both ceramic and glass yield the same calibrated data as demonstrated in the plot. A loose spring may result in unreliable data as shown by the blue curve. In that case, the sample and the probe may not even have good contact. The drop of the real impedance at very low frequencies is the contamination from the contact resistance. Fig. 2.10 shows the sheet impedance for one of the InOz films calibrated at room temperature. In the plot, the real impedance is demonstrated by solid curves and the corresponding imaginary part is shown by a dashed line with the same color scheme. The set screw stayed at the same position for the four sets of measurements. Three 49 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER 10 nm Al using glass as open 10 nm Al using ceramic as open 10nm Al with spring 0.25 mm tighter 10nm Al with spring 5 mm tighter 10nm Al with spring 0.25 mm looser 10nm Al with spring 0.5 mm looser guild to the eye of the expected impedance 8 10 Frequency (GHz) Figure 2.9: Impedance of a 10 nm Al film at room temperature with different spring configurations. 1500 - 1000 — InO, in air lnO*when we started pumping the IVC lnOxwhen we pumped the IVC for 10 mins lnO„ in vacuum - 500 - -500 - 10 Frequency (GHz) Figure 2.10: Impedance of InOz at room temperature in the range of oj/2-k — 0.05 20 GHz. Solid lines are the real parts of the sheet impedance at four different trials and the dashed lines are the imaginary parts. 50 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER calibration standards were all measured when the IVC was open (glass, 20 nm NiCr film and bulk copper). The difference in each data set was the environment of the InOz film. The black line was taken when InOx stayed in the atmosphere. After we close the IVC and start pumping, we took another set of data and calibrated it by the same error coefficients. This set of data is displayed by the green curves. Data in blue show the calibrated InOx data after we pumped the IVC for about 10 mins and both the real and imaginary impedance for this run yield the same results as the green lines. After pumping the IVC overnight (~ 12 hours) by a turbo pump, InOz film was in a high vacuum environment. We took another room temperature data of InOx and calibrated them with the same error terms. The calibrated data are displayed by the red curves. The set of calibrated InOx data definitely has more wiggles after the IVC was pumped over night as shown in Fig. 2.8. After we pumped the air out, the cryostat was compressed a little bit and also the air dielectric in some adapters changed to vacuum. These caused a shift in the sample reference plane, thus introduced a small calibration error in the final data. We also notice that the impedance for InOx has very strong frequency dependence. However, InOx film in the normal state at room temperature is a very disordered metal and its scattering rate (usually in the range of THz) is much higher than the upper limit of the frequency range we can access in this setup. According to the Drude model for a dirty metal with a very high scattering rate, this implies that the impedance of the InOx sample should be flat in our frequency range. Therefore, 51 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER the frequency dependence of the calibrated data can only come from the substrate contribution. Although Si substrates have no effects on 40 nm NiCr films and Al films, the assumption of £s ~ 0 is no longer valid for this InOx film. We obtained the substrate contribution as discussed in the previous section. An example of the substrate contribution is displayed in Fig. 2.11 where we demonstrate the magnitude of Z§ub obtained using equation 2.3. 3 I £ % ie I 0.1 Figure 2.11: Magnitude of Z§ub from one of the low temperature measurements. 2.3.2 Low tem perature calibrations Low temperature calibrations are more complicated than room temperature cal ibration procedures, especially after we added the superconducting cable. One ex 52 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER perimental challenge when we were characterizing the system was the repeatability of the measurement of each sample since all the error coefficients have strong tem perature dependence. To correctly remove the cable contributions, we have to make sure the temperature profile along the transmission line be the same for the three calibration standards and the sample. For that reason, the repeatability of the cool ing down procedures for the three calibration standards and each measurement of the sample is essential. The detailed information of the experimental procedures can be found in Appendix B. In this section, we mainly discuss the characterization of the spectrometer with the superconducting cable. 2.3.2.1 Effects of the superconducting cable We added the superconducting NbTi cable to better thermally isolate the sample from the heat flow down the inner conductor of the transmission line. There is some tradeoff between a lower base temperature and better repeatability in the design of the whole transmission line. We originally had an approximate 20 cm NbTi cable in the system and had a faster scan in temperature. As a result, we could reach a very stable base temperature at about 290 mK. However, it took a very long time for this 20 cm superconducting cable to reach a thermal equilibrium. To characterize the cables’ response, we looked at SJ? = |S„|e<* (tp is in radian for the following graphs) as a function of temperature for a standard which should have little low temperature dependence. Therefore, the changes in S'™ as we scan the temperature are mainly 53 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER the contributions from the cables to the reflected signals. In Fig. 2.12, we display the temperature dependence of l^nl and ^ at 7 GHz for the bulk copper for two different warming up and two cooling down procedures with the same temperature controlling parameters. These measurements are faster scans in temperature than the standard procedures as described in Appendix B. For the two warming up scans, the difference in |5 n | is about 2 %. For the two cooling down procedures, this difference can be as big as 4 %, especially at high temperatures. However, we concerned most about the discontinuity in the signal at about 2 K in which the l^n j signal can be different up to 8 %. This is caused by the long relaxation time the superconducting cable needs to reach its thermal equilibrium as well as the different level of liquid helium in the dewar for each scan (the order for each scan can be found in the plot of iSnl in Fig. 2.12). This required us to figure out a strategy so that we can continuously scan from base temperature to 6 K or higher. In that case, we just need to maintain the same cryogenic environment in the dewar and the same temperature setting for all the measurements. Knowing that the superconducting cable demands a long time to reach its thermal equilibrium, we shortened the cable by half and carefully thermally anchored all the connections as detailed in the experiment overview section. The base temperature now is 300 mK, 10 mK higher than before. We strictly follow the experiment pro cedures in Appendix B for each measurement cycle where we show how we control the temperature repeatedly and reliably from 300 mK up to 10 K. Microwave data 54 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER 0.45 0.44 0.43 of - 0-42 0.41 0.40 0.39 - 0.20 - 0.22 - 0.24 - 0.26 - 0.28 - 0.30 - 0 5 10 Tem perature (K) 15 20 Figure 2.12: Four temperature scans of the bulk copper at 7 GHz with a 20 cm superconducting cable in the transmission line. Red and black lines are data for two different warming up procedures and green and blue ones are for two cooling down scans. The order in which these scans were taken is showed by the number next to the curves in the plot of |S n|. 55 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER 0.42 first warming up cooling down second warming up 0.41 0.40 0.39 0.38 0.37 0.36 2.50 2.48 2.46 2.44 2.42 2.40 2.38 0 5 10 Temperature (K) 15 20 Figure 2.13: Magnitude and phase of .S1^ of Si as a function of temperature at 7 GHz with a 10 cm superconducting cable in the transmission line. Different runs are indicated by the color legend. 56 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER at zero field were taken 3 times per cycle: first warming up (from base temperature to about 4 K), cooling down (from 20 K to 2 K) and second warming up (from base temperature to around 10 K). We always start the first warming up at helium level 55 cm (~ 20 liters liquid helium in the dewar), the cooling down scan at helium level 50 cm and the second warming up at a helium level roughly around 26 cm. To re produce the same temperature profile, we wait the same amount of time before each measurement for every sample. We also found out that the bulk copper might be a cause for some negative responses in the real conductance of InOz. After those initial characterizations of the spectrometer, we switched to a superconducting Nb film below 6 K as a short standard for low temperature measurements. Since the Nb film might have some temperature dependence at low temperatures, especially around 6 K when it becomes superconducting, we focused on S™ of Si to characterize the response of the cables for different runs. In Fig. 2.13, we show the temperature dependence of iS'nl and ip of 51 standard for the three measurement runs in one cycle. The two warming up scans have a difference in |S'n| that can be close to 2 %, especially at low temperatures. This is caused by the difference in the helium volume in the dewar: the helium level for the second warming up is almost half of the one for the first warming up. Therefore, the cryogenic environments for the two scans are different. The overall temperature of the coaxial cables for the second warming up is usually slightly higher than the first warming up. However, this also indicates that our data should not have very 57 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER strong dependence on the helium level. The maximum difference in jS'n | is about 2 % for a huge change in the helium level for the two warming up scans. If the helium level differs by 1 or 2 cm for the second warming up for different standards, the error introduced in this case is small and should be negligible. This conclusion may also be supported by the fact that all the coaxial cables (except the one connecting the network analyzer) sit in high vacuum without a direct contact with the liquid helium bath. The difference in |S n| between the cooling down and second warming up reduces to 1 % or less over the whole temperature range. The overall temperature dependence of S’™ for the two runs feature the same shape except some offset in the y axis. Both sets of data have little dependence in temperature above 9 K indicating the cables have little thermal response in this temperature range. Below 9 K, the NbTi cable becomes superconducting, hence introduces a change in S™. Fig. 2.13 demonstrates how the NbTi cable reaches its equilibrium for one scan. Its response can be reproduced despite some offset in the absolute value. For data analysis, we use standards from one scheme to calibrate the sample measured in the same fashion. It is in general easier during the second warming up for us to control how long the system stays at the base temperature to eliminate some external disturbance (we need to transfer liquid helium before the first warming up). Our data in Chapter 3 and Appendix F were taken at the second warming up and were calibrated by standards measured also at the second warming up scans. Although 58 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER 0.4 lnOxfilm first warm ing up lnOxfilm cooling down lnOxfilm seco n d warm ing up 0.3 tn 0.2 0.1 3 -2 - 0 5 10 15 20 Temperature (K) Figure 2.14: Magnitude and phase of S™ of InOx as a function of temperature at 7 GHz before calibration. Different runs are indicated by the color legend. 59 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER there is some difference in Sft in first warming up, cooling down and second warming up scans, the error due to the calibration procedures should be way less than 1 % for the same type of scan in temperature with almost the same helium level in the dewar. The InOx film that we present data in this thesis has a very sharp feature in the raw 5"! data for its finite temperature superconducting transition (see Fig. 2.14). The change in the reflection signal in the InOx film is much more dramatic than the changes in the three standards. We still can see the changes in the raw S™ due to the existence of the superconducting cable around 8 - 9 K, but this change is very small compared to the overall signal change in the InOx film. So error introduced by the possible difference in the helium level in the dewar should be totally negligible in the final analyzed data and do not affect the physical interpretation. 2.3.2.2 Error coefficients After the measurements of the three standards, we obtain three equations between S fx and S'™ for the three calibration standards using the formula 2.1. Three unknown error coefficients E r , Es , and E D can be determined by the three equations and are plotted as a function of temperature for the exemplary frequency of 9 GHz in Fig. 2.15. The data are from a low-temperature calibration using a bulk copper as short, a blank Si substrate as open, and a 20 nm NiCr film as load. The data suggest that Er, and Ed change very little with temperature below 6 E r , K. Although the transition CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER 0.4 49 c 0 0.3 1O w § 0.2 111 o T«J 1 C 0.1 o> « X - 0.0 Ik 0.5 F f49 c .2 00 io « w § Ul o S £<0 a. -1.0 - 1.5 •2.0 Ik 0 1 2 4 3 Temperature (K) 5 6 Figure 2.15: Temperature dependence of the magnitude and phase of the three error terms at 9 GHz. 61 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER temperature of the superconducting cable is around an equilibrium below 6 8 — 9 K, it more or less reaches K. Therefore, a short only calibration might be possible for low temperature measurements (see reference [33] for more details) that one only uses a short standard to calibrate the sample’s reflection coefficients. This short only procedure would greatly reduce the errors due to different measurement runs of different standards. If the sample under study is a very good superconductor (with high superfluid density), one in principle can use the sample in the superconducting state as a short standard. In this case, no extra measurements are needed. However, we cannot use the InOx sample in the superconducting state as a short standard since the film is very disordered with a low superfluid density compared with other clean superconductors such as A1 films. The weak temperature dependence of the three error terms demonstrate the potential possibility of performing a short only calibration in our spectrometer and could be investigated in the future. 2.3.3 M icrowave m easurem ents in a perpendicular m agnetic field The detailed dimensions of the He3 cryostat coupled with the magnet can be found in Appendix D. The superconducting magnet consists of superconducting NbTi wires. A power supply supplies a current to energize the magnet. The magnet can reach 8 Tesla with a current of 47.62 Amperes. Contour plots of the magnetic field 62 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER distributions in the axial and radial directions as well as the magnitude of the field are also displayed in Appendix D. The magnet can stay in a “persistent” mode. Once we reach a desired magnetic field, we turn off the persistence switch heater which is effectively a superconducting short across the power leads within the cryostat. Once the persistence mode has been entered, we ramp down the current in the power supply quickly. The magnet still stays at the desired value even after the current in the power supply becomes zero. Our microwave scans in temperature are performed at constant magnetic fields that the magnet is set to stay in the persistent mode. This way, the magnet will be stable throughout the whole measurements. Also, the current in the power supply of the magnet is zero, thus does not affect the readings of all the other electric components that are close by such as temperature controllers and the lock in amplifier. We found some small effects from the current leads on the temperature and lockin readings when we were charging the magnet. A magnet tends to trap magnetic fluxes during magnetic field sweeps and thus shows hysteresis. The nominal “zero field point” is not true zero due to the trapped flux. To perform measurements at true zero magnetic field, one has to oscillate the magnetic field to 0 to remove any possible trapped flux. In experiments, we need to estimate the magnitude of the hysteresis to make sure that it has little effect on the experimental data. To do so, we swept the magnetic field at a fixed temperature and searched for the minimum point of the resistance reading of a film. In Fig. 2.16, we show the sheet resistance of an InOx sample as a function of field at 2 K. This 63 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER 700 A field scan from -1 T to 1 T A field scan from 1 T to -1 T too 800 § I S aP 300 200 100 • 1.0 -0.5 0.0 1.0 urn Figure 2.16: The hysteresis of the magnet in our system. The curve in red is a scan of the InOx film in field from -1 Tesla to 1 Tesla at 2 K. The curve in blue is a scan of the InO* film in field from 1 Tesla to -1 Tesla at 2 K 64 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER shows that the hysteresis of the magnet in our system is about 0.036 T, which is small enough and can be safely ignored for our measurements. 0.8 GHz 0.251 4 GHz 0.453 - 0.035 0.280 0.482 0.259 0.030 2 0.259 < 0.257 0.481 - 0.025 0.480 0.255 0.470 0.255 7 GHz 0.020 55x10' 12.5 GHz 0.434 2.02 0.310 0.433 2.01 -£ 0.432 2.00 « 2.00 0.431 0.300 - 2.00 0.430 2.07 0.420 2.00 0 2 0 4 ■«D 2 4 0 > r o Figure 2.17: Magnitude and phase of S'™ of a Si standard as a function of field at 300 mK for 0.8, 4, 7, and 12.5 GHz. Different frequency values are indicated in the legend for each plot. In all the plots, data in red are the magnitude of S™ and data in blue are the phase of S'™ (in radians) for that particular frequency. For calibration purpose, we need to find out the field dependence of the coaxial cables, especially because the superconducting cable in the system might be affected by the applying magnetic field. One can in principal, examine this dependence by looking at S'Jj of a Si standard for a magnetic field scan at fixed temperatures. Fig. 65 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER 1.20 o, (T * 0.3321 K) calibrated by standards at 2 T/o, (T « 0.3323 K) calibrated by standards at 3.5 T o, (T » 2.6658K) calibrated by standards at 2 T/o, (T = 2.5652 K) calibrated by standards at 3.5 T 1.16 1.10 I 1.06 1.00 0.06 L _ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ L 0 2 4 0 0 10 12 14 10 Frequency (OHz) Figure 2.18: Ratio of real conductance of InOx measured at 3.5 Tesla but calibrated by standards measured at 2 Tesla and 3.5 Tesla. The blue dashed line marks the expected ratio when the calibrated data from both calibration sets are the same. One ratio was calculated from data at about 330 mK and another was obtained from data at about 2.67 K. 2.17 shows the magnitude and phase of S™ of a Si standard as a function of field at 300 mK for 4 frequencies. Since the Si standard should have little temperature and magnetic field dependence, the changes in S'™ showed in Fig. 2.17 for frequencies at 0.8, 4, 7, and 12.5 GHz are mainly the contributions from the cables to the reflected signals as we sweep field. Both the magnitude and phase of S'™ are not linear in field and they show minimums at different fields for different frequencies. However, the overall changes in the response of the cables are still small compared with the change in InOx signal showed in Fig. 2.14. Fig. 2.18 plots the ratios of the real conductance of InOx measured at 3.5 Tesla but calibrated by measurements of standards at 2 Tesla and 3.5 Tesla. As one can see an interpolation in field can still yield reasonable calibrated data as long as the sample under study has a very large change in signal 66 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER at different magnetic fields. This validates the calibrated InOx data at 4 Tesla using an effective calibration interpolated from 3.5 Tesla and 5 Tesla calibration standards in Chapter 4 due to a missing set of calibration curves. 2.4 Review of the Corbino spectrom eters from other groups The broadband Corbino spectroscopy was introduced by Anlage’s group in Uni versity of Maryland in the 90s [34,35]. They used this technique to study thin films of high temperature superconductors YBCO [36-39] as well as colossal magnetoresistive manganites [40]. Later, a group in University of Virginia used the same technique to study the microwave ac conductivity spectrum of a doped semiconductor in its Coulomb glass state [41]. This technique has also been used to study the dielec tric response of liquids and soft condensed matter [42] in a group in Leiden. Those experiments went down to 4 K or just liquid nitrogen temperature. The limit in temperature was pushed down to 1.7 K by Marc Scheffler at University of Stuttgart in Germany [33,43]. There, they reported the frequency dependence of microwave conductivity of the heavy fermion metal [44] and superconducting A1 films [45]. Kitano et al. also constructed a Corbino spectrometer [46] to investigate the critical behavior of LSCO [47,48] as well as NbN films [49]. Recently, Pratap at Tata institute also has constructed a Corbino spectrometer that goes down to 2.3 K 67 CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER to study NbN films [50]. The Corbino spectrometer in our group is the first of this kind that can go down to 300 mK with an applied magnetic field up to 8 Tesla. As I have shown in this chapter that we can repeat the measurements in a reliable fashion and we are the first group reporting reproducible data at 300 mK and 8 Tesla. We have applied our setup to investigate 2 D superconducting InO* films, graphene and 2D quantum phase transition [51-53]. 68 Chapter 3 AC studies of the zero field phase transition In this chapter, I will discuss the dynamic effects of the KTB phase transition probed by the home-built microwave spectrometer. I will start this chapter by dis cussing the expectations of such a transition. 3.1 Superconducting fluctuations In Fig. 3.1, sheet resistance in the unit of Ohms of one InOx film is displayed as a function of temperature. This plot is a guide to understand superconducting fluctuations in different regimes showing how the sample evolves from the normal state towards the superconducting state at low temperatures. The superconducting 69 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION transition is much broader than the 3D case due to the existence of strong fluctuations. Following Ginzburg and Landau, a pseudowavefunction ip(r) — A ellf is the order parameter which describes how deep the system is into the superconducting phase. A and <p are the magnitude and the phase of the complex order parameter. Within conventional wisdom, one expects that the sample evolves to a superconducting state from the normal state by first entering an amplitude fluctuation region, which can be well described by the GL mean field theory. At some temperature scale Tco, we expect the amplitude of the order parameter is well defined to support fluctuations in the phase (p. In this regime, the resistance is still finite due to phase fluctuations. At T = T k t b >the transverse phase fluctuations are frozen out and KTB phase transition takes place. The system enters a superconducting state with bound vortex-antivortex pairs. To establish the temperature Tc0 where the superconducting transition might oc cur without phase fluctuations, we fit the DC resistance curves to the AslamazovLarkin form for the fluctuation conductivity for two dimensional systems. The total conductivity in the amplitude fluctuation regime can be written as atotai = < tn + &a l gal is given by equation 1.18. Converting the expression to the measured sheet resistance Rmeas>one obtains = = (31) Here R n = l / a Nd is the sheet resistance for normal electrons. In Fig. 3.2, we fit the R m from Rmeas of an InO* film using as a fitting parameter. Curves in black and 70 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION 1000 800 □ 600 £ 400 200 1 2 3 T(K) 4 6 6 Figure 3.1: Ra as a function of temperature for one of the InOx films. 71 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION 1*00 1400 1200 1000 800 — 600 Manured DC atiaat Resistance Effective Normal sto a t Rasistanca with Tj, ■ 2.76 K Effective Normal sto a t Rasistanca with T . « 2.58 K 400 200 1 3 2 4 S 6 T (K) Figure 3.2: Measured sheet resistance as a function of temperature along with two fitted effective normal sheet resistances. blue are the fittings for Tc0 = 2.55 K and T& = 2.75 K. Under the assumption that R m should not have a minimum in the temperature range we measured, Tc0 is determined as the fitting parameter when the minimum in fitted R n first disappears. In this fitting analysis, one actually only obtains a minimum value for T ^. It is technically possible that T& could be larger. However, this Tco is just a parameter to show the possible crossover behaviors. Equation 1.18 is only valid when the ensemble average of |i/>|2 is small. That is why the extracted R n {T ) in the data has a notable upturn which is obviously not physical. For this particular sample, Tco is 2.75 K and is marked by the dashed vertical green line in the plot. Note this temperature Tco is way above the temperature T k tb ~ 2.4 K, when Rmeas is indistinguishable from the contact resistance. Below Tco, we 72 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION consider the superconducting fluctuations are mostly fluctuations in the phase of the superconducting order parameter. 3.2 Dynamics of KTB phase transition The remarkable properties of superconductors and superfluids arise from the macroscopic quantum-mechanical coherence of their complex order parameter (OP), \J) — A el<t>. In conventional superconductors, fluctuations of the OP amplitude and phase occur in temperature regions only infinitesimally close to Tc. In contrast, in disordered materials with reduced dimensionality, the situation may be considerably different. Their low superfluid density gives a small phase stiffness and phase fluc tuations that may be particularly soft [54]. In such systems, phase plays the role of a dynamic variable and may result in a situation where the transition results from a phase disordering of the order parameter while its amplitude remains finite. Effects such as zero resistivity are lost when the phase is no longer ordered on all lengths. However, phase correlations may remain over finite length and time scales resulting in significant precursor effects above Tc. As discussed in Chapter 1, in strictly two dimensions (2D), such a transition has been proposed [17,55] to be of the Kosterlitz-Thouless-Berezinsky (KTB) variety, as in 4He films [8,9,12,56]. In such a transition, thermally excited free vortices are not possible below the transition temperature 73 T ktb as the vortex-antivortex binding CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION energy increases logarithmically with separation. However, above Tktb it becomes entropically favorable for vortices to unbind. Vortex pairs with the largest separation unbind first and the phase stiffness measured in the long-length and low-frequency limit suffers a discontinuous drop. Vortex unbinding reduces the global phase stiffness and renders the system increasingly susceptible to further vortex proliferation. At temperatures just above Tktb such systems can be described as a two-component vortex plasma and may be realizations of the 2D XY model. Because free vortices are the topological defects of the phase field, their spacing plays the role of a Ginzburg-Landau correlation length £, which diverges as T —>T k t b The role of free vortices as topological defects and their finite energy cost give an exponentially activated vortex density (nF oc l/£ 2). Asymptotically close to the transition, this results in an unusually stretched exponential dependence of f on temperature, £ ~ es/ T'^ T~TKTB\ which is in stark contrast to the power laws typically expected near continuous phase transitions [9]. Similar dependence is expected in the “critical slowing down” of the phase correlation time 1/D, which in a vortex plasma, is proportional to the time £2/D required to diffuse the intervortex spacing (where D is the vortex diffusion constant) [17]. Although the conventional wisdom is that a KTB-like transition occurs in ultrathin superconducting films [55], the issue is still in fact controversial. For instance, it has been proposed that, unlike 4He films, superconducting films are in a regime of low core energy (the high “fugacity” limit), that causes the transition to acquire 74 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION a nonuniversal character or even be first order [12,57,58]. There has been a great deal of work looking for KTB physics in the linear and nonlinear dc transport char acteristics of thin films superconductors [21,59,60]. But it is not clear to what extent these experiments are influenced by inhomogeneous broadening [61] and if even KTB physics would be detectable in such experiments [12]. In contrast, finite-frequency measurements can directly probe temporal correlations and can be explicitly sensi tive to phase fluctuations right above Tc. Important information has been gained from measurements at discrete frequencies [21,25,47,62], but only true spectroscopic measurements can give important information concerning critical slowing down. In this paper, we present a comprehensive study of the complex ac conductance of effectively 2D amorphous superconducting InO* films. We make use of our recent development of a broadband Corbino microwave spectrometer, which can measure the explicit frequency dependence of the complex conductance of thin films over a range from 0.21 - 15 GHz at temperatures down to 350 mK. These unique measurements allow true spectroscopy in the microwave range at low temperatures. We explicitly measure the temporal correlations of the fluctuation superconductivity and demon strate the manner in which their time scales diverge on the approach to the transition. The temperature dependence of the critical slowing down is consistent with a contin uous transition induced by the freezing out of vortex-like phase fluctuations. 75 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION 3.3 Sample Details For these measurements, high-purity (99.999 %) 1^03 was e-gun evaporated un der high vacuum onto clean high-resistivity silicon substrates held at liquid nitrogen temperature to a thickness of approximately 30 nm. InOa, samples were prepared at Sambandamurthy’s lab at University of Buffalo. Their synthesis derive from the work of Ref. [63], where it was shown that amorphous InO* can be reproducibly made by a combination of e-beam evaporation of 1 ^ 0 3 with optional annealing. Es sentially similar films have been used in a large number of recent studies of the 2D superconductor-insulator quantum phase transition [24,62,64-67]. We believe that the films are morphologically homogeneous with no crystalline inclusions or largescale morphological disorder because of the following: TEM-diffraction patterns are diffuse rings with no diffraction spots, AFM images are completely featureless down to a scale of a few nanometers (the resolution of the AFM), and R versus T curves when investigating the 2D superconductor-insulator transition [24,62] are smooth with no reentrant behavior that is the hallmark of gross inhomogeneity. In Fig. 3.3, we show the AFM image, TEM diffraction pattern and TEM image for a co-deposited insulat ing InO* film showing that InOx films evaporated in this fashion are amorphous and homogeneous. The temperature dependence of the sheet resistance of a granular InOz film can be found in Fig. 3.4, which shows the very noticeable reentrant behavior of a granular film. The in-plane penetration depth — the so called Pearl length (2A|D/d) [16] — can 76 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION O Figure 3.3: The AFM image, TEM diffraction pattern and TEM image of a co deposited insulating InOx film. The TEM diffraction pattern shows diffuse rings indicating that there are no crystalline inclusions. Both AFM and TEM images show that films prepared by the quench condensed e-gun evaporation do not include mesostructure and are not granular as well. 6000 A granular inaulating lnOxfilm 6000 5 m i 4000 JZ O 7 3000 2000 1.0 1.6 2.0 2.6 3.0 3.6 4.0 T(K) Figure 3.4: Sheet resistance of a granular InOx film showing the noticeable reentrant behavior. 77 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION be calculated from the data below to be approximately 6 mm near Tc, which is well in excess of any sample dimension. Vortices and antivortices are thus held together by logarithmical confining potentials. In this case, superconducting films are similar to the case of 4He films. 3.4 Results Experiments were performed in the home built broadband Corbino microwave spectrometer described by the previous chapter. We concentrate on a particular In 0 2 film with a Tc = 2.36 K, but the data is broadly representative of samples with this normal-state resistance. In what follows, Tc is defined as the temperature at which the simultaneously measured dc resistivity becomes indistinguishable from zero (shown in Fig. 3.6). A small ± 5 mK uncertainty in this determination does not affect our conclusions. In Figs. 3.5 (a) and 3.5 (b) we plot the real (G\) and imaginary (G2) conductance as a function of frequency at different temperatures. Well above the transition, Gi is flat and featureless and G 2 is small, as one expects for a highly disordered metal at low frequencies. When the sample is cooled toward Tc, the real conductance initially becomes enhanced and its spectral weight shifts to lower frequencies. At lower temperatures, the imaginary conductance grows dramatically and its frequency dependence becomes close to 1/u. This is the low-temperature behavior expected for a superconductor. As seen clearly in plots of the same data 78 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION 0.5 1.0 1.5 2 .0 2 .5 3.0 T (K) 0 .5 1.0 1.5 2.0 2 .5 3.0 T (K) Figure 3.5: (a) and (b): frequency dependence of the real and imaginary conductance in the ranges uj/2 t: = 0.21 —27 GHz and T = 0.35 —4 K. A color scale representing different temperatures is displayed in (a). The black curves are the conductance at Tc. The features in (a) at about 22 GHz are residual features imperfectly removed during calibration, (c) and (d): temperature dependence of the real and imaginary conductance in the frequency range uj/2n = 0.660—15 GHz. A color scale representing different frequencies is displayed in (d). The dashed black lines mark Tc = 2.36K. 79 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION as a function of temperature (Fig. 3.5 (c) and (d)), the region immediately above Tc is dominated by superconducting fluctuations. As shown by comparison to the dashed line, the real and imaginary conductance begin to show an enhancement in the temperature region above Tc. Our measurements are explicitly sensitive to temporal correlations. This is seen for instance in the fact that the near-Tc “dissipation peak” in Fig. 3.5 (c) is exhibited at lower frequencies for lower temperatures; the maximum in dissipation is expected when the characteristic fluctuation rate Q /27r is of the order of the probing frequency uj/2n. Above the transition temperature, due to the thermal energy, one expects short range fluctuations with length £ which persists for time r. On approaching the tran sition point, both £ and r become larger and larger. The characteristic fluctuation frequency 0 , which is defined as 1 / r , becomes smaller and smaller and vanishes in the vicinity of the transition. This phenomenon, so called critical slowing down, is generic to any continuous phase transition. One interesting feature in our very raw data is that we can see the signature of the critical slowing down from the movement of the peak in real conductance in temperature as shown in Fig. 3.5 (c). A particularly important quantity for quantifying fluctuations is the phase stiff ness, which is the energy scale to twist the phase of the OP. The phase stiffness, Tg, is proportional to the superfluid density and can be defined (in units of degrees Kelvin) through the imaginary conductance G2 as , , G2 kBTg(u) - — N (u)e2Hd - ——— — , 80 (3.2) CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION 1000 800 73 600 % ZJ 400 3 " □ 200 0 1.0 2.0 3.0 4.0 T (K) Figure 3.6: Temperature dependence of the phase stiffness at o;/27r = 0.21 — 15 GHz plotted against the vertical axis on the left. A color scale representing different frequencies is displayed as well. The black curve shows resistance per square of the same sample plotted with the vertical axis on the right. The dashed pink line is the KTB prediction, 4Tktb = Tg, for the universal jump in stiffness. The dark purple A markers are Tg s obtained via the scaling analysis described in the text. Tc is marked by the black dashed line. where G q — ^ is the quantum conductance for Cooper pairs and N (u) is a frequency- dependent effective density. In Fig. 3.6, we plot the stiffness versus temperature measured at frequencies between 0.21 to 15 GHz. Tg(u) defined through Eq. (3.2) measures the stiffness on a length scale set by the probing frequency, which is typically proportional to the vortex diffusion length during a single radiation cycle, A is a constant on order of unity and D is the diffusion constant. At temperatures well 81 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION below Tc, there is essentially no frequency dependence to the phase stiffness, consis tent with the scenario that the phase stiffness is rigid on all lengths. At temperatures slightly above Tc, the phase stiffness is largest at high frequencies. In the fluctuation regime, the system retains a phase stiffness on short length scales. Plotted alongside the stiffness data is the co-measured resistance per square, R q. Within experimental uncertainty, the phase acquires a frequency dependence at the temperature where the resistance appears to go to zero. In keeping with our discussion in the introduction, this is reasonable as a superconductor can only exhibit zero resistance when its phase is ordered on all lengths. KTB theory predicts that at the transition temperature, TKt b , the stiffness in the zero-frequency limit will have a discontinuous jump to zero with a magnitude Tg = 4Tktb- However, because finite frequencies set a length scale, the ac stiffness should go to zero continuously. We generally expect that a signature of the discontinuity will manifest in a strong frequency dependence in the stiffness that onsets at Tk t b - In Fig. 3.6, the dashed diagonal line gives the prediction [9,17] for the universal relationship between Tktb and the stiffness. It crosses the stiffness curves very close to where they start to spread. This, along with the fact that the resistivity goes to zero at this temperature, leads us to assign the transition to a vortex unbinding transition of KTB-like character. Note that a careful inspection of Fig. 3.6 on linear scale reveals that the stiffness is in fact approximately 30% (see Fig. 3.7) greater at Tc than the universal prediction. We cannot be sure at this time whether this is a 82 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION 20 4 8 g )/2 jc 15 12 (GH z ) T6 critical “^ p r e d ic te d 5- ok 2.0 Figure 3.7: A plot of Fig. 3.6 on a linear scale near Tc reveals that the stiffness is approximately 30% greater at Tc than the universal prediction. systematic deviation (due perhaps to the dissipative motion of bound vortex pairs or evidence of a non-universal jump [1 2 ]) or a small calibration error. Above Tk t b , the conductance due to fluctuating superconductivity is predicted [17,48,68,69] to scale with the form g <3 _ (* s 3 )s (u /n ). ' m (3.3) In this scaling function, all temperature dependencies enter through f!(T), the char acteristic relaxation fluctuation rate, and Tg, an overall amplitude factor related to the total spectral weight in the fluctuating part of the conductivity. Note that in Eq. 83 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION co/n(T) Figure 3.8: (a) Phase of G (u /fi) as a function of frequency, (b) Magnitude of G(ui/Q) as a function of reduced frequency. A color scale representing different temperatures for both plots is displayed in (b). (3.3) the prefactors T f and Q are real quantities, so that the conductance phase angle, V?, must be equal to the phase angle of S(u/Q ). In Fig. 3.8 (a), we plot phases of the measured conductance at different temperatures above the transition temperature for this particular sample from 460 MHz to 10 GHz in a temperature range from 2.398 to 2.993 K. Those phases are extracted directly from the measured conductance. Phases below 2.398 K are not shown because one needs to accurately determine the three error coefficients to higher precision near and also below the critical temperature. Thus phases obtained after calibration at those temperatures are a little off from the actual phases of the sample. To collapse all the phases into one single universal curve, we divide the probing a; by a different D(T) at each temperature, thus extract the characteristic fluctuation frequency fi for each individual temperature. In Fig. 3.8 (b), we show the magnitude of conductance for all the temperatures as a function of 84 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION reduced frequency u/Q . The amplitude of the universal function l^l can be directly obtained by multiplying the amplitude of the conductance by M I /^ b T^G q ). 1.0 0.8 ^ £S 0.6 * 0.4 0.2 0.0 0.01 2 4 6 0.1 2 4 6 1 2 4 6 0.01 co/fXT) 2 4 6 0.1 2 4 6 1 2 4 6 co/n(T) Figure 3.9: (a) Phase of S(w/Q) normalized by 7r/ 2 as a function of reduced frequency w/O. (b) Magnitude of S(u/Q ) as a function of reduced frequency. A color scale representing different temperatures for both plots is displayed in (a). Each plot is comprised of data measured at temperatures from 2.398 to 3 K and frequencies from 0.46 to 10 GHz. In Fig. 3.9 (a), we show this phase angle, <p, collapsed into a function of reduced frequency ui/Q for each temperature. The data collapse reasonably well down to 38 mK above Tc. At lower temperatures, the fluctuation frequency begins to enter the low-frequency end of the spectrometer (about 500 MHz for this set of data). In the low scaled-frequency limit, which corresponds to high temperatures and normal-state response, the phase approaches zero as expected. In the high scaled-frequency limit, which corresponds to low temperatures and the superconducting state, the phase approaches 7r/ 2 also as expected. This analysis allows us to extract fi(T). Here, 85 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION we have isolated the fluctuation contribution to the conductance by subtracting off the dc value from well above Tc (at 5.6 K). Having determined U(T), we adjust Tg and normalize the magnitude of conductance by l jO to get the magnitude (|Sj) of S(u/Q.) so that they fall onto one curve as demonstrated in Fig. 3.9 (b). In 2D, Tg is equivalent to the high-frequency limit of the stiffness. We plot it alongside the finitefrequency stiffness in Fig. 3.6. One can in principal compare the scaling forms for the 1.0 1 0.8 ^ 2.4 2. 5 2.6 2.7 2. 8 2.93.0 0.6 I * CO 0.4 2DAL 0.2 0 . 0.0 0.1 1 10 100 oVQ(T) 0.1 1 10 100 w/Q(T) Figure 3.10: Comparison of the scaling forms with the 2D AL formalism. InOz film with the 2D AL formalism (see Fig. 3.10). The explicit expression of the frequency dependent complex conductance was calculated by Schmidt [70]. Unlike the NbN films [49], one can see that we cannot fit the phase and magnitude of the scaling function simultaneously by the 2D AL expression. This also suggests that the transition in our InOx cannot be explained by the 2D AL formalism only. 86 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION The monotonic decrease of D(T) [Fig. 3.11 (a)] as Tc is approached from above, is an indication for the critical slowing down expected near a continuous transition. In Figs. 3.11 (b) and 3.11 (c), we fit Q(T) to the stretched exponential form expected near a KTB transition D0 exp(—y /4 T '/(T —Tc)) as well as to a generic power-law form D0(l — The fits were performed over different temperature ranges from Tc on up (147 and 112 mK, respectively) such that the same reduced x 2 is achieved on both fits. The stretched-exponential fit gives coefficients of Qq/2 tt — 181 GHz and T ' = 0.23 K, while the power-law fit gives Q,q/2 tt = 90 GHz and zu — 1.58, which are all rea sonable parameters. Unfortunately it is hard to distinguish definitely between these scenarios in our data. However, we do favor the KTB scenario since the stretchedexponential fit covers a wider temperature range and fits better near Tc. For instance, within the ansatz of Ref [17]. it is predicted that V — 7 (1 ]*} —TKtb)i where a constant of the order of unity. Ref. [61] predicts more specifically that 7 7 is = 4a2, where a is the ratio of the vortex core energy, //, to the vortex core energy in the 2D XY model, fixy. Viewed in this regard, our value of V is consistent with a reasonably small value of the core energy. For both functional forms, one expects that the prefactor D0 will be of order of the inverse time needed to diffuse a vortex with core size fo- Using the BardeenStephen [17] approximation for D, one can derive the expression hfi0 = 2ttXh D / ^ = 27tA^A:bTc, where A is a constant of the order of unity and G n is the normal-state 87 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION dc conductance. For the present sample, this gives Q0/27t rj 48A GHz, which is consistent with both fits. Due to its larger fitting range and its consistency with the universal jump, we favor the stretched exponential form, but in practice, it is difficult to definitively exclude power-law dependencies. However, we can note a number of additional aspects consistent with a vortex plasma regime. Over a more extended temperature range above Tc, one expects that Q(T) will obey the relation n 0 exp(—^-). Here, T* is half the energy needed to thermally excite a free vortex-antivortex pair. In Fig. 3.11 (d), we plot T* = Tln(Do/U). As expected this quantity appears to diverge as T —» Tc. It reaches a high-temperature limiting value of about 0.27 K. One expects [59] that kr>T* — n + \ksTg ln(£/f0) as the logarithmic interaction has a cutoff at f. Here, (jl is the vortex core energy and the second term is the vortex interaction energy. At temperatures well above Tc where £ is of the order of £0> the logarithmic term is negligible and the excitation energy should be proportional to the core energy alone. Within the BCS model, the core energy can be shown [59] to be approximately fcflT^T)/ 8 . A comparison with T# from Fig. 3.6 gives an estimate of ii/k B ~ 0.3 K in this temperature range. The agreement with experiment is essentially exact, but one should not take the exactness too seriously as there are a number of neglected factors of the order of unity. 88 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION 3.5 Conclusion We have presented a comprehensive study of the complex microwave conductance of amorphous superconducting InOx thin films. Our data explicitly demonstrate critical slowing down close to the phase transition and, in general, the applicability of a vortex-plasma model above Tc. This technique opens up the possibility of studying dynamic scaling of phase transitions at low temperatures and frequencies of a number of material systems. 89 CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION <t*N20 JN 2.4 2.6 T (K) 2.8 2 5 8F * 010 £ a£ 4 5 6 7 8 9 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 oVO(T) Figure 3.11: (a) Fluctuation frequency as a function of temperature from 2.398 to 2.993 K. In (b) and (c), we plot U (T)j2 ti versus l /y /T — Tc and T —Tc, respectively, along with the fitting (black curves), (d) Excitation energy in units of degrees Kelvin as a function of T —Tc. 90 Chapter 4 2D field tuned superconductor-m etal quantum phase transition A quantum phase transition (QPT) is a zero temperature change of state as a function of a non-thermal parameter. The two dimensional (2D) superconductorinsulator transition (SIT) is an emblematic example of a QPT and has been the subject of many theoretical and experimental studies [22,71]. As conventionally en visioned, superconductivity can be suppressed by applying magnetic fields beyond some critical value whereupon the system transitions to an insulating state with a diverging resistance at T = 0. One possible scenario is that this transition occurs via a “fermionic” mechanism by destroying the amplitude of the superconducting 91 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION order parameter. At the transition point, superconducting correlations are strongly suppressed. Another possibility is a “bosonic” mechanism in which the Cooper pairs become localized through quantum disordering of the order parameter’s phase. In this case, one expects pairing to exist on both sides of the transition. Within the latter bosonic description, a universal resistance of order the quantum resistance for Cooper pairs Rq = h / 4e2 ~ 6450 il is expected at the quantum critical point (QCP) [22]. Although some indications for superconducting correlations in the insulating state have been reported [24-26], there has been little definitive evidence in favor of this pure bosonic model. Some systems do exhibit a resistance of order Rq at the transition; however, others show a much smaller critical resistance [72]. More over, instead of a direct transition to an insulator many materials appear to exhibit an intervening metallic region that features a small yet finite saturated resistance at the lowest measured temperatures [73-78]. This effect is usually more pronounced in low-disorder films. In these cases the transition appears to be one from a supercon ductor to a strange metal with superconducting correlations. A true metallic phase as such may be surprising because one might naively expect that delocalized bosons would ultimately condense at the lowest temperatures. The physics is still unclear despite various theoretical efforts to demonstrate the possibility of a zero-temperature dissipative state with superconducting correlations [30,79-82]. On the experimental side, the possibility exists that the apparent zero-temperature DC dissipation could be a consequence of insufficient cooling of the carriers despite careful experimental 92 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION checks. For these reasons, it is important to utilize experimental probes other than DC transport to investigate this problem. Microwave spectroscopy gives an advantage in studying the 2D SIT in that one can be explicitly sensitive to temporal correlations. AC measurements provide detailed information about the critical slowing down of the characteristic frequency scales approaching a transition, which may reveal the true location of the QCP. Furthermore, we can study the dynamics of the possible intervening metallic state. Through the imaginary conductance, AC measurements of superconductors also allow access to the phase stiffness Tg(u), which is directly related to the phase coherence on a length scale set by the probing frequency. 4.1 Dynamics of 2D quantum phase tran sition In this chapter, we present novel measurements of the frequency, temperature and field dependence of the complex microwave conductance on a particularly low-disorder superconducting InOx film through its QPT. From the simultaneously measured DC resistance, a well-defined field BcrOSS is identified as the crossing point of different isotherms R(B). In most interpretations of similar data, Bcr0as is taken to be the location of the QCP. However, quite contrary to expectations for the slowing down of fluctuations near the presumed continuous transition at Bcr0SS, the relevant frequency 93 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION scales extrapolate to zero at a much smaller field B sm. The phase stiffness Tg at the lowest frequency vanishes from the superconducting side at B « Bsm, while the high frequency limit approaches zero at B « Bcr0SS. Our data support a scenario in which B sm is the true QCP for a transition from a superconductor to an anomalous metal, while Bcross only signifies a crossover to a regime where superconducting correlations are strongly suppressed. Samples are morphologically homogeneous InOx films prepared by e-gun evapora tion of In 2 0 3 to a thickness of approximately 30 nm onto high-resistivity Si substrates as described in Chapter 3 . The nominal 2D SIT in InOx can be tuned by applying perpendicular magnetic fields [24,25,66,67]. Broadband microwave experiments were performed in a home-built Corbino microwave spectrometer coupled into a He-3 cryostat. We measured the complex reflectivity of the sample in the microwave regime, from which complex impedance and conductance can be obtained. Three calibration samples with known reflection coefficients (20nm NiCr on Si, a blank high-resistivity Si substrate and a bulk copper sample) were measured to remove the contributions from the coaxial cables to the reflected signals [38,41,44,48,51]. Two terminal DC resistance can be simultaneously measured via a bias tee. The DC resistance without the microwave illuminations was used to carefully check and correct for any microwave induced heating. The sample at 5 Tesla had a impedance closely matched to the cable thus had a maximum absorption of microwave radiations. The lowest temperature for that field was 426 mK under microwave radiation. The heating effects at other 94 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION fields were negligible. Calibrations were performed at each displayed magnetic field unless otherwise specified. With substrate corrections 2.3, the true response of the InOx film can be isolated at all fields and temperatures. 0 1 2 3 4 0 T(K) 2 4 6 8 B (T) Figure 4.1: (a) Temperature dependence of sheet resistance Rq at different fields as indicated by the color legend, (b) R q as a function of field at 6 fixed temperatures as shown by the color legend. The crossing point of the two lowest temperature isotherms is approximately 7.5 Tesla. In Fig. 4.1 (a) we plot the two-terminal sheet resistance Rq as a function of temperature at fixed magnetic fields. The particular InOx film in this paper shows a transition to a zero resistance state at Tc = 2.36 K at zero field. Previous microwave studies have demonstrated that its zero-field thermal fluctuations are consistent with a 2D Kosterlitz-Thouless-Berezinsky transition [51]. The normal state resistance per square R n is about 1200 fh This number is far below R q and is comparable to R n for thin films like a-MoGe [72]. It implies that this film has a much lower disorder level (kpl is in the range 3 ~ 6 [83]) compared to the InOx films used in many previous 95 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION QPT studies and falls into the same class of lower-disorder thin films superconductors such as a-MoGe [30]. At low temperatures, the slopes of the resistance curves change sign at about 7.5 Tesla (see Fig. 4.1 (a)). For the two lowest temperatures, the data also exhibit an isoresistance crossing point Bcr0sa at 7.5 Tesla. This field is conventionally interpreted as the location of QCP. Like a-MoGe, this sample exhibits an exceedingly weak “insulating state”, with barely a 10 % rise in the resistance from 4 K to the lowest measured temperatures at B > Bcr0gs as shown in Fig. 4.1 (a). This is again very different from strongly disordered InOx films that show an enhancement of the resistance upwards of 109 Q at similar magnetic fields at low temperatures [24,66]. Like other weak disordered superconducting films, the DC data show an apparent trend towards saturation at low temperatures for fields above 3 Tesla. This saturation was confirmed in separate two-terminal measurements of this sample down to dilution fridge temperatures. A separate measurement of the same sample down to 60 mK was carried out roughly half a year after the microwave measurements. Unfortunately, a direct com parison between the two DC data sets cannot be made as the InOx sample anneals even at room temperature over the course of intervening months. As shown in the supplementary information, although the sample changed with a systematic move ment of curves with the same resistance to higher fields, the data sets are overall very similar with continuing trend towards saturation at low temperatures. The sample has been kept in the dry box the whole time and has lost some oxygen over the course 96 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION of intervening months. In Fig. 4.2 (a), we display Ra as a function of temperature at different fields for the sample after a few months aging. The level of disorder has changed with a new Tc = 2.68 K at zero field. We show R q as a function of field at 75 mK and 150 mK in Fig. 4.2 (b) and one can clearly see that BcrOSS changes to 7.86 Tesla. In Figs. 4.3 (a) and (b), we plot the real (Gi) and imaginary (G2) conductance respectively as a function of frequency at the base temperatures for each field (« 426 mK for 5 Tesla and « 300 mK for all other fields). Due to a missing set of calibration curves, the data at 4 Tesla were calibrated using an effective calibration interpolated from 3.5 Tesla and 5 Tesla calibration standards. We estimate that the error introduced by this interpolation is less than with a slope of -1 1 %. As shown by the straight line on the log-log plot in 2 (b), at zero field and 300 mK, G 2 shows the 1/u frequency dependence expected for a superconductor at frequencies below the gap. This dependence is consistent with Gi = 6(uj) via the Kramers-Kronig relation. Indeed, Gi at zero field is small with a value that is at the limit of our experimental sensitivity, thus is not plotted. At B <tc BcrOSs, G% remains linear with the same slop in the log-log plot but its magnitude drops dramatically as the field is increased. This implies that the 5-function in G\ is preserved, although its spectral weight (proportional to the superfluid density) is greatly decreasing. At intermediate field strengths (B ~ 3 Tesla), a maximum in G 2 appears. Accord ing to the Kramers-Kronig relation, this implies that a significant spectral component 97 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION 1400 r 1200 1000 □ 0T 800 2T 3. 5 T 4T 5T 6T 6. 5 T 600 400 7. 5 T 8T 11 T 14 T, 200 4 3 2 1 0 T (K) 1400 1200 75 mK 150 mK 1000 cross 800 £ 600 400 200 0 4 6 8 BCD 10 12 14 Figure 4.2: (a) Temperature dependence of the R q measured in the range T = 0.06 - 4 K at fields represented by the color legend, (b) Ro as a function of field at two fixed low temperatures. 98 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION Figure 4.3: Frequency dependence of (a) real (Gi) and (b) imaginary (G2) conduc tance respectively in the ranges u /2 n = 0.08 - 16 GHz. Gi and G2 have the same color legend at finite fields except that G\ at zero field is not plotted. The dashed grey line in (b) is a guide to the eye of G2 oc l /u . in G\ has a finite width. As shown previously [51] , the frequency of the maximum in G2 corresponds to the characteristic fluctuation rate D in a fluctuating supercon ductor. The decrease in the frequency of the peak in G2 as the field is reduced is an unambiguous signature of critical slowing down of the fluctuation frequency while approaching a transition. However, the peak in G2 is developed at a field that is well below Bcross and the fluctuations are clearly speeding up as we approach Rc-oss from below. This behavior is inconsistent with the conventional wisdom for QCP phenomenologies if Reross is a QCP because one generally expects a slowing down of the internal fluctuation frequency scales as one approaches a continuous transition. When B ~ Rcross> we cannot distinguish the superconducting signal from the normal state background as Gi is flat and featureless and G2 is small. The data is reminiscent of what one expects for a disordered metal with a high scattering rate [51]. 99 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION 50M H z 1 125M Hz 500M H z 1G H z 5G H z 10G H z 14G H z &/2n (GHz) B (T) Figure 4.4: (a) Frequency dependence of the phase stiffness in the ranges u/2 ir = 0.08 - 16 GHz at the base temperature for each field. The color legend for fields is the same as in Fig. 4.3 (a), (b) Phase stiffness as a function of field at different frequencies at the base temperature for each field. An essential quantity for analyzing superconducting fluctuations is the phase stiff ness Tg , which is the energy scale required to twist the phase of the superconducting order parameter. Within a parabolic band approximation, Tg <x N, the superfluid density. More precisely (and in a model independent fashion), it is proportional to the spectral weight in the zero frequency delta function and can be measured through Gi Tg as Tg(ui) = , where Gq = 1/ R q . This relation expresses the energy scale in degrees Kelvin, and gives the phase stiffness on a length scale set by the probing frequency. Fig. 4.4 (a) shows T g (u ) above for each field. At low field, at the respective base temperatures described Tg shows essentially no frequency dependence, which suggests that the phase is ordered on all lengths. We see a dramatic drop in Tg at B -C Bcross. For intermediate fields, 100 Tg starts to acquire a strong frequency CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION dependence at low u/, which reflects that Cooper pairs have short-range correlations that can be resolved at high probing frequency while the long-range correlations are suppressed. At high u the frequency dependence becomes less pronounced showing that one approaches a well-defined high frequency limit. The rapid decrease in the overall scale of (b) where we display field dependence of Tg Tg can be clearly observed in Fig. 4.4 at several frequencies cuts from Fig. 4.4 (a). Above 2 Tesla, the curves start to spread, indicating the superconducting correlations gain a length dependence. At the lowest frequency (50 MHz, which probes the longest length scale), Tg drops dramatically around 3 Tesla indicating that long range ordered phase coherence is suppressed by increasing fields. Note the strong suppression in Tg in this field range; at some frequencies the suppression in followed over 7 orders of magnitude. Unlike the low frequency behavior, Tg Tg can be at high frequency extrapolates towards zero near Bcr0as. This latter finding greatly differs from previous microwave cavity measurements on a much more disordered InOx film (Bcross = 3.68 Tesla) [25]. In that work the finite-frequency Tg was non-zero well past the phenomenologically defined Bcross into the strongly insulating phase. This was interpreted as a state that while strongly insulating on long length scales, has superconducting correlations on short ones. In contrast, Tg for our low-disorder film vanishes on approaching Bcross instead of staying finite well beyond it. This indicates that the superconducting correlations do not survive appreciably through 5cro«s and the superfluid density is indistinguishable from zero into the weakly insulating state 101 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION at the lowest temperatures we can access. To form a more quantitative understanding of the fluctuations, we fit G\ and G2 to a model where the fluctuation contribution is given by a zero-frequency Lorentzian lineshape [84]. This is simpler, but essentially equivalent to the scaling analysis we performed in Chapter 3 to obtain the characteristic fluctuation rate approaching a finite temperature transition at zero field [51]. Lorentzian fits are equivalent to assuming that time correlations are exponentially diverging while approaching the transition. The fits agree well with the data, thus justifying this assumption. The fit of G\ and G2 to a Lorentzian lineshape goes as , Nie2d n 1 N2e2dr2 1 G = od= ---- :------ 1----------------- :---m 1 —iuti m 1 —tur2 Essentially, we use two Drude models to describe the combined contributions from the normal electrons and superconducting fluctuations to the complex conductance. The scattering rates of the normal electrons for this film are much bigger than Q and exceed our frequency range. One can estimate the scattering rate r for the InOx in the normal state from the normal resistance by using the equation R N = l / G N = l/C2^ ^ ) . Here, d = 30 nm is the film thickness. Using n ~ 1020 cm-3 [63,85,86] and R n « 1000 Ohms, we estimate the scattering rate for the normal electrons is on order of 100 THz. Therefore, the Drude term of the normal electrons just makes a constant contribution to G\ and a negligible contribution to G2. In Fig. 4.5, we show the fitting to G\ and G2 at B = 3.5 Tesla, T = 850 mK as an example and the fitting agrees well with our experimental data. For this particular data set, the complex conductance can be 102 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION 12x10 B = 3.5 Tesla T = 850 m K T M E £ 0 Fitted G, Fitted G , 9 £ §3 1 o 0 2 4 6 8 10 a>/2n (GHz) 12 14 16 Figure 4.5: Experimentally measured real and imaginary conductance and fitted data to a Lorentzian lineshape model. fitted as, _ + 0.0094 l-i(£ /1 .2 3 9 )G H z 0.00172 l- i( £ / 2 2 .7 8 7 ) G H z In this model the fitted width of G\ is the characteristic fluctuation rate D(T), while its integrated area is equivalent to the high frequency limit of Tg (in appropriate units). In Fig. 4.6 (a), we plot D(T) for fields up to 6 Tesla. Data above 6 Tesla exhibit fluctuation rates that axe far above our accessible frequency range. At zero field, Q goes to zero when T approaches T c from above showing the critical slowing down that confirms our previous results [51]. Q drops in a much slower fashion at finite fields and 103 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION even begins to saturate to a finite value at the lowest temperatures for B is greater than or equal to 2 Tesla. Fig. 4.6 (b) is a contour plot of in field and temperature. It is safe to conclude that the D = 0 contour falls below the lowermost curve that has n = 0.2 GHz. In general, small fi contours at low temperatures extrapolate to zero at a field less than 3 Tesla, which is again far below Bcross- To form a global view, we bring a number of these quantities together in a single phase diagram in Fig. 4.6 (c). For all quantities, energy scales and frequencies have been converted to energy units (in degrees Kelvin). Upward and downward triangles show the low and high frequency limits of Tg in our setup’s accessible frequency range. Squares are f2(T) taken at base temperatures. Circles demonstrate the temperature when Rq = 0.3% R n , where R n is the sheet resistance at 4 K. It denotes a region in phase space where the resistance is “small” . It is clear from this plot that Tg in the zero-frequency limit (shown by the bold line in Fig. 4.6 (c)) and Q converge towards zero at B « 3 Tesla. This “V” shaped phase diagram is exactly what one expects near a QCP where energy scales extrapolate to zero from either side. Again, BcrOSS, which is conventionally considered to be a QCP, appears to be completely unrelated to the actual critical behavior. One can see that B ^oss is the field scale where the high frequency Tg is suppressed. Due to the lack of evidence for a diverging R q at B sm < B< Bcross in the zero-temperature limit, one reasonable interpretation of the phase diagram is that this low-disorder InOx film has a true QCP located at B sm « 3 Tesla between a superconducting and an anomalous metallic state. In this picture BCTOSS only marks a crossover in behaviors 104 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION 0.2 5 10 15 20 25 cross . 1 .0 1 .5 2 .0 2 .5 0 T(K) T dt R qt s 0.3 A> Rn Te at 50MHz Te at 14GHz Q in Kelvin cross Te at 0 Hz Figure 4.6: (a) Temperature dependence of Cl at different magnetic fields, (b) Contour plot of f! in temperature and field. Solid vertical black lines are the actual B and T values where data were taken. Color indicates the magnitude of interpolated values of Cl from the fitted data, (c) A phase diagram of all the quantities converted to unit of Kelvin. The dashed vertical black lines in (b) and (c) mark Bcross- (d) An alternative scenario for contour plot of Cl assuming Bcross is the QCP. 105 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION between a dissipative state with strong superconducting correlations on short length scales and one with vanishing such correlations. The view that B sm instead of Bcross is the true QCP in weakly disordered films runs counter to much prevailing dogma in the field. If one is to insist that Bcross controls the critical behavior, it requires at least two additional conditions to explain our data, both of which we consider unlikely. First, one has to posit that the contours of constant Q must have particular shapes like those shown in Fig. 4.6 (d) and what we are seeing is a finite temperature effect. If, as discussed above, a true superconducting state only exists for T ^ 0, B = 0 and T = 0, B ^ 0, the true Q = 0 contours are the two red lines lying on the B = 0 and T = 0 axes. All the finite Q contours should intercept the two axes at T > Tc, B = 0 or B > B^oss, T = 0. In our case, this means that these contours would have to have exceedingly long tails extending over Bcross• Although we cannot exclude this possibility, it is a challenge within the current theoretical framework to explain why the contours would pick up this particular shape. Secondly, if Bcross is the actual QCP, the AC dynamics must be completely insensitive to the existence of the QCP as we have seen no evidence of quantum critical signatures in our AC data extrapolating towards Bcross• This may be possible, but only in a scenario where the QPT is a transition associated with pure classical percolation [30,87]. 106 CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM PHASE TRANSITION 4.2 Conclusion To conclude, we find evidence for a possible scenario where a 2D QPT occurs at a field B am between a superconductor and an anomalous metal with superconducting correlations. The lowest temperature Q and Tg at the lowest frequency extrapolate to zero from both sides of B sm. The lack of evidence for finite-frequency Tg surviving Bcross shows that BcrOSS is a crossover above which superconducting fluctuations make a vanishing contribution to both DC transport and AC measurements. A careful and complete investigation of a more disordered film is needed to compare the effects of different disorder levels, which is a future direction of our project. 107 Chapter 5 Summary We have studied the dynamics of the superconducting transition in two dimen sions, of both thermal and quantum types. To carry out the experiments, a broadband Corbino microwave spectrometer was developed. To the best of my knowledge, this is the first time this type of spectrometer was applied to study 2D quantum phase transition. We were able to push the temperature limit of the spectrometer of this kind to 300 mK, which enable us to access a regime where Hu < k g T (1 GHz « 50 mK). The InOx sample we reported data in this thesis features a broad transition in sheet resistance for B = 0 transition. We fit the sheet resistance to the AslamazovLarkin form in 2D to determine Tco, the temperature where Cooper pairs start to form. Below Tco, phase fluctuations dominate. By doing a true spectroscopy, we obtain the dynamical information about the superconducting fluctuations and find 108 CHAPTER 5. SUMMARY evidence for critical slowing down along the T axis. We also demonstrate that the thermal phase transition is a transition of KTB type and can be described by a vortex plasma physics. We can extract T tk b as the temperature where the superfluid stiffness acquires a frequency dependence. However, the superfluid stiffness at Tktb is about 30% greater than the prediction of the universal jump of the KTB physics. It needs a more systematic study on 2D superconductors with different transition temperatures to determine whether this deviation is caused by small calibration errors or effects of non-universal jump nature of the KTB transition. The study of thermal fluctuations helps with our understanding of the fluctuations across the quantum critical point. Our microwave data of the low-disorder InOx sample look very much like a thermal transition, for example, the evolution of the real and imaginary conductance at 300 mK in magnetic fields looks similar to the transition in temperature at B = 0. One can compare Fig. 3.5 and Fig. 4.3 and easily come to the same conclusion. We reveal that while the DC transport is consistent with previous measurements of SIT with a crossing field BCTOSS which is the nominal quantum critical point, we find no evidence of the phase stiffness past this field. More anomalously we observe the characteristic fluctuation rate is “speeding up” as we approach BcrOSS from below. Our data show that the true quantum critical point might be at a field B am where both the low frequency phase stiffness and the fluctuation rate extrapolate to zero. This field J3sm is far below jBc-oss and it seems 109 CHAPTER 5. SUMMARY that the sample displays some intermediate metallic behavior above B am. Compared with the microwave cavity data from R. Crane et al [25], where they showed that the superfluid stiffness is finite above Bcross in a high-disorder limit, we observed that the low-frequency superfluid stiffness vanishes well below B^-oss in the low-disorder limit. Interesting questions to ask are that there has be to one disorder level where superfluid stiffness vanishes at 5 CTOSS, what is the significance of this disorder value? Is it possible this disorder value is the critical parameter? We might be able to find answers to those questions once we take spectroscopic measurements on InOx films with higher disorder values. This is planned for future measurements. 110 A ppendix A Transmission line model This chapter is dedicated for readers who want to know how to obtain impedance and conductance from the reflection coefficients. A transmission line exhibits proper ties of capacitance, inductance, resistance, and conductance and can be schematically represented by Fig. A.I. R is the distributed resistance of the conductors. The mag netic field produced by the currents contains magnetic energy as an inductor L. The capacitance C describes the electric field between the conductors. G represents the conductance of the dielectric material separating the inner and outer conductors. • nnrr\_A^_ Figure A.l: A schematic plot of the transmission line model. Ill APPENDIX A. TRANSMISSION LINE MODEL In the transmission line model, it is more convenient to use voltages (V) and cur rents (I) instead of electric and magnetic fields to describe the propagating waves since the characteristic impedance of a line is always the ratio of V and I, but some times not directlythe ratio of E and B fields. This modelbest suits transmission lines carrying TEM waves. V and I fora transmission line depend on the position and time as V(z,t) and I(z,t). The telegrapher equations for the voltage and current [88] are: = W ^ H KC + L G ) ^ + RGI{z,t) The time dependence of the voltage and current are V (z,t) = V(z)eiut I(z,t) = /(z)ei<Jt The telegrapher equations thus simplify to = 0 (A.l) = 0 (A.2) where 7 = y /(R + iuL){G -I- iuC) = a + i/3. 112 APPENDIX A. TRANSMISSION LINE MODEL The solutions to the telegrapher equations are of the form V{z) = V’+e-7* + V~e+,yz I ( z ) = ~ e ~ lz - ^ - e +7Z = 7+e~7* + 7~e+7* Zq (A.3) (A.4) Zq where ZQ = y/(R + iijjL)j{G + iuC) is the complex characteristic impedance of the transmission line. V + and 7+ are the amplitudes for a wave propagating along the positive z direction and V~ and 7~ are those for a wave propagating along the negative z direction. A .0.1 Im pedance m atching In a homogeneous line, the characteristic impedance is independent of z and thus is a constant for a TEM wave. If there is a discontinuity in the line, the impedance changes and a reflection occurs. This discontinuity may come from a change in the line dimensions and materials which has a different impedance from the original line. Suppose a load terminates the line at z = 0. The characteristic impedance of the line is Z0 and the impedance of the load is Z i. According to equations A.4, a voltage wave propagating toward the load is expressed as F +e~"72, and the corresponding current wave is I +erlz (Z0 = V +f I +). In general, only part of the propagating wave power is reflected and the other part is absorbed. The reflected voltage and current waves are V~e+,yz and I~e+lz with Z q = V~ j l ~ . V +, I +, V~ and I~ are all complex amplitudes. At z = 0, the voltage and the current of the line and those of the load 113 APPENDIX A. TRANSMISSION LINE MODEL must be equal: = vL + i+- r = iL Note that 7+ and I~ have opposite directions. Using Z L — Vl / I l , the equation for current can be written as Y1_Y=. = Yl Zq Zq Zi The reflection coefficient of the load is defined as S = V ~ / V +. Eliminating V ~ , we obtain the equation for refection from a load as _ ZL - Zq Z l + Zq For the geometry of the samples we use in our experiments, we have Z L = [ln{b/a)/2Tx]Zs, where surface impedance Zs is defined as Zs = Er/H$. The surface impedance of a bulk material is related to its conductivity as Z \f = For a film with thickness d, the effective impedance is Zs = Z f coth(/cd) where k = y/i^Qua. In the thin film limit where the thickness is much smaller than the skin depth, one obtains Z$ 1/{ad). In general, the confusion comes when we try to calculate the conductance from the impedance. We need to know whether we should take the conjugate of the final results. If we define E = the conductivity is defined as <r = <Ti —zcr2 from the Drude model. What if we define V (z , t ) = V(z)e~VJt for the transmission line model? The 114 APPENDIX A. TRANSMISSION LINE MODEL voltage and current now can be written as V (z,t) = Vi(z)e~tu>t, I(z, t) = Ii(z)e~lu>t, which satisfy the following equations: « - « . > - » where 71 = \ / ( R —iojL)(G —iuiC) = a —i/3 = 7 * Complex conjugate of equations A.2 are written as: - 0 - 0 Then the solutions to the above equations are the conjugate of the solutions to equa tions A.2: Vi(z) = V*(z), h (z ) — I*(z). Therefore, the reflection coefficients under this convention is S = r t = ■ The conductivity is defined as a — o\ + i a 2 from the Drude model in this convention. In the thin film limit, Zs is still proportional to 1/a. Therefore, if we take the time dependence to be elu)t, a = o\ — i a 2. If we take the time dependence to be e~lu)t, what we calculate from the reflection is Z*L and a = 0 \+ i a 2. And we obtain a correct sign of a 2 for both cases. 115 Appendix B Experimental procedures B .l Measurements at zero field As discussed in the main content, experiments for the three standards and samples under study have to be carried out in the same manner repeatedly. Our cryostat is a one shot He3 system. When the sorption pump (charcoal pump) is heated over about 25 K, it releases enough He3 gas so that we can condense liquid He3 into the He3 pot by keeping the temperature of 1 K pot below 2 K. After the temperature of the He3 pot reaches below 2 K for over 20 minutes, we can start cooling down the charcoal pump. For our system, once the temperature of the charcoal pump becomes less than 10 K, it functions as a very good pump and pumps on the condensed liquid He3 in the He3 pot. After charcoal pump reaches its base temperature (usually at about 4.3K) for about half an hour, He3 pot would approach its base temperature ~ 116 APPENDIX B. EXPERIMENTAL PROCEDURES 300 mK. First warming up Cooling down Second warming up £ £ I & £ .2 olE 0 100 200 300 Time (arbitrary unit) 400 Figure B.l: Examples of temperature profiles of the sample stage for the standardized three zero field measurements. Each data point was recorded every minute for this set of experiment. Therefore, the X axis is proportional to time. The total time span of the X axis is about 450 mins. So in general, for T > 2 K, the system is cooled by He4. From 2 K to the base temperature, the system is cooled by He3. It is difficult to control the temperature well from 1.5 K to 2 K and this was a challenge for us since we need to be able to reproduce the temperature profile for each sample. We could not simply control the temperature by controlling the heater on the He3 pot. Once the liquid He3 starts evaporating, the whole condensed He3 liquid would be gone in just a second and we would get a jump in the sample temperature. Nor could we control the temperature through purely controlling the 1 K pot temperature cause we could not get down to 117 APPENDIX B. EXPERIMENTAL PROCEDURES 300 mK in that way. The most applicable way to control the temperature of our He3 pot is to actually control the temperature of the charcoal pump. If we slowly warm up the charcoal from about 4.3 K (lowest it can reach) to 30 K, the He3 exchange gas is slowly warmed up and it would gradually warm up the sample stage. We use a very slow warming up procedure and one warming up scan usually takes a couple of hours to reach 5 to 6 K. But as one can see from Fig. B.l that the temperature profile is very reproducible for two different warming up procedures with different liquid He4 levels in the dewar. B .1.1 O perating sequence It is very important for us to reproduce the cryogenic conditions in every temper ature sweep due to the sensitivity of the calibration procedure on cryogenic environ ment. The full procedure we followed is below: Day One: 1. Pump the vacuum jacket of the dewar and also the vacuum jacket of the liquid He4 transfer tube to high vacuum (HV) if it’s necessary. The turbo pump we use is Pfeiffer vacuum turbo pump TPH 062 or similar. It’s highly recommended to pump the vacuum jackets every a couple of months. We usually pump them overnight (more than 12 hours) or longer for the vacuum jacket of the dewar (at least 24 hours) since it has a huge surface area. 118 APPENDIX B. EXPERIMENTAL PROCEDURES Day two: 1. Attach the samples into the sample stage. Use a caliper to fix the length of the set screw. For a sample of thickness 380 microns, the length of the set screw outside the sample stage is 4.5 mm. Adjust it accordingly if the sample thickness is different so that the spring force is the same for every single sample so that the references plane is the same. (Note the length might change after we changed the Corbino probe in the summer of 2012.) Check the DC reading to make sure that the Corbino probe and the sample are in good electric contact. Also check the microwave spectra at room temperature. 2. Attach the IVC cylinder to the insert and seal the IVC with an indium wire. 3. Pump the IVC to HV using a turbo pump. We usually pump the cylinder overnight (more than 12 hours). Day three: 1. Leak check at room temperature. 2. Fill the IVC with N2 gas. We do not have a pressure gauge for the IVC. In general the regulator for the N2 gas tank reads about 1 or 2 Psi. We usually fill the IVC with two pump tubes of gas N2 (the pump tube is connected to the IVC from the turbo pump and it’s about 2.5 meters long). 3. Back flow the 1 K pot and the charcoal pump with He4 gas since during this step, the dewar is still cold in most cases. 119 APPENDIX B. EXPERIMENTAL PROCEDURES 4. Lift up the dewar. This is usually a two persons’ job. One person cranks up the dewar and another person needs to make sure that the baffles of the insert are not hitting the baffles in the dewar. The tolerance between them are very small. Attach the dewar to the insert which is fixed onto a frame. We leave the platform of the lift up to provide extra support to the dewar. Check the pressure reading of the He3 tank. It usually reads 220 Psi at room temperature. The back flow to the 1 K pot and the charcoal pump is on all the time during this process. 5. Fill the dewar with liquid nitrogen. Once liquid nitrogen starts to accumulate at the bottom of the dewar, stop the back flow to the 1 K pot and the charcoal pump and start pumping them. Check the flow for both absorption lines for the 1 K pot and the charcoal pump and make sure they are not clogged. Close the flow meter of the charcoal pump. 6. Stop transferring once the level of liquid nitrogen reaches about 12” — 15”. 7. When 1 K pot reaches about 100 K, fill the 1 K pot with He4 gas.Turn off the pumps for the 1 K pot and the charcoal pump. 8. Leave the system overnight to reach thermal equilibrium at 77 K Day four: 1. Pumping out the gas nitrogen in the IVC. 120 APPENDIX B. EXPERIMENTAL PROCEDURES 2. Attach the IVC to a leak check. Push out the liquid nitrogen in the dewar by overpressurizing the dewar with He4 gas while leak checking. The regulator of the He4 tank reads about 1 or 2 Psi. But we never use pressure higher than 3 Psi. The whole IVC is immersed in He4 gas environment during this procedure. 3. When all the liquid nitrogen in the dewar is pushed out, flush the 1 K pot and the charcoal pump with He4 gas for about 10 seconds. Overpressurize the dewar again to get rid of any possible residue of liquid nitrogen. 4. Transfer liquid helium. Once liquid helium starts condensing at the bottom of the dewar (when the He level meter yields positive reading. It usually takes about 5 — 10 mins when the pressure in the liquid He tank reads about 1.5 Psi. But the time is longer after we installed the magnet.) start the pumps for the 1 K pot and the charcoal pump. Open the flow meter for the charcoal pump and the needle valve for the 1 K pot. Check the readings for both to make sure they are not clogged. 5. Stop transferring when the helium level monitor reads about 55 — 60 cm for the initial cool down. 6. Start cooling the sample stage to base temperature. Set the set point to 32 K and turn on the heater in medium range for the charcoal pump. Open the needle valve for the 1 K pot between 1/2 and 2/3 turn open. Since we do not have exchange gas in the IVC, it takes about 2 — 3 hours for the sample stage 121 APPENDIX B. EXPERIMENTAL PROCEDURES to reach 2 K from 77 K. We maintain the temperature of the 1 K pot below 3 K during the process by either opening or closing some amount of the needle valve. 7. Once the temperature of the sample stage reaches 2 K, we set the set point for the charcoal to 25 K and keep the 1 K pot below 2 K. After condensing He3 for about half an hour, turn off the heater and open the flow meter for the charcoal pump to cool down the charcoal pump below 5 K. Close the needle valve of the 1 K pot. After about half an hour, the sample stage reaches its base temperature, usually at about 300 mK. 8. Close the flow meter of the charcoal pump and turn off the pumps. Leave the system overnight to reach thermal equilibrium at low temperatures. Day Five: 1. Come in the early morning. Sample stage usually stays below 1 K. 2. Transfer up to 56 cm liquid helium in the dewar. 3. Set the set point of the charcoal pump to 25 K and control the 1 K pot below 2 K. 4. After condensing He3 for about 25 mins, start cooling down the He3 pot to the base temperature by turning off the heater and opening the flow meter of the charcoal pump. Close the needle valve of the 1 K pot. 122 APPENDIX B. EXPERIMENTAL PROCEDURES 5. We always start the first warming up when the helium level in the dewar hits 55 cm. Start the labview program to record the temperature, resistance and microwave data. Close the flow meter for the charcoal pump. Set the set point of the charcoal pump to ramp from 3.5 K to 35 K at a ramping rate 0.1 K/min. PID values for the heater are P = 30, I = 20, D = 0. Turn on the heater for the charcoal pump and the heater setting is medium. When the set point gets 6 K, we turn off both rough pumps to the 1 K pot and the charcoal pump and open the needle valve for the 1 K pot 1 turn open. To make sure the helium in the dewar has almost the same boil off rate, the helium level monitor measures the liquid helium level every 15 mins. 6. Stop the program for the first warming up measurement when the helium level becomes 50.5 cm. Usually the sample stage is at 4 K. First warming up in general takes a little more than 4 hours. 7. Prepare for the cooling down procedure by setting the set point of the charcoal pump to be 25 K. Turn on the pumps for the 1 K pot and the charcoal pump. When helium level gets to 50 cm, warm up the sample stage to 20 K. PID values for the heater are P = 30,1 = 100, D = 0. 10 mins after the sample stage reaches 20 K (to reach temperature stability), set the set point of the sample stage to ramp from 20 K to 2 K at a ramping rate 0.05 K/min. Heater range is set to be 2.5 Watt. Start the program to record data for the cooling down procedure. 123 APPENDIX B. EXPERIMENTAL PROCEDURES We keep charcoal pump at 25 K and 1 K pot between 2.35 K — 2.55 K (needle valve about 1/2 to 2/3 turn open) during the whole cooling down scan. 8. Stop the program for the cooling down measurement when the sample stage reaches below 2 K. Turn off-the heater for the sample stage. This measurement takes about 6 hours. Helium level usually reads at about 27 —27.8 cm. Prepare for the second warming up measurement. 9. The charcoal pump still stays at 25 K and the 1 K pot stays below 2 K. Condense He3 for about 30 mins (count from the time when the sample stage gets 2 K from the previous cooling down procedure). Turn off the heater of the charcoal pump and close the needle valve for 1 K pot. Cool down to base temperature by opening the flow meter to cool the charcoal pump. Start the second warming up measurement 52 mins after we start cooling down to the base temperature. Start the program to record data for the second warming up procedure. Close the flow meter and set the set point of the charcoal pump to ramp from 3.5 K to 35 K at a ramping rate 0.1 K/min. PID values for the heater are P = 30, I = 20, D = 0. Turn on the heater for the charcoal pump and the heater setting is medium. When the set point gets 6 K, we turn off the pumps and open the needle valve for the 1 K pot 1 turn open. The helium level monitor measures the liquid helium level every 15 mins. The helium level usually is between 24 — 25 cm. Leave the system to slowly warm up overnight. 124 APPENDIX B. EXPERIMENTAL PROCEDURES Day Six: 1. Stop the program when the sample stage reaches the desirable temperature (usually above 10 K). 2. remove the cryostat from the insert. Fill the IVC with gas nitrogen. Wait until temperature of the insert is close to room temperature. Vent the IVC and remove the cylinder. Remove the samples. Cover the dewar to prevent ice forming inside. One complete measurement cycle on one sample can be performed within a week. As one can see, we have more controls in the second warming up. Also, we do not need to transfer liquid helium for that measurements. As it turns out, the temperature of the He3 pot changes a little bit when we step on the frame where the insert is attached to. That is why we normally analyze data from the second warming up. However, as demonstrated in the main text, the difference from the three different temperature scans is minor. Since our insert is sitting in HV and has no direct contact with the liquid helium bath, the small deviation in helium level for the second warming up for different samples has insignificant effects. B.2 M easurements at finite fields We purchased the magnet through the Janis company and successfully installed it in the He3 cryostat. The dimension for the whole cryostat and the magnet can 125 APPENDIX B. EXPERIMENTAL PROCEDURES be found in Fig. D.l. The whole magnet has to be immersed in liquid helium to avoid possible magnetic quench which happened once when we were charging the magnet for a measurement on the low-disorder InO* film. We transfer enough liquid helium in the dewar to make sure the magnet is still covered by liquid helium after the temperature scan. All the helium transfers have to be done when the magnet is off so that there would not be any local heating effects to cause a possible quench. The IVC cylinder was made from copper before we installed the magnet. We changed it to a stainless steel one for measurements in magnetic fields since the eddy currents generated by a changing magnetic field in the copper would be too large. The sample stage is also made of copper. Therefore, when we scan field at fixed temperatures, to maintain temperature stability, the sweep rate cannot be too fast. The sweep rate in current for magneto-resistance measurements is 0.0079 A/S from 0 to 5 Amperes, 0.0158 A/s from 5 to 45 Amperes and then 0.0079 A/S again from 45 to 47.62 Amperes. The scan time is about an hour from 0 to 8 Tesla and the temperature is pretty stable during the whole scan. B .2.1 O perating sequence For measurements at finite magnetic fields, since the whole magnet has to be covered by liquid helium, we cannot perform the three measurement scans described in the previous section at one field. However, we know that as long as we can reproduce the temperature and control the time the system staying at the low temperature 126 APPENDIX B. EXPERIMENTAL PROCEDURES environment, errors due to different runs are negligible. A measurement cycle at one field takes about 12 hours. We generally take data at two magnetic fields per day. For the data we demonstrate in Chapter 4, it takes about a week and half to finish the measurements for one sample. Day 1 ~ Day 4: the same as the procedures for zero field measurements. Day Five: 1. Refill the liquid helium level to about 50 cm. Let the system stay at low temperature for extra a number of hours. After the initial filling the cryostat has to sit in cryogenic environment for at least 24 hours. 2. Prepare for measurements in magnetic fields after the system stays in the liquid helium environment for at least 24 hours. 3. Transfer liquid helium up to 55.2 ~ 56 cm. The amount of transferred liquid helium depends on the strength of the desired magnet field. The persistent heater stays on longer when we are charging the magnet to higher magnetic fields thus boils off more liquid helium. We normally transfer to 55.2 cm liquid helium for measurements at B = 0.1 Tesla and 56.2 cm for B = 8 Tesla. For B = 0 Tesla, we transfer up to 56.3 cm liquid helium since we need to oscillate the field to zero to get rid of any possible trapped flux. By doing so, we start our data acquisition typically at a helium level between 53 — 53.6 cm. 4. 20 mins after we finish transferring liquid helium, turn on the pumps and keep 127 APPENDIX B. EXPERIMENTAL PROCEDURES the charcoal pump at 25 K and the 1 K pot below 2 K (needle valve about half turn open). Condense He3 for 30 mins (count from the time when the sample stage gets 2 K). Open the needle valve a little more to warm up the 1 K pot to 2.2 K (make sure we have enough liquid He4 in the 1 K pot when we cool the system to the base temperature) and then close the needle valve. Turn off the heater and open the flow meter for the charcoal pump slightly over range. Cool the system to the base temperature of the cryostat. When the temperature sensor of the sample stage reads 400 mK, turn on the persistent heater and ramp up the field to the desirable value. Typical ramping rate is 0.02 A/S from 0 to 40 Amperes, 0.0158 A/S from 40 to 45 Amperes and then 0.0079 A/S from 45 to 47.62 Amperes. Turn off the persistent heater and put the magnet in persistent mode after the magnet reaches the set point. 5. 2.5 hours after we finish transferring liquid helium, start the program to record all the data. Close the flow meter and set the set point of the charcoal pump to ramp from 3.5 K to 35 k at a ramping rate 0.1 K/min. PID values for the heater are P = 30, I = 20, D = 0. Turn on the heater for the charcoal pump and the heater setting is medium. When the set point gets 6 K, we turn off the pumps and open the needle valve for 1 K pot 1 turn open. 6. Sample reaches about 6 K roughly after 6 hours. Stop the program. The helium level normally is at 40 cm. Ramp down the magnetic field. Proceed to the next 128 APPENDIX B. EXPERIMENTAL PROCEDURES measurement at another magnetic field or prepare to remove the cryostat from the insert. One thing I need to point out is that the magnetic field has very strong effects on the silicon diode temperature sensors. The silicon diode temperature sensor at the charcoal pump reads 3.5 K at 8 Tesla, while it reads 4.35 K at 0 Tesla. So the warming up procedure is in general shorter for measurements at high fields. However, this is the same for every sample since we calibrate the system at every single field. To avoid a sudden change in the output of the heater of the charcoal pump, we typically set the set point of the charcoal pump to be 3.5 K, turn on the heater and then start ramping the set point. This order, however, does not m atter for measurements at low fields since 3.5 K is normally much lower than the reading of the charcoal pump’s base temperature. 129 A ppendix C Reflection coefficients for zero-field measurements As we discussed in length in Chapter 2, the difference for the first warming up, cooling down and second warming up scans for zero field measurements are not that significant. In this appendix, I would like to present the raw S™ for a superconducting Nb film (Fig. C.l), a 20 nm NiCr film on Si (Fig. C.2), and a glass substrate (Fig. C.3). We can clearly see the Nb film becomes a superconductor at about 6 K in Fig. C.l. For the three graphs, one can again see the effects from the NbTi superconducting cable at around 8 — 9 K. The |5 n | for open and short standards have almost the same magnitude. We found that the differences are not significant at all for the three scans. 130 APPENDIX C. REFLECTION COEFFICIENTS FOR ZERO-FIELD MEASUREMENTS Nb film first warming up Nb film cooling down Nb film second warming up 0.42 0.40 0.38 0.36 -0.15 - 0.20 -0.25 0 5 10 Temperature (K) 15 20 Figure C.l: Magnitude and phase of of one Nb film as a function of temperature at 7 GHz. Different runs are indicated by the color legend. 131 APPENDIX C. REFLECTION COEFFICIENTS FOR ZERO-FIELD MEASUREMENTS 0.106 NiCr film first warming up NiCr film cooling down NiCr film second warming up 0.104 0.102 CO 0.100 0.098 0.40 0.35 0.30 0 5 10 Temperature (K) 15 20 Figure C.2: Magnitude and phase of S'™ of one NiCr film as a function of temperature at 7 GHz. Different runs are indicated by the color legend. 132 APPENDIX C. REFLECTION COEFFICIENTS FOR ZERO-FIELD MEASUREMENTS 0.42 G la ss first w arm ing up G la ss cooling dow n G la ss s e c o n d w arm ing up 0.40 CO 0.38 2.60 2.55 2.50 0 5 10 Temperature (K) 15 20 Figure C.3: Magnitude and phase of S'™ of a glass substrate as a function of temper ature at 7 GHz. Different runs are indicated by the color legend. 133 A ppendix D M agnetic field distributions The center of the magnetic field is carefully adjusted to be the exact location of the sample. Knowing the field distribution in the system is essential to estimate the effects of the magnetic field on the superconducting cable, all the microwave connections and also on the temperature sensors. We also notice that readings for both RuO* temperature sensors (one located on top of the He3 pot and another one located at the back of the sample stage close to the sample) also were suppressed by the increasing magnetic field. Knowing the field strength at the location of those temperature sensors enables us to correct the temperature reading under applied magnetic fields. Data of magnetic field dependence on the axial axis (Z direction) and the radius axis (R direction) from the center of the magnet were provided by Janis, the manu facturer company of our cryostat. We plot them in a fashion of contour plots. Fig. 134 APPENDIX D. MAGNETIC FIELD DISTRIBUTIONS D.2 is the contour plot of the B z in axial and radial directions. Fig. D.3 displays the contour plot of the B r . B z is much stronger than B R at locations that are close to the field center so that the magnitude of the total field is about the same as B z in that region as showed in Fig. D.4. We roughly estimated the distance of the two RuOx sensors from the magnet center. When B = 8 Tesla at the center, the field at the sensor closer to the sample is about 8 Tesla, and the field at the sensor on top of the He3 pot is about 3.6 Tesla. The temperature therefore can be corrected by a universal correction of temperature in a magnetic field [89] for RU600 series temperature sensors as: T(actual) = (1 + 0.009B) * T (B ). 135 (D.l) APPENDIX D. MAGNETIC FIELD DISTRIBUTIONS mi mbtm (2) HNH CURRENT LEADS 3K M R M M M I WUC MU CHARCOAL OOOUNt WL« it m tu e w K jL wwiowi * BJECHSCAL FEEOIHROUOt ( n » MAGNET) N W -2SFUN 0E 30' Hi ~ 300 P 9 PRESSURE GAUGE 1*1 IK P O T M OPERATOR port ‘“ [Sf XU U M VENT W / OUCNGH fCUEF PORT ON N W -40 FLANGE 4 PW OSCRBCAL FEED1HR0UO1 FOR HEUUN LEVEL SENSOR <2) UFTVM U M S HdJUM n a BELLOWS VALVE » / VCO ADAPTER msx "'wsarsL »«*«<*« TOP VIEW me wucm cam POT PUMPOUT HW-2 5 FLAM E 8.0 OIA. 10.0 OIA. 090 0 0 . 9 0 UL 1.5 (T IP .) •>7.79 OIA. 0,90 OD. 937 OH. IVC CAN ►a a it oil a n olo. o a (C») « / is o it] OW L m XO OA * U T AA1E IM t LEVEL PROBE QLmET CAflUARV k po t m « M POT JAMS RESEARCH CO. CONFIDENTIAL ANO PROPRIETARY H E -J-S S V /N A flN E T $na-i Figure D.l: The design of our cryostat with the magnet from Janis company. Cour tesy of Janis company. 136 APPENDIX D. MAGNETIC FIELD DISTRIBUTIONS 30 0.2 25 0.4 0.6 Axial (cm) 20 0.6 0.8 1.6 15 2.4 3.4 1 = 3 .6 10 4.4 3.8 5.4 E 6 .4 6.6 2.8 3.2 52 4.4 6.8 6.6 7.4 7.2 5 7.6 2.2 3.6 7.6 7.8 8.2 0 3 4 Radius (cm) Figure D.2: Contour plot of B z . 137 6.4 APPENDIX D. MAGNETIC FIELD DISTRIBUTIONS Axial (cm) 30 Radius (cm) Figure D.3: Contour plot of B R. 138 APPENDIX d . m a g n e t i c f i e l d d i s t r i b u t i o n s Axial (cm) M a g n itu d e = ®±=66 = 6.8 n Radius (cm) Figure D A Contour plot of the magnitude of the magnetic field. 139 A ppendix E Microwave conductance of InOx at finite fields In this appendix, I present almost all the data we took for the measurements at finite fields of the low-disorder InOx film. I show the complex conductance and the phase stiffness at B = 0, 1, 2, 3.5, 4, 5 and 6 Tesla. One can notice the suppression of the complex conductance with an increasing magnetic field at the lowest measured temperatures. The sharp universal jump feature in stiffness shown in Fig. E .l (d) also gets smeared for higher and higher magnetic fields. 140 APPENDIX E. MICROWAVE CONDUCTANCE OF INOx AT FINITE FIELDS o O 1 0 '1 E .c O 10’2 (D 10-3 10-4 8 12 <ii>/2jt (G H z) co/271 4 w/271 (G H z) 8 12 (G H z ) Figure E.l: Complex conductance and phase stiffness at B = 0 Tesla. Frequency dependence of (a) real and (b) imaginary conductance are shown in the ranges u;/27r = 0.05 - 16 GHz. The color legend for temperature for (a) and (b) is shown in (a). Temperature dependence of (c) real conductance and (d) phase stiffness respectively in the ranges T = 0.3 - 4 K. The color legend for frequency for (c) and (d) is shown in (d). 141 APPENDIX E. MICROWAVE CONDUCTANCE OF INO* AT FINITE FIELDS o)/2rc(GHz) to/2n(G H z) Figure E.2: Complex conductance and phase stiffness at B = 1 Tesla. Frequency dependence of (a) real and (b) imaginary conductance are shown in the ranges o;/27t = 0.05 - 16 GHz. The color legend for temperature for (a) and (b) is shown in (a). Temperature dependence of (c) real conductance and (d) phase stiffness respectively in the ranges T = 0.3 - 4 K. The color legend for frequency for (c) and (d) is shown in (d). 142 APPENDIX E. MICROWAVE CONDUCTANCE OF INO* AT FINITE FIELDS 1.0 2 .0 3.0 ' C2 _c T (K) S ■ 10-4 PP w 10 X 1 0 ‘3 O 1 * 2 0.5 4 1.0 6 8 10 12 14 16 <u/2ir (GHz) 1.5 2.0 T (K) 2.5 1 1 11 ll 6 8 co/2jt (GHz) 3.0 Figure E.3: Complex conductance and phase stiffness at B = 2 Tesla. Frequency dependence of (a) real and (b) imaginary conductance are shown in the ranges a;/27r = 0.05 - 16 GHz. The color legend for temperature for (a) and (b) is shown in (a). Real conductance and phase stiffness as a function of temperature are plotted respectively in (c) and (d). The color legend for frequency for (c) and (d) is shown in (d). 143 APPENDIX E. MICROWAVE CONDUCTANCE OF INO* AT FINITE FIELDS 0.51.0 1.5 2.0 2.5 3.0 T( K) 2 4 6 8 10 12 14 16 g>/2ji (GHz) 8 0.1 2 4 6 8 2 4 6 8 4 8 1 C0/27C(GHz) 0 10 12 co/2rt (GHz) ^10 0.5 1.0 1.5 2.0 2.5 3.0 T (K) Figure E.4: Complex conductance and phase stiffness at B = 3.5 Tesla. Frequency dependence of (a) real and (b) imaginary conductance are shown in the ranges w/27r = 0.08 -1 6 GHz. The color legend for temperature for (a) and (b) is shown in (a). Real conductance and phase stiffness as a function of temperature are plotted respectively in (c) and (d). The color legend for frequency for (c) and (d) is shown in (d). 144 APPENDIX E. MICROWAVE CONDUCTANCE OF INO* AT FINITE FIELDS _i 0.5 i i l t 1.0 1.5 2.0 2.5 -1 3.0 T (K) L_i i ii ii. in 1111 11 mi iin ii 1 1.0 2.0 3.0 4.0 T (K) Figure E.5: Complex conductance and phase stiffness at B = 4 Tesla. Frequency dependence of (a) real (G i) and (b) imaginary (G2) conductance respectively in the ranges u)/2n = 0.08 - 16 GHz. The color legend for temperature for (a) and (b) is shown in (a). Real conductance and phase stiffness as a function of temperature are plotted respectively in (c) and (d). The color legend for frequency for (c) and (d) is shown in (c). 145 APPENDIX E. MICROWAVE CONDUCTANCE OF INO* AT FINITE FIELDS 1.0 2.0 3.0 T(K) 2 4 6 8 10 12 14 16 4 6 8 2 4 6 1 (o/2n (GHz) <o/2 k (G H z) 4 2 8 12 co/2n (GHz) 0.5 1.0 1.5 2.0 2.5 3.0 T (K) Figure E.6: Complex conductance and phase stiffness at B = 5 Tesla. Frequency dependence of (a) real (Gi) and (b) imaginary (G 2 ) conductance respectively in the ranges uj/ 2 tx = 0.12 - 16 GHz. The color legend for temperature for (a) and (b) is shown in (a). Real conductance and phase stiffness as a function of temperature are plotted respectively in (c) and (d). The color legend for frequency for (c) and (d) is shown in (c). 146 APPENDIX E. MICROWAVE CONDUCTANCE OF INO* AT FINITE FIELDS 10'3 - 0.5 1.0 1.5 2.0 2.5 3.0 T (K) 0.5 1.0 1.5 2.0 2.5 3.0 T(K) Figure E.7: Complex conductance and phase stiffness at B = 6 Tesla. Frequency dependence of (a) real (Gi) and (b) imaginary (G2) conductance respectively in the ranges w/27r = 0.1 - 16 GHz. The color legend for temperature for (a) and (b) is shown in (a). Temperature dependence of (c) real conductance and (d) phase stiffness respectively in the ranges T = 0.3 - 3 K. The color legend for frequency for (c) and (d) is shown in (c). 147 Appendix F AC conductance of CVD grown graphene Graphene is a material consisting of a single atomic carbon layer arranged in a 2D honeycomb lattice, discovered in 1969 [90] and studied extensively since then [91] by the surface science community. It has attracted wide-spread interest for both its novel electronic properties and Dirac band dispersion as well as its broad application potential [91-98]. Due to its high mobility, it also has been proposed to show great promise for high speed switching in microwave and terahertz devices [99-101] and terahertz plasmon amplification [102]. 148 APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE F .l Electrodynamics of single layer graphene Recently it has been demonstrated that large-area monolayer graphene films can be grown by chemical vapor deposition (CVD) on copper foils [103], following the precipitation-based growth of somewhat non-uniform few-layer graphene films on Ni foils [104,105]. This method [106-109] allows the growth of large scale graphene films that can be transferred to various substrates, which will be essential for any practical device applications. The availability of large area uniform graphene also allows access to these materials from a greater variety of experimental techniques including studies of their long wavelength electromagnetic response. Their complex microwave and terahertz frequency dependent response are of particular interest and their understanding is crucial in order to use graphene for fast electronic devices. However, it has traditionally been difficult to get significant broadband spectral in formation in these frequency ranges, particularly in the microwave regime. Microwave experimental techniques are typically very narrow band and may at best allow the characterization of materials at only a few discrete frequencies. In this study, we make use of our newly-developed microwave “Corbino” spec trometer to measure the broadband microwave response in the frequency range from 100 MHz to 16 GHz of CVD grown graphene at temperatures down to 330 mK. This technique allows one to gain broadband spectral information in the microwave 149 APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE regime. Microwave techniques are typically very narrow band. We present data for both the sheet impedance and complex conductance. The measurement of the in trinsic impedance of a single atomic layer film on an insulating substrate presents a number of experimental challenges in the microwave regime. As a non-resonant tech nique, the Corbino spectrometer requires an intricate calibration procedure. More importantly, any attem pt to measure the intrinsic impedance of graphene will be af fected by capacitive coupling to its dielectric substrate. At microwave frequencies, the impedance associated with the substrate can be a substantial fraction of the graphene impedance. We detail the manner in which substrate effects may be calibrated for, as well as a number of other difficulties peculiar to this frequency range that must be overcome. We also compare our data to that we measure at higher frequencies using time-domain terahertz spectroscopy. Both measurements techniques are capable of measuring the complex optical response functions as a function of temperature and frequency, without resorting to Kramers-Kronig transforms. CVD grown Graphene was prepared by Yufeng at UT Austin. The samples were prepared by methane and hydrogen at pressures of 1.5 mbar over a 25 /xm thick Cu foil. The graphene films are coated with PMMA and then the Cu foils are dissolved in an aqueous solution of FeCl3 . The graphene is rinsed several times with de-ionized water and can then be scooped out of solution onto a 380 jum thick clean high resistivity Si substrate. High purity Si was used as it has a purely dielectric contribution to the impedance of the graphene-silicon multilayer. The sample is allowed to dry and 150 APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE 300 ^ 250 I 200 2D 150 ‘<5 jj c - 100 50 0 1500 2000 2500 3000 Raman shift (cm'1) 2400 — R d before an n ealing — Rc a fter an n ealing for 24 ho u rs ^ (/> E .c 2200 + R n after an n ealing for 24 h o u rs a t 260M Hz O 2000 □ CU 1800 0 50 100 150 200 250 300 T em perature (K) Figure F.l: (a) Raman spectra of graphene on Si. The Raman signal from a bare Si substrate has been subtracted. The Raman spectra were taken by Yufeng at UT Austin, (b) Resistance per square Ro for one of the CVD grown graphene samples as a function of temperature. The black crosses are the microwave data at 260 MHz. After the same sample was annealed at about 60 C in vacuum for about 24 hours, the temperature dependence of its resistance shifted as shown by the red curve. 151 APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE adhere to the Si and then the PMMA is removed by acetone. The resulting films are verified to be of high-quality, predominantly single layer graphene from the intensities and positions of the G- and 2D-band peaks in their 532 nm Raman spectra [110] (Fig. F .l (a)). To isolate the impedance of the graphene layer, we use Eq. 2.3 with Z§ub from our previous independent study of thin superconducting films on identical Si substrates [51]. In that study, we used amorphous metal films with scattering rates (« 100 THz) so high that the intrinsic AC impedance of the film itself in the normal state was purely real and could be deduced from the DC resistance exactly. Thus, Z§ub can be calculated by comparing the measured AC impedance of the metal film with its known value. Knowing Z fu6 one can isolate the intrinsic broadband impedance of the graphene layer. In addition to measurements at microwave frequencies, the Corbino spectrometer system can measure the two contact DC resistance simultaneously using a lock-in amplifier and a bias tee. Multiplying the measured resistance between the inner and outer conductors of the coaxial cable by the geometric factor g, we obtain resistance per square. As shown in Fig. F .l (b), the resistance per square as a function of tem perature is approximately temperature independent ( « 3% over the range) with only an upturn below 30K as the principal distinguishing feature. The sample properties are changed only slightly by annealing in vacuum for about 24 hours at 60 C degree. As shown in Fig. F.l(b) the resistance per square increased by 5 % with almost the 152 APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE | 4000 8 2000 JZ T = 296 K Without substrate correction With substrate correction 8 c (0 ■8 g -2000 </) E -C O 3000 2000 © O c (0 1000 S a. E 20 15 a O 3 5K 19 K 330 mK 15 K 296 K 10 . 0 2 4 6 8 10 12 14 16 Frequency (GHz) Figure F.2: (a) Calibrated impedance with (black) or without (red) correction for the substrate contribution at room temperature. Corrected (b) impedance and (c) conductance (normalized by G q = ire2/2h) as a function of frequency in the range of 100 MHz to 16 GHz at different temperatures. Different colors in both panels indicate different temperatures. In all the panels, solid lines are the real parts while dashed lines are for the imaginary parts. 153 APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE same temperature dependence. We ascribe the change in the overall scale due to a change in carrier density by driving off absorbed gases. In Fig. F.2, we present the results of our broadband microwave measurements on one particular graphene sample from 100 MHz to 16 GHz at temperatures down to 330 mK. The small oscillations are the residual effects of standing wave resonances in the transmission line that have been imperfectly removed by the calibration procedure. In Fig. F.2(a), we compare the effective impedance at the sample surface calculated from Eq. 2.2 with the impedance of the sample after the substrate correction described above. One can see that the correction becomes significant at higher frequency where the effective capacitance of the dielectric substrate plays a larger role. After correction the impedance becomes primarily real and frequency independent as expected for a conductor with a scattering rate larger than the measurement range. One can see that it is essential to perform such a correction to quantify the impedance correctly. In Fig. F.2 (b), we plot both real (Zi) and imaginary (Z2) sample impedance corrected for the substrate contribution as a function of frequency at 5 different temperatures. We can see that the frequency dependence of impedance at the base temperature of 330 mK and room temperature are almost the same. The vertical dif ference in those temperatures matches with the difference of measured DC resistivity. As clearly seen from Fig. F.2 (b), the real and imaginary parts of impedance have lit tle dependence on frequency down to low temperatures. This can also be seen in Fig. F .l (b), where the resistance at the low frequency of 260 MHz at 4 different tempera- 154 APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE tures is also plotted. These data follow the DC values closely indicating a consistency of AC and DC measurements. Also, the real and imaginary parts of conductance have little dependence on temperature which means that the Drude response of electrons in graphene does not change a lot over this wide range of temperatures (from room temperature to 330 mK). It also indicates that the scattering rate r - the average time between scattering events - bears little dependence in temperature. 0.5 0.4 T=15 K 0.3 ^ 0 0.2 - 0.1 8x10 Frequency (Hz) Figure F.3: Ratio (red) of real and imaginary parts of conductance as a function of frequency in the range from 700 MHz to 9 GHz at 15 K before annealing. The black curve is a linear fit with zero y axis intercept. In Fig. F.2 (c), we plot the complex conductance obtained by using the inverse of the data Fig. F.2 (b). Since in the thin film limit, the complex conductance is the reciprocal of complex impedance, it also has almost no dependence on frequency. Here we have ratioed this data to the quantum of conductance 155 Gq = ire2/ 2 h — 1/ APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE 16433 Ohms-1 expected for a graphene sample with its chemical potential tuned to the Dirac point. Its large dimensionless scale shows that the carrier density for this sample is high with the chemical potential far from the Dirac point. Although the imaginary part of the conductivity is very small in Fig. 2, it is not zero, which gives a measurable r from the data. Within the Drude model, the complex conductivity from a charge responding to a time varying external oscillating electromagnetic field with frequency u> is o(u) — where n is the electron density. Inspection of this equation shows that the ratio of the imaginary to real parts of 0 2 / ( 7 1 gives w r a s a function of c j such that r can be determined from its slope. In Fig. F.3, we plot this ratio for the conductances (G = od) as a function of frequency at 15 K. G 2 /G 1 at other temperatures give similar results as both G2 and G\ have little dependence on temperature. The slope of G2 /G 1 gives us an estimate of the relaxation time, which is about 25.7 ps. A rough estimation of scattering rate is then r = l / r = 38.9 GHz. This is quite close to the value for the Drude scattering rate of 36.4 GHz obtained independently through fitting our data using a Drude-Lorentz model [84]. Using this extracted scattering rate, we can estimate the mean free path I = Vp * t = 28.3 fan with Fermi velocity vp = 1.1 * 106m /s [111]. However, we should note that this small value of the scattering rate is interesting as it is at odds with that inferred from previous studies using higher frequency time-domain terahertz spectroscopy [112] or far-infrared reflectivity measurements [113]. The small value of the scattering rate is also different from that inferred from our 156 APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE 6K 7K 8K 9K 10 K 15 K 20K 25K 90K 100K 150K 200K O o 0 -10 0.2 0.6 0.4 0.8 1.0 Frequency (THz) Figure F.4: Real (solid) and imaginary (dashed) parts of normalized conductance as a function of frequency in the range from 150 GHz to 1.0 THz at different temperatures for a similar graphene sample. own time-domain terahertz spectroscopy (TDTS) measurements on another similarly prepared sample. In TDTS an ultrafast laser pulse excites a semiconductor switch, which generates an almost single cycle pulse with frequencies in the terahertz range. The transmitted terahertz pulse’s electric field is mapped out as a function of time. The ratio of the Fourier transform of the transmission through the sample to that of a reference (usually the substrate on which the sample is deposited) gives the complex transmission function of the film under study. This can be inverted using standard formulas in the ‘thin film approximation’ T(u) = Y+n+z^ilJjd6^'' t° Se^ the complex conductivity. Here is the phase accumulated from the small difference in thickness between the sample and reference substrates and n is the substrate index 157 APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE of refraction. We have measured in terahertz frequency range from 150 GHz to 1.0 THz at temperatures down to 6K with flowing He4 gas. It is possible that the He4 gas environment for graphene sample in TDTS might make a slight difference for its scattering rate compared with the Corbino system where the sample is sealed in high vacuum. In Fig. F.4, we show terahertz complex conductance data for a sample from a different batch. This sample has a higher carrier concentration as evidenced by its larger conductance. The sample also shows little dependence on frequency of the real part of the conductivity and a small imaginary conductivity in this range. As the conductance is expected to drop off dramatically around the frequency of the scattering rate within the Drude model, this sample has a scattering rate greater than 1.0 THz. This is consistent with previous work [112,113] on CVD grown graphene and on few-layer epitaxial graphene [114]. F.2 Conclusion The two different scattering rates found in our microwave and TDTS measure ments show either the limitations of the Drude model for describing the fine details of electron transport in graphene at low frequencies, or alternatively the inherent difficulties of extracting out the precise complex impedance of graphene. In the first case it may be that the details of scattering Dirac electrons give a conductivity line- 158 APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE shape that is not precisely Lorentzian. Then the slope of G 2 /G 1 cannot be taken as a measure of r. 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McEuen, E. D. Minot, and Y.-S. Lee, “Terahertz imaging and spectroscopy of large-area single-layer graphene,” Opt. Express, vol. 19, no. 1, pp. 141-146, Jan 2011. [113] J. Horng, C. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. Bechtel, M. Mar tin, A. Zettl, M. Crommie et al., “Drude conductivity of dirac fermions in graphene,” Phys. Rev. B, vol. 83, no. 16, p. 165113, 2011. [114] H. Choi, F. Borondics, D. Siegel, S. Zhou, M. Martin, A. Lanzara, and R. Kaindl, “Broadband electromagnetic response and ultrafast dynamics of fewlayer epitaxial graphene,” Appl. Phys. Lett., vol. 94, p. 172102, 2009. 177 Vita Wei Liu was born in Le’an, a small town in Jiangxi Province in China. After high school, she moved to Beijing for her undergraduate and master degrees. She attended Beijing Normal University, receiving a B.S degree in Physics. She then enrolled in the master program in physics at Peking university, working on Monte Carlo simulation of lattice QCD and tried to understand the properties of elementary particles. She moved across the world to Baltimore to enroll in the Ph.D. program in physics at Johns Hopkins University. Her research has mostly focused on applying a homebuilt microwave spectrometer to study the thermal phase fluctuations and the twodimensional superconductor-insulator quantum phase transition in disordered InO* films. 178

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