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Coherent tunneling of lithium defect pairs in KCl crystals.

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Ann. Physik 6 (1997) 263-286
Annalen
der Physik
0 Johann Ambrosius Barth 1997
Coherent tunneling of lithium defect pairs
in KCl crystals
R. Weis ', C. Enss, A. Wiirger2, and F. Luty3
I
Institut fur Angewandte Physik, Universitat Heidelberg, Albert-Ueberle-Str. 3-5,
D-69120 Heidelberg, Germany
Institut Laue-Langevin, B. P. 156, 38042 Grenoble Cedex 9, France
' Department of Physics, University of Utah, Salt Lake City, USA
Received 8 August 1996, final version 27 January 1997, accepted 27 January 1997
Abstract. The tunneling of two lithium ion impurities on next-nearest neighbor sites in potassium
chloride are investigated both experimentally and theoretically. The strong dipolar interaction leads
to coherent tunneling motion of the two defect ions between degenerate off-center positions. Comparing data of rotary echo experiments for impurity pairs 'LG7Li, 6Li-6Li, and 7Li-6Li with theory
permits a thorough investigation of the isotope effect and of the effect of the interaction on the tunnel states. Our findings confirm the tunneling model with (1 11) off-center states to be valid even
for strongly interacting impurities. Using degenerate perturbation theory in terms of two-particle
states, we obtain essentially exact expressions for the tunneling spectnim and the dynamical susceptibility which agree well with the measured data.
Keywords: Tunneling states; Rotary echoes; Isotope effect
1 Introduction
Certain substitutional defects in alkali halides provide model systems for the study of
quantum tunneling of ions or molecules in a crystalline environment. As examples
we note Li+, CN-, and OH- in potassium chloride and similar crystals. Due to their
misfit in size or shape such defect ions are confined to a few equivalent positions in
degenerate potential wells of the host crystals. At low temperature thermally activated barrier crossing is inhibited, and the particle rather passes through the barrier
by quantum mechanical tunneling. For the mentioned examples the tunnel splitting
takes values of about 1 Kelvin.
In the extremely dilute case the defects may be considered as isolated from each
other, and the tunneling frequency provides the only energy scale. With rising concentration, there is a small but finite probability to find nearby pairs of defects whose
electrostatic or elastic interaction results in coherent tunneling of the two atoms. At
still higher concentration each defect ion interacts with many neighbors, and one
faces a complicated many-body problem.
As to the single-impurity case, lithium-doped potassium chloride has attracted
much attention, both experimentally and theoretically [ 1-41. By comparing various
measured data with calculated tunneling spectra, it was shown that a lithium ion oc-
264
Ann. Physik 6 (1997)
cupies eight off-center positions in the (1 11) crystal directions, resulting in a fourlevel energy spectrum with degeneracies 1:3:3: 1 [4].
Experiments on samples with a defect concentration of about 100 ppm showed deviations with respect to the dielectric [5-81 and thermostatic [9, 101 behavior which
clearly indicated the importance of interaction. Including interaction between nearby
defects was considered to be too complicated to be treated as such; thus the actual
problem of two ions disposing of eight states each was replaced by a pair of twostate systems [11-131. Much work was devoted to both static and dynamic properties
of this so-called pair model [14-161.
From experiment it was clear that at higher concentration a more thorough treatment was required, and that interaction of many defects would give rise to collective
relaxation phenomena [5]. Only recently some progress was made in this direction
and permitted understanding of the defect motion at higher concentration [8, 171.
In this paper we consider lithium defects in potassium chloride at moderate concentrations between 60 and 70 ppm. In this range most defects are only weakly affected by interaction; a few of them, however, are involved in strongly coupled pairs.
The present work deals with next-nearest neighbors ("2)
whose distance vector lies
along a (100) crystal direction.
In Section 2 we present a theory for such pairs which does not rely on the twostate approximation for a single impurity ion. For sufficiently close defects the interaction is much larger than the tunneling energy; thus the ground state splitting can be
calculated in high accuracy by degenerate perturbation theory. In our experiments we
investigate rotary echoes which arise from a suficiently strong time-dependent electric field. In Section 3 we give details of the experimental technique, and in Section 4
we present the measured data and compare them with theory. Finally we summarize
our results in Section 5.
2 Theory
2.1 A single tunneling defect
Due to its smaller ionic radius and its interaction with the surrounding in KCI, the total system finds it energetically more favorable, if the Li+ stays in off-center positions, with respect to the lattice site of the potassium which it substitutes for. The fcc
symmetry of the host crystal imposes eight equivalent positions lying along the
(1 11) symmetry directions,
with ai= fl.These eight positions form a cube of side length d 1.4 A,as is schematically illustrated in Fig. 1. The tunneling motion between these states has been studied in detail by Gomez et al. [4]. According to these authors there are three different
amplitudes for tunneling along the edges of the cube, along a face diagonal, and along
a space diagonal. For our purpose, only the former is relevant, the latter being smaller
by approximately a factor 4 and 50, respectively [18]. Denoting the quantum state localized at position (1) by laxaya,),and the tunneling amplitude by +Ao, we have
265
R. Weis et al., Coherent tunneling of lithium defect pairs in KC1 crystals
cI0 K+
0
Fig. 1 The elementary fcc cell of
KCI, in which the center K-ion is
replaced by a Li-ion. Ionic radii are
reduced by a factor 1/5 for visibility. The arrow indicates the direction
of the dipole moment
'
Li+
4
6.3A
where we have used standard basis operators and the notation 6 = --a. (The localized states IaPy) are assumed normalized and perpendicular on each other.)
The energy eigenstates of (2) are irreducible representations of the point group o h ;
they are constructed as superpositions of the localized states,
with appropriate phase factors
C, T
0.
~
) ,take the values f1; thus we have fl = - 1
The entries of the vector cr= ( c T ~
for a = 1 = D andf,O = 1 else.
The eight states (3) are arranged in four levels; indicating both the corresponding
representation of o h and the label 6,and using = - 1, we have
3
E(A2u)= ~ A :o (111)
1
: ( I l l ) , ( I T I ) , (111)
2
1
E(T~,)= - - A ~ : (iii),(Tii),( i i i )
E(Tzg)=
-A0
2
3
E(Alg)= --bo : (iii).
2
(5)
266
Ann. Physik 6 (1997)
2.2 Tunneling of a coupled defect pair
2.2.1 Dipolar interaction
Because of its net charge q, the tunneling lithium ion in KCl carries a bare dipole
moment
(6)
P = ’4
with an absolute value p = (&/2)qd. From paraelectric resonance measurements
p = 2.63 Debye has been determined [18].
The random configuration of the lithium ions on the host lattice leads to a finite
probability for the occurrence of pairs of nearby impurities. Their dipolar interaction
adds a potential term to their tunneling energy,
(7)
H = T+T’+ W = K + W ,
where T’ has the same structure as T ; in order to account for an isotope effect, we
permit for a different tunneling amplitude Ah,
Considering two impurities with dipole moments p and p’ and distance vector R, the
interaction potential W reads
The coupling energy is largest for nearby neighbors; at very small distance the discrete values of the vector R lead to well separated energy levels.
NN1
“3
“2
Fig. 2 Classical ground state
configurations for different
pairs of Li+ impurities “1,
“2, and ”3. The dipole
orientations are projected onto
the (001)-plane. The squares
indicate the space accessible
to the Li+-ions in KCl. The
positive and negative z-component of the dipole moments
are indicated by filled and unfilled arrows, respectively.
Further details are explained
in the text.
R. Weis et al., Coherent tunneling of lithium defect pairs in KCI crystals
267
Fig. 2 shows the classical ground state configurations with respect to the dipolar
energy (9) for three particular defect configurations; in terms of the lattice constant a
their distance is given by m
a (NNl), a (NN2), and m
a ("3);
the number
of degenerate states is 4, 8, and 2, respectively. (Note that there are four potassium
sites per unit cell.) Dynamic experiments involve dipole transitions between energy
eigenstates. Recently Weis et al. [19] have pointed out some important selection rules
for tunneling between the states shown in Fig. 2. Before undertaking a rigorous calculation, we qualitatively discuss the resulting ground state tunnel splitting. Note that
for the discussion of the selection rules the applied electrical field is assumed to be
parallel to x in Fig. 2.
The tunneling operators T and T' account for single moves along the crystal directions x,y , or z. From Fig. 2 we deduce that passing from one configuration NN1
to another requires two moves of each defect. The intemediate states are separated from
the ground state level by energies of the order of magnitude of the dipolar interaction.
(Tunneling in z-direction, i.e. perpendicular to the plane, is not taken into account, since
the corresponding dipolar transitions are forbidden for symmetry reasons.) As to the
configurations "2,
the transition from one state to another takes one move of each
defect; for those of configuration NN3 three moves of each are necessary.
A perturbation expansion with respect to K involves powers of A o / J , where J denotes the energy scale of the dipolar interaction (9). The amplitude for dipolar transitions between degenerate states in turn determines the energy scale of the tunnelin
spectrum; the above reasoning ields a ground state splitting of the order of A i / J
for "1,
A i / J for "2,
and A (7
0 / J 5 for "3.
For KC1:Li we have Ao/J M &j for all
three cases and thus configurations NN1 and NN3 provide much smaller tunneling
energies than "2.
We should note that the spectra for NN1 and NN3 are probably determined by the matrix elements for face or space diagonal tunneling, which have been
neglected in the present work. Again, since these tunneling amplitudes are much smaller
than that retained in (2), the resulting ground state splitting is small compared to A i / J .
In view of experimental restrictions on both frequency and temperature for the
measurements presented in Section 3, the most interesting case arises from defect
pairs with the configuration "2.
Choosing the distance vector proportional to the
[ 1 001-direction, R = Re,, the corresponding dipolar energy reads
!?
w=--1
4
1 (-2PxPL
~ R3~ 8
+PYP; + P X )
The sequel of this paper is confined to this particular case describing the configuration "2.
The cases NN1 and NN3 will be treated elsewhere [20]. The perturbation
analysis given below is valid for any pair along a [loo] direction with sufficiently
small distance in order to assure A0 << J . When discussing our rotary echo experiments we will consider two impurities on next-nearest neighbor sites.
Each ion disposes of eight different positions; hence there are 64 product states for
the coupled pair,
Greek letters taking the values f l ; T acts on the first triplet of variables, and T' on the
second one. With p = &qd(cu,P,y ) and p' = Iqd(a',$, y') it is clear that the dipolar
potential is diagonal in these states. Noting lp\ = Ip'J = ( 4 / 2 ) q d and defining
268
Ann, Physik 6 (1997)
1 q2d2
I=--4 7 1 ~ 0R3
~
we find the 64 states to be distributed over 5 levels,
whose degeneracies read 8: 16:16:16%.
2.2.2 Ground state splitting
For a pair of nearby impurities the dipolar energy J is by several orders of magnitude larger than the tunneling amplitude Ao. Thus the kinetic energy will merely
cause a tiny splitting of the degenerate levels given in (13). Only the ground state
multiplet is of experimental interest; for notational convenience we put
The Hamiltonian (7) displays a lower symmetry than the cubic one of a single impurity, (2). Besides the fourfold symmetry about the x-axis, H is invariant under the inversion operation r -+ -r and r'
-r'. The corresponding point group is given by
the direct product of C4 and inversion Ci [21],
--f
C4h = c
4
x
ci,
(15)
which has four one-dimensional representations and two two-dimensional ones. Their
behavior under rotations about the x-axis and inversion is shown in Table 1.
The energy eigenstates corresponding to these representations are formally identical to those of a single defect, (3), and may be labelled by the same quantum numbers 6,
Table 1 Character table of C4h
R. Weis et al., Coherent tunneling of lithium defect pairs in KC1 crystals
269
(Using T = -1, we have indicated the o-label of the corresponding states in Table 1).
The spectrum, however, will be different from that shown in (5); because of the lower symmetry we expect a partial removal of the degeneracy.
Diagonalization of the Hamiltonian is achieved by solving the eigenvalue equation
det(E - H ) = 0.
(17)
Here we evaluate the ground state level splitting by means of second-order perturbation theory with respect to the tunneling part K ; we recall that W is diagonal and K
small. Degenerate perturbation theory may be quite a tedious matter. The present
case, however, is greatly simplified by the fact that the Hamiltonian is diagonal in
the ground states (14); hence the perturbation analysis can be done in these states.
With the projections
and Q = 1 - P, we find for the unperturbed ground-state energy
Eo = P W P = -J.
(19)
There is no first-order term, since PKP = 0; the second-order term reads
Eqs. (18-20) provide the second-order approximation with respect to K for (17),
(
det E
- EO-
) = 0.
(21)
The small parameter is given by the ratio A o / J , as is obvious from (2, 12, 20).
in the basis IGapy). It turns out
Now we calculate the matrix elements of
convenient to split H ( ~in) two parts,
where the first one contains the diagonal terms and the second one the remainder.
The particular structure of the tunneling Hamiltonian and of the ground states results
in the simple relations
270
Ann. Physik 6 (1997)
The first term describes tunneling of one particle from one position to another and
back to the original one. The non-diagonal part implies one tunneling movement of
each impurity; from (14) it is clear that H i ) is finite only between states differing in
one label, i.e. that both particles move alon the same axis.
We begin with the non-diagonal part Ifnd.First consider the case where the tunneling motion occurs in x-direction, corresponding to a change of the label a in (14).
Inserting (14) and (24) we find
6)
the factor 2 arises from the two channels in (24), which pass both by an intermediate
state with energy E2.
Transitions involving the second label correspond in real space to one defect moving in positive, the other in negative y-direction. Proceeding as above we find
note that these transitions involve intermediate states with energy El. Because of the
symmetry about the x-axis the same result holds for tunneling along the z-axis. Matrix elements involving more than one label vanish in the present second-order approximation with respect to K .
Now we turn to the diagonal contribution (23). Since initial and final states are
identical, the three terms for tunneling along the different axes have to be added. Proceeding as above we find
Putting together the above results and using (1 3) and the definitions
the Hamiltonian matrix reads
where, as usual, Greek letters take the values 4 ~ 1and the bar changes the sign,
a: = -a.
could be calculated by diagNow, in principle, eigenvalues and eigenstates of
onalization of the 8-dimensional matrix (27). From the above symmetry considerations, however, we already know the eigenstates (16). We merely have to apply (27)
27 1
R. Weis et al., Coherent tunneling of lithium defect pairs in KCl crystals
in order to find the tunneling levels. Noting that the phase factorsf:
-1 for ai = I = ai and 1 else, one easily finds the spectrum
E(B,)= Eo-;qo+;y
E(B,) = EO- i q o + i q
E(E,) =
E(E,) =
E(A,) =
E(A,) =
:
:
take the values
(111)
(ill)
E ~ - ; Y ~ + :; ~ ( i T i ) , ( i i T )
E~ - ;qo - ;q :
(TIT), (iii)
~
~2 V o -- z y I 5
E~ - ; T o - ; q
.
(iii)
:
(ITT);
we have indicated both symmetry and
written as
(T
labels. In terms of the latter, (28) may be
(Note that for A0 = A; we have yo = q.)
2.2.3 Finite asymmetry energy
In the previous section we have assumed the eight off-center positions of each impurity ion to be degenerate. This degeneracy may be partially removed, however, by
elastic strain or by residual interactions with other impurities. We thus add a one-particle potential V = v(r) v’(r’) to the Hamiltonian,
+
H=K+V+W.
(30)
For sufficiently distant sources, V may be expanded in terms of the position operators, r and r’, and the potential gradients will be identical at both impurity sites,
-F = Vv = V’v’; in linear order we have
Obviously V is diagonal in the states (14); moreover one finds that only the x-component yields a finite bias
LA
2 = F,d.
(32)
After inserting (1) and (32), Eq. (31) reads in terms of the coordinates
(GaPyl V IGKA/-h)
1
=
5 MCtKJpaJyp
(6K1
- ‘rci
)’
(33)
(Note that the asymmetry energy of each impurity is equal to A/2; the quantity A is
the sum of both terms.)
The potential (33) breaks the inversion symmetry; accordingly a, as defined in (3)
is no longer a good quantum number. For A << J , we may treat V K as a small
+
272
Ann. Physik 6 (1997)
perturbation. Note that the perturbation analysis may still be done in the basis (14),
since V is diagonal with respect to these states. With PVQ = 0 and
the second-order approximation for the eigenvalue equation (17) now reads
+
In order to diagonalize H(’)
it is convenient to define new states
+
For the effective Hamiltonian H = EO H(’)
elements
+
we find the diagonal matrix
and the off-diagonal ones
(The states (36) have been chosen such that
3 is diagonal with respect to the labels
aY and 02.)The resulting eigenvalue equation involves diagonalization of a 2 x 2-matrix; different values for the quantum numbers oy and
shift of the energies.
Diagonalizing (37, 38) yields both the energy levels
and the corresponding eigenstates
where the coefficients are given by
(T,
lead merely to an overall
273
R. Weis et al., Coherent tunneling of lithium defect pairs in KCl crystals
Pi=
/*.
Inserting (36) finally yields the eigenstates in terms of the basis (14),
ThL energy (39) resembles very much (29), the only difference involving th_e teim
arising from tunneling along the x-axis. In the limit A + 0 the coefficients f," tend
towards fl as given by Eq. (4). In the opposite case q / A + 0 we find f$ + fi
and J" t 0, which corresponds to states where the two impurities are localized at
either a, = 1 = a: or a, = -1 = a:. Fig. 3 shows the energy spectrum according
to (28) for Ah = A0 and for the two cases A = 0 and A # 0. The arrows indicate the
allowed dipolar transitions for an electrical field in (100)-direction.
2.2.4 Response function
From the energy spectrum calculated in the previous section one easily obtains the
statistical operator
with the partition function Z = Ccn-Ea/'cST
at temperature T . Static properties as
the specific heat are determined by the internal energy U = (I?),
where (...) denotes the thermal average tr(p...) and where the system is confined to
the ground state multiplet (Fig. 3 ) . Dynamical quantities are described by the timedependent response function with respect to an external field,
Fig. 3 Tunneling level scheme of a pair of Li+...........
E 41
.............
.....
A.-'
ions in next nearest neighbor position for both
"
cases A = 0 and A # 0. The arrows indicate allowed transitions for electrical fields in (100)-di- EQ-57 ............. n,i,ii ...........
......
rection.
.......
.......
274
Ann. Physik 6 (1997)
x ( t - t’) = (i/fi>([P(t)>P(01),
with t
(451
2 t’ and the total dipole moment of the two impurities
P = p + p’.
(46)
We start by calculating the matrix elements of P in the basis (14),
P is diagonal, and its only non-zero component is along the x-axis. Now it is straightforward to derive the matrix elements with respect to the energy eigenstates (42),
as an important selection rule we note that dipolar transitions occur only between
states with the same labels oy and o,, i.e. only the x-component of the external field
induces transitions.
With (40, 41) and a;, = o; we find after some algebra the diagonal elements
Inserting (49) in (48) yields the static dipole moment
which vanishes for zero asymmetry energy and takes opposite values in the 8 states
(T,
= & I . (50) leads to a constant term in the correlation function (P(t)P(t’));
yet it
does not contribute to the response function (45).
we obtain with
Putting d, = -0, = i?x
the amplitudes for non-diagonal dipole transitions
With (50, 52) the magnitude of the total dipole moment is given by \PI = qd.
Since the energy difference for states connected by dipolar transition is given by
we may perform first the trace over the variables oy and 0, in
the constant
m,
R. Weis et al., Coherent tunneling of lithium defect pairs in KCl crystals
275
(45). Thus calculation of the response function reduces to a two-level problem with
the states a, = *I. Inserting
p ( t )= e i H t / h p e - i H t / h
and the spectral representation
in (45), and using the orthogonality relation ( ~ ( d
= bad,
)
we obtain with (48, 51) and
E=
dv2+ A2
(53)
the response function
~ ( z=
) 2 ( q d )2 -tanh
v2
E2
(2>c,:
-
sin
6)
.
(54)
2.2.5 Rabi frequency
Rotary echo experiments cannot be described in linear-response or first Born approximation with respect to the external field Fo exp(iwt). They rather require to consider a
non-linear response function, which in general is quite an involved object. One of the
rare tractable situations arises from a two-level system in resonance with the external
field, E = hco, whose energy is much larger than the coupling energy, E >> IP . Fo(.
Starting from a 64-dimensional Hilbert space, our description of an impurity pair
reduced to an effective eight-dimensional problem whose solution, in turn, is identical
to that of an asymmetric two-level system with dynamic dipole moment (52) and energy splitting E; the former determines the prefactor of the response function (54),
and the latter temperature and time dependence. These quantities yield both the resonance condition E = hw and the Rabi frequency
which are essential for the discussion of our experiments.
3 Experimental technique
3.1 Samples
The three samples used in our experiments were KC1 single crystals containing between 60 and 70 ppm of Li [22].After pretreating the starting materials to reduce the
OH concentration to typically less than 0.2 ppm the samples were grown from the
melt at the Crystal Growth Laboratory of the University of Utah in Salt Lake City.
276
Ann. Physik 6 (1997)
The main difference between these samples are their Li isotope composition, The one
which is labeled 60 ppm 7Li was doped with the natural isotope mixture containing
7.4 % 6Li and the sample labeled 70 ppm 6Li contains over 95 % 6Li. In contrast to
these nearly isotopically pure samples the crystal with 66 ppm Li was doped with a
5050 mixture of both isotopes.
We have cleaved the crystals along the (100) direction to obtain flat disks about
2 mm thick and 15 mm in diameter, necessary for our echo experiments.
3.2 Cavity
The echo experiments were performed using reentrant microwave cavities with
ground mode frequencies between 0.7 and 1.2 GHz screwed to the mixing chamber
of a dilution refrigerator. The samples were placed between the center post and the
bottom of the cavity, since in this region a large portion of the electric field energy
of the cavity is concentrated. Because of the diameter of the center post (15 mm), the
field is rather homogeneous.
Using a loop at the end of semi-rigid coaxial cable the r.f. signal is coupled into
the cavity. This loop can be rotated in order to adjust the degree of coupling, To allow experiments with high time resolution the coupling was adjusted to an overcritical coupling keeping the Q-factor of the cavity low ( Q x 1.50).
An important point in the analysis of our data is the knowledge of the absolute
field amplitude in the sample. Unfortunately the determination of the absolute field
amplitude is not a trivial matter. However, a special effort has been made to determine the field amplitude within 10%.For a detailed description of this procedure we
refer to Ref. [231.
3.3 Echo generation
The generation of a rotary echo is schematically illustrated in Fig. 4. At low temperatures (kgT < hco,) the majority of the resonant tunneling systems are in the ground
state. An rf-field with the frequency o / 2 n induces transitions between the two levels
if the resonance condition is fulfilled. The periodic change of the occupation number
of the two level is accompanied by an harmonic oscillation of the macroscopic electrical polarization with the effective Rabi frequency
R = $22,
+ co;.
Here md = o - w,. denotes the difference between the frequency of the applied field
and the resonance frequency of the tunneling systems. In our experiments cod can at
most vary within the bandwidth of the cavity.
If tunneling systems with different Rabi frequencies contribute to the signal, the resulting macroscopic polarization vanishes rapidly because of superposition of the individual contributions. In many cases it is impossible to detect the polarization of the
sample at the beginning of the pulse sequence because the amplifiers are saturated
for a few ps after switching on the r.f. signal. But since a phase inversion of the driving field at t = tp is equivalent to a reversal of time evolution of the polarization, the
277
R. Weis et al., Coherent tunneling of lithium defect pairs in KCI crystals
b)
Fig. 4 Formation of a dielectric rotary echo.
a) Phase of the applied r.f. field b) Time evolution of the occupation number difference of
the two levels of an ensemble of identical
tunneling states. c) Time evolution of the
corresponding electrical polarization.
d) Superposition of the polarization of five
tunneling systems with slightly different Rabi
frequencies. e) Time evolution of the resulting macroscopic polarization of tunneling
systems with a Lorentzian distribution of
Rabi frequencies.
PI
I
c)
I
I
I
I
I
4
c pi
e)
'
A
I V
0
I
I
I
I
^
"
_
I
l
I
-
I
tP
Time --+
Rabi oscillations will appear again around t = 2t,. This phenomenon is called the rotary echo.
3.4 Phase-sensitive detection
An overall schematic of the electronic circuitry necessary to generate and detect dielectric echoes is shown in Fig. 5.
The phase-sensitive detection of dielectric echoes requires an r.f. generator with
high frequency stability. We therefore used a microwave synthesizer (Rhode &
Schwarz SMH) to provide a continuous r.f. signal. A power divider splits up this signal for the receiver and the transmitter branches. The amplitude of the r.f. signal is
adjustable by a variable attenuator and pulses with a typical length of some microseconds are produced by a p-i-n-diode switch with an on/off transit time of about 15 ns.
After the diode switch, the signal was split again, and a delay corresponding to a
phase shift of 180" was introduced into one branch. By switching from one to the
;zsy
-$(
I
Fig. 5 Electronic circuitry for the
generation and phase-sensitive detection of rotary echoes.
hybrid coupler
p i n diode
27 8
Ann. Physik 6 (1997)
other position of the following double throw p-i-n diode switch, the signal could be
phase inverted within less than 1 ps. Both p-i-n diode switches are controlled by an
eight channel digital pulse and delay generator with a time resolution of 5 ps (Stanford Research, DG 535).
After the double throw p-i-n diode switch the r.f. signal is split again and one part
is fed through a circulator into the cavity inside the cryostat. The other part can be
adjusted in phase and amplitude to cancel the reflected signal from the cavity resonator. Therefore only contributions to the signal originating from nonlinear phenomena
will appear at the preamplifier (Aertech, Gain 20 dB) in the receiver branch.
A local oscillator (Marconi Instruments 6070 A cavity oscillator) provides a reference signal for two mixers. By mixing the output of the local oscillator with the signal from the cavity mixer 1 produces a 60MHz-signal for an intermediate frequency
amplifier (RHG EST6010). Mixer 2 generates a 60MHz phase reference by mixing
the synthesizer output and the signal of the local oscillator. Finally, mixer3 works as
a phase-sensitive demodulator by mixing the output of the IF amplifier with the
phase reference.
Mixer3 is followed by a 500MHz digital oscilloscope (HP 54520A) which adds
up subsequent echo cycles and thus performs signal averaging. Between 1000 and
10000 averaging cycles were necessary to reach an acceptable signal to noise ratio in
our experiments.
4 Experimental results
4.1 KC1 doped with 70 ppm 6Li
The main purpose of these experiments was to measure the distribution of the Rabi
frequencies and to determine the tunnel splitting of impurity pairs. Since the energy
of the contributing tunneling states E = hco,. is fixed within the bandwidth of the microwave cavity and the electrical field amplitude Fo is set by the external field, a distribution of the effective Rabi frequency f
2 may arise only from a distribution of the
tunnel splittings y or of the dipole moment P.
For frequency of about 1 GHz only next-nearest neighbor pairs ("2)
are to be
seen in our experiment. Thus we have a fixed value for the dipolar interaction
J = 137 K and with Eq. (26) a fixed tunnel splitting y. According to (53), the resonance condition w = 0,.
singles out pairs with asymmetry energy A = d m , resulting in a fixed value for the non-diagonal part of the dipole moment (52). Thus
we expect a narrow distribution of Rabi frequencies
which requires hco 2 q, of course. (Note that we have assumed a continuous distribution for the asymmetry energy A.)
The insert of Fig. 6 shows a typical rotary echo pattern for a KCI crystal containing 70 ppm 6Li. From a Fourier transformation of this signal the distribution of the
effective Rabi frequencies can be obtained. The resulting curve is shown in Fig. 6.
For a direct comparison of the Fourier spectra of different samples, all curves have
been normalized relative to the amplitude and the frequency at their maxima.
279
R. Weis et al., Coherent tunneling of lithium defect pairs in KCl crystals
1.2
KCI?Li 70 ppm
T = 10mK
Fig. 6 Fourier transform of the amplitude of a
rotary echo measured in KCl:"i+. The solid
line represents a numerical fit with a Lorentzian
distribution of s2/s2,. The insert shows the rotary echo used for the Fourier transformation.
The echo has been obtained at a frequency of
1175 MHz and a field strength of 54 V/m.
7
0
1
2
Rabi frequency
8
9
3
Q/Q,
As expected we find a very narrow distribution of the effective Rabi frequencies
for this sample. We have fitted the Fourier spectra as in Fig. 6 assuming a Lorentz
distribution for R,
<
denotes the width of the Lorentz distribution, and Qo the average frequency. We
find N 0.3, which represents an upper limit for the distribution of the tunnel splitting 7. This width is similar to the one observed in measurements on KBr crystals
containing a few percent of CN--molecules [23] and is an order of magnitude smaller than the results of analogous experiments on structural glasses [24]. Considering
that noise and restrictions in the time domain of our experiment certainly contribute
to the width of the curve in Fig. 6 we conclude that the rotary echoes in KC1 containing 70 ppm 6Li are most likely associated with rather well-defined species of tunneling states.
This same conclusion can be drawn from rotary echo experiments at different resonance frequencies. Fig. 7 shows the dependence of the Rabi frequencies on the amplitude of the driving field obtained with two different microwave cavities. As
expected from Eq. (57) one finds in both cases a linear variation of the Rabi frequency with the field amplitude, and the slopes of the two curves are proportional to
u-'.This means that the product of all other factors in Eq. (57) is constant; hence
the pairs which are observed at the two different resonance frequencies have the
same tunnel frequency y / h , but different energy splittings E .
<
4.2 KCl doped with 60 ppm 7Li
A set of identical experiments have been performed with KC1-crystals containing 70
ppm 7Li. Fig. 8 shows the distribution of the effective Rabi frequencies Q for this
sample [25]. As described before, this curve has been obtained after performing a
Fourier transformation of the original echo pattern shown in the insert of the figure.
280
Ann. Physik 6 (1997)
I
730 MHz
0
0 1175MHz
T = 10mK
50
100
150
Field amplitude Fo (V/m)
"O
n
KCI:'Li
200
Fig. 7 Rabi frequencies Cl of KCk6Li+ as a
function of the local electrical field strength Fo
and for two different resonance frequencies.
After Ref. [19].
60 ppm
T = 10mK
F n = 201 V/m
3.5
4.0
4.5
Time (ps)
Fig. 8 Fourier transform of the amplitude of a
rotary echo measured in KCL7Li+. The solid line
represents a numerical fit with a Lorentzian distribution of ClR/Ro. The insert shows the rotary echo
used for the Fourier transformation. The echo has
been obtained at a frequency of 1125MHz and a
field strength of 201 Vlm.
Lb
0.0 0
Robi frequency R/R,
Again we find a very narrow distribution of Rabi frequencies. From a fit with
Eq. (58) we find a width 5 N_ 0.3, which is identical with the value found for the
6Li-doped crystal. Moreover, the measurements of the field dependence at two different resonance frequencies (Fig. 9) look very similar to the ones obtained with the
lighter isotope. Again we find two straight lines, whose slopes vary proportional to
up',in accord with Eq. (57). We therefore conclude that we observe well-defined
states for the 7Li-doped sample which are most likely of the same type as those
which gave rise to the rotary echoes in the 6Li-doped sample.
4.3 KCl doped with 33 ppm 'Li and 33 ppm 7L1
The idea to investigate samples which are doped with both isotopes in approximately
the same proportion was to study pairs of Li-tunneling states which consist of a 'Lit
and a 7Li+ ion. According to Eq. (26) one would expect a tunneling splitting which
is in between the one for isotopically pure pairs. Measurements on samples contain-
28 1
R. Weis et al., Coherent tunneling of lithium defect pairs in KCl crystals
v)
v
700 MHz
1125 MHz
T = 10 mK
Fig. 9 Rabi frequencies !2 of KCl:7Li+ as a
function of the local electrical field strength FO
and for two different resonance frequencies.
10
Field amplitude Fo (V/m)
ing both isotopes should prove that we observe pairs of strongly interacting Li+ ions,
because any other configuration should lead to a different isotope dependence. In particular, this rules out the (unlikely) possibility that the states observed in our experiments are single Li+ ions whose energy spectra are strongly modified by some additional interaction.
In rotary echo experiments with samples doped with both isotopes it should be
possible to detect the Rabi frequencies of three types of pairs. For statistical reasons
one expects twice as many mixed configurations as pure 7Li or pure 6Li pairs. Due
to the width of distribution for each type, the individual lines for the different pairs
could not be resolved in our experiments. Nevertheless the maximum of the distribution should originate from pairs formed by a 6Li+ and a 7Li+ ion. We therefore measured the field dependence of the Rabi frequency for this sample. The result is
shown in Fig. 10. Again we find a linear field dependence, as above for the isotopically pure pairs.
I
4-
0
987 MHz
Fig. 10 Rabi frequencies 51 of KCI doped with
33 ppm hLi and 33 ppm 7Li as a function of the
local electrical field strength Fo and at a resonance
frequency of 987 MHz.
T = 12mK
Field amplitude Fo (V/m)
282
Ann. Physik 6 (1997)
-4
N
0)
v
ID3
0
r
\2
4:
3
1
0
0
100
200
300
400
Field amplitude F, (V/m)
Fig. 11 Product R o as a function of the local
electrical field strength Fo for different isotopes.
The lines are linear fits to the data.
4.4 Isotope effect
To prove the validity of the predictions of the theory presented in Section 2 the isotope dependence of the Rabi frequency is a key property. In order to compare experiments at different resonance frequency we have plotted in Fig. 11 the product a m .
Clearly a strong and systematic isotope dependence is found in these measurements.
Since the dipole moment is the same for both Li isotope, the ratio of the tunnel splitting of the three types of pairs and the absolute value of the tunnel splitting can be
derived from the different slopes in Fig. 11. The results are gathered in Table 2 topairs from Eqs. (26) and (57). Within the
gether with the values expected for "2
experimental error of about lo%, which is mainly due to difficult determination of
the filling factor, the measured numbers agree well with the theoretical values. Note
that the theoretical values for the tunnel splittings depend on the magnitude of F used
to determine the interaction energy. Here we have assumed e ~ a / 2= 2.25. However,
we like to emphasize that the ratio of the tunnel splittings and therefore the isotope
effect does not depend on the exact value of c.
The good quantitative agreement between theory and data proves that we observe
coherent tunneling oscillations of two strongly coupled Li+ ions with next-nearest
In addition it shows that the properties of such pairs are
neighbor configuration "2.
well described in the framework of the theory in Section 2.
Table 2 Experimental and theoretical values for the ratio of the tunnel splitting of different types
of Li pairs and the absolute values of their tunnel splitting.
Experiment
q (%-'Li)/q ( k k 7 ~ i )
q ( 6 ~ i - 7 ~ i )(/' q~ i - ' ~ i )
q (6Li-6Li)/v( 7 ~ i - 7 ~ i )
q ('Li-'Li) (mK)
y (7Li-7Li)(mK)
q (6Li-7Li)(mK)
1.6
1.5
2.4
20.5
8.5
12.9
Theory
1.54
1.54
2.38
22.0
9.3
14.3
R. Weis et al., Coherent tunneling of lithium defect pairs in KCl crystals
283
5 Discussion
5.I Validity of perturbation theory
The ground state multiplet with average energy EOhas been calculated to second order in terms of the perturbation K V ; the unperturbed Hamiltonian is given by the
dipolar interaction W . Because of PVQ = 0 the potential V contributes a first-order
term only; noting PKP = 0 on the other hand, it is clear the first non-vanishing term
arising from the tunneling part is of second order in K .
The eigenstates) . 1 are confined to the subspace spanned by the ground states
IGap?). Formally this corresponds to a zeroth-order approximation; the first-order
corrections would yield an admixture of excited states with energies El and EZ and
amplitudes of the order of magnitude Ao/ J . The resulting corrections to the susceptibility are immaterial for our purpose.
The perturbation analysis relies on the small parameters Ao/J and A/J; there is
no restriction on the ratio Ao/A. The most relevant range, however, involves an
asymmetry energy of the order of T,I = A i / J or smaller, i.e. A (< do. For lithium impurities in KCl in next nearest neighbor positions one finds for the small parameter
Ao/J x &. Thus perturbation theory should yield essentially exact results.
Degenerate perturbation theory in an eight-dimensional subspace may yield quite
involved expressions for the energy spectrum and the eigenstates. Because of the cubic symmetry the present problem factorizes with respect to tunneling motion in the
three crystal directions; as a consequence the wave function may be written as a
product of three factors, each depending on either x or y or z only. (This is seen
most clearly in (42) where e.g. the wave function of the coordinate ay is given by
CCy,=*,
J;: .) When attempting to go beyond the perturbation analysis, a more explicit use of the factorization property might prove advantageous 1261.
+
5.2 The interaction energy
The coupling term (12) is correct for sufficiently distant impurities; our using it for
nearby defects gives rise to several flaws. First, at such small distance higher terms
of the multipole expansion are not negligible; calculating the energy spectrum from a
point charge model, however, yielded levels essentially identical to Eo, El, E2 given
in (13).
Second, we have assumed tunneling states and amplitudes to be independent of
the strong dipolar interaction. Strictly speaking we cannot exclude the presence of a
nearby impurity to cause some modification of the tunneling Hamiltonian. In particular the off-center positions may deviate from (1 1 1) more towards (100). Yet for the
following reason the isotope effect observed for rotary echoes rules out any major
change of the tunneling states. The experiment confirms the ratio of the tunneling
amplitudes of isolated defects, 6 A 0 / 7 A ~= 1.54. Any deformation of the one-impurity
potential would significantly modify that ratio. Given that the interaction energy J is
much larger than the barrier the particle has to tunnel through, one might be surprised
by the good agreement displayed in Table 2. Nevertheless these data strongly support
the validity of our description.
Third, the dielectric constant is a macroscopic quantity which is not well defined
on an atomic length scale. Because of the incomplete screening at small distances
284
Ann. Physik 6 (1997)
one expects some effective value between that for vacuum, E = 1, and the macroscopic value for a KCI crystal, E K C ~= 4.49; in the previous section we have used
E K C ~=
/ ~2.25. Note that the NN2 pairs are separated by just a single chlorine ion. In
this respect it is interesting that the purely electronic contribution coo to the dielectric
constant of KCl is 2.20 [27], because E , is mainly determined by the relatively polarizable chlorines. However, contributions from the surrounding host lattice cannot
be completely excluded, since the changes of the level population occur at the rather
slow Rabi frequencies.
5.3 Two-state approximation
Coupled pairs of tunneling defects have been treated in various approaches [13-161.
As a common feature all these works relied on a two-state approximation, where the
actual eight-state system formed by a single impurity is replaced with a spin-4 system. Comparing the present results with those derived for a pair of two-level systems, we find the two-state approximation to be essentially correct.
The most obvious deviation involves the free energy (44) which is more complicated
than that for a two-level system; as a consequence the specific heat C = d U / d T is not
described properly in two-state approximation. Moreover there is an ambiguity as to the
definition of the excitation energy in terms of the dipolar interaction J . The second-order Hamiltonian fd2)involves excitations to the first and second levels El and E2
which are separated from the ground states by J / 2 and J , respectively. In two-state approximation there is a single excitation energy of the order of J ; an arbitrary factor
arises from the precise two-state truncation scheme. Thus the expression used in
Ref. [19] is too large by a factor of two, lJiil = 2J. In the somewhat different context
of Ref. [S], a coupling energy of I JV1 = J has been used.
As a most surprising fact we note that (54) is identical to the dynamical response
function of a single two-state system with tunneling amplitude 7 and asymmetry energy A. As a consequence in linear-response approximation a coupled impurity pair
may be substituted with a two-level system with energy splitting E and non-diagonal
dipole moment (52); this holds true for the non-linear response as long as the field
energy is smaller than the effective tunnel splitting, [P. Fo/ << 7. This permitted us
to use the well-known theory of rotary echoes for two-level systems, which will
greatly simplify the discussion of the experimental results.
2
5.4 Role of the asymmetry energy
Other impurities and lattice imperfections may remove the degeneracy of the eight
off-center positions of a single defect. For a sufficiently distant source, this additional
potential may be linearized in terms of the position operators r and r’, resulting in a
perturbation of the form (31). Because of the particular form of the ground states, the
contributions from the two impurities add along the x-axis and cancel for the remaining directions. For a fixed value of the tunnel splitting 7, the random configuration
of the sources of this perturbation leads to a continuous distribution for the asymmetry energy A, and hence for the energy splitting E.
Both energy spectrum and eigenstates are labelled by three quantum numbers 6.
For zero asymmetry, A = 0, the variable ox denotes even (a, = -1) or odd (0,= 1)
R. Weis et al., Coherent tunneling of lithium defect pairs in KCl crystals
285
superpositions of the states localized at x = f d / 2 = x'; similar statements hold for
0,. In the case of a finite bias A, the meaning of the label a, changes; then it
denotes two different superpositions with amplitudes f,", which in the limit of large
asymmetry tend towards the partially localized states IGa,; oyaz).
o,,and
5.5 Mass dependence
The availability of two stable isotopes 6Li and 7Li constitutes a most appealing aspect of lithium tunneling. Defect pairs may involve either two ions of the same mass
or two different isotopes. According to the WKB formula A0 =
tzwo exp( -( d / ( 2 t z ) ) J m ) the tunneling amplitude depends exponentially on the
defect mass; thus we have accounted for different tunneling frequencies Ao/h and
AL/h in our theory. Both dipole moment p and off-center shift d are the same for the
two isotopes [28]. This is why in our model neither the asymmetry A nor the coupling to an external field depend on the defect mass.
involves yo (25) and merely causes an insignificant shift
The diagonal part of
of the energy spectrum (39); the more interesting non-diagonal part results in the tunneling splitting which is determined by the quantity y = AoAL/J. Considering all
combinations of 6 A and
~ 7Ao yields the isotope effect for both resonance energy (53)
and off-diagonal dipolar transition amplitudes (52).
6 Summary
In this paper we have investigated rotary echoes of pairs of strongly coupled lithium
impurities in KCl. In the theory part we have dealt with two eight-state tunneling
systems with strong dipolar interaction. Second-order perturbation theory provides essentially exact results, corrections being quadratic in the small parameter Ao/J M
Experimentally, we have observed Rabi oscillations of three types of impurity
pairs, involving either two 7Li ions, or two 6Li, or one 6Li and 7Li. The resulting isotope effect is essential for the understanding of impurity interaction. The oscillations
in real time shown in the inserts of Figs. 6 and 8 correspond to coherent tunneling
between the odd (ox = 1) and even (ox = -1) collective eigenstates of the two defects; this tunneling motion is driven by the time-dependent external field. Here we
summarize our main results and conclusions.
First, the dipole moment P and the single-ion tunneling energies 6Ao and 7A0 are
known; the asymmetry energy is fixed by the resonance condition E = hco. As to the
interaction energy, there is some uncertainty arising from the appropriate value for
the dielectric constant at small distance and from the possibility that the Li+ may deviate from the normal (1 1I ) off-center position more towards (100) due to the large
interaction energy. Even so we find a remarkably good agreement between the measured data and theory; note that the isotope effect displayed in Table 2 does not allow
for any adjustable parameter. From this we conclude on the validity of our description for a defect pair.
Second, the static asymmetry arising e.g. from lattice imperfections is small; for
the "2
defect pairs observed in the present work we find values for A of less than
30mK. This is in agreement with the fact that the relaxation peak in the dynamical
susceptibility found at higher lithium concentration does not arise from a static asym-
&.
286
Ann. Physik 6 (1997)
metry energy but is of different origin, contrary to the situation encountered for tunneling systems in glasses [8].
Third,the theory of Section 2 yields results similar to those obtained from the pair
model of two coupled two-level systems, and thus shows the two-state approximation
for interacting tunneling impurities to be essentially correct. This is of interest for the
description of higher concentrated samples where one faces a complicated many-body
problem and where the two-state approximation can hardly be avoided.
The authors would like to thank A.C. Anderson for providing two crystals previously investigated
at the University of Illinois. We are grateful to H. Homer, S. Hunklinger, M.v. Schickfus and
G. Weiss for many useful discussions. This work has been supported by the Deutsche Forschungsgemeinschaft (Contract: En299/1-1) and the National Science Foundation (Grant: DMR 92-23230).
One of us (EL.) acknowledges support by the Alexander von Humboldt Foundation.
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