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Calculation of profile of charge carrier concentration in modulation doped structure with a wide potential well.

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Ann. Phys. (Berlin) 18, No. 12, 928 – 934 (2009) / DOI 10.1002/andp.200910399
Calculation of profile of charge carrier concentration
in modulation doped structure with a wide potential well
L. Yu. Shchurova∗
P. N. Lebedev Physical Institute of Russian Academy of Science, 119991 Moscow, Russia
Received 1 September 2009, accepted 12 September 2009
Published online 11 December 2009
Key words Modulation doped structure, potential well, carrier concentration profile.
We investigate the equilibrium state of interacting electron system with Fermi statistics in modulation doped
structure with a wide quantum well. The model is formulated for the carrier system with a sufficiently high
density, such that the de Broglie wavelength of electrons is smaller than the width of the quantum well.
Due to a significant interaction of electrons with electric field of the doped layer, a state with stronglyinhomogeneous density of electrons is formed. Within the hydrodynamic approach, we set up formalism for
calculating the electron density across the width of the potential well. We have obtained the exact solution
of the equations, which is expressed in terms of hypergeometric functions. Based on the deduced formulas,
we performed numerical computations for the profile of carriers’ concentrations in a potential well in the
modulation doped Si/SiGe/Si structures.
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
Modulation doped Si/SiGe structures are promising for nano- and micro-electronics. In such structures,
doped layer is separated from the quantum well by an undoped spacer to reduce the influence of impurity
scattering on carrier mobility. This leads to very high carrier mobility, higher by more than an order of
magnitude than the corresponding Si MOSFETs carrier mobility [1]. Modulation doped Si/SiGe structures
also represent interest for physics research.The effect of giant magnetoresistance in a strong parallel (to
two-dimensional layers) magnetic field presents particular interest, as observed in the modulation doped
Si(B)/SiGe/Si structures with wide quantum well. This effect differs for structures with different doped
levels. In [2] it was observed that for these structures, whose doping level differs by 2.5 times, magnetoresistance changes by up to three orders of magnitude.
In structures with varying doping levels, the distribution of carriers’ density across the width of the
quantum well may differ completely. Quantum confinement levels are formed at sufficiently low concentrations of carriers, under the conditions L ≤ λ, L > l (where L is the width of the quantum well, λ is the
de Broglie wavelength, and l is mean free path of charge carriers). A system of particles in such quantum
well is considered as effectively two-dimensional. With the increase in the charge carrier concentration,
condition L ≤ λ may be violated, and the quantization of energy levels does not happen. Since the motion
of particles in one direction is limited, we can not consider such a system of particles as two-dimensional,
but we cannot consider it as explicitly three-dimensional either. For this structure, a state with inhomogeneous density of electrons is formed due to the interaction of electrons with ions of the doped layer. The
problem of the distribution of the electron density in such system has more than half a century history.
The questions of such distributions were solved by numerical methods, for example, in [3]. Approximate
analytical solutions require many simplifying assumptions, in particular the assumption that the screening
∗
E-mail: ljusia@gmail.com
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Ann. Phys. (Berlin) 18, No. 12 (2009)
929
length is much smaller than the width of the potential well [4]. The screening length is the characteristic region of heterogeneity. However, there is particular interest in electron systems with a characteristic
screening length comparable to the width of the potential well. In such structures, electron mobility happens largely due to a weaker scattering of electrons on the ions of the doped layer. In addition, for structures
with lower doping level we can expect more significant effects of magnetoresistance [5].
In this paper, we investigate the system of charge carriers in modulation doped structure with a wide
potential well L ≤ λ. We have formulated a model for calculating the density of charge carriers in such
structure, taking into account their interactions with the impurity layer. Within the proposed model, we
obtained the exact solution of the equations to calculate the electron density across the width of the potential
well. These formulas are suitable to describe the inhomogeneous distribution of electron density at any ratio
between the screening length and width of the well. Then, we present numerical estimates for the profile
of electron concentrations in some structures.
2 Model
We investigate the properties of strongly correlated systems of charge carriers with the Fermi statistics in
modulation doped structures with a wide potential well. All carriers in the two- dimensional conductance
channel are formed due to the ionization of impurity in the doped layer, separated from the potential well
by the spacer with a width L. We consider the system of charge carriers with sufficiently high density, with
de Broglie wavelength smaller than the width of the quantum well (λ ≤ L). The equilibrium state with
inhomogeneous density of the charge carriers forms due to the significant interaction of charge carriers
with ions of doped layer.
The exact Schrödinger equation for the correlated system of particles in a potential well, which includes
all electron-ion and electron-electron interactions, depends on a great number of parameters and is too
complicated. However, in the mean-field approach, density is used instead of the wave function to describe
the interactions. The wave equation becomes much simpler, however, it becomes nonlinear. Traditionally,
to calculate the states in the quantum well (for λ ≥ d), the self-consistent solutions of the Schrödinger
equation and the Poisson equation are carried out. If λ ≥ d, the movement of particles across a quantum
well (in z direction) corresponds to a discrete energy spectrum. However, at low temperatures the lowest
energy state normally occurs. In this situation, only one wave function along z direction determines the
distribution of carrier concentration.
We consider the problem of carriers’ movement in the situation when quantum confinement levels do
not occur, and the wave equation includes a lot of wave functions. However, at a distance greater than de
Broglie’s wavelength, information on the structure of the wave function is eroded. Therefore, hydrodynamic approximation, can be used instead of the quantum-mechanical approach.
Within the hydrodynamic approach, we calculated the equilibrium distribution of three-dimensional
concentration, n(z), on the basis of the thermodynamic condition of total energy conservation:
μ + Eint = const,
where μ = μ(z) =
2/3
(3π2 )
2
(1)
2
2/3
mh
n(z)2/3 is the chemical potential of the generated system of charged
carriers, Eint = Eint (z) is the interaction energy, which depends on coordinate z and three-dimensional
concentration of particles, n(z), in the potential well.
The energy of electrostatic interaction is Eint = Eint (z) = e · Φ(z), where e is the elementary charge,
Φ is the electrostatic potential, determined by the Poisson equation
d2 Φ
4π · e
n(z),
=
dz 2
χ
(2)
and χ is the permittivity of the medium.
www.ann-phys.org
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
930
L. Yu. Shchurova: Calculation of profile of charge carrier concentration
We are interested in modulation doped structures, where the impurity layer is located far enough from
the potential well, so that the distance between the ions and charge carriers is greater than the distance
between neighboring ions of doped layer. In this case, we can assume with exponent accuracy that charge
carriers interact with the whole layer of ions, instead of with each isolated ion. To be definite, we will
assume that the doped layer is to the right of the quantum well. Then from Eq. (2) should be
⎛
⎞
z
4π
·
e
dΦ
= e ⎝Ei −
· n(ζ) · dζ ⎠ ,
e
(3)
dx
χ
0
where Ei is the unscreened field of the ions’ layer, z = 0 is a coordinate of the interface from the side of
the doped layer, z is the coordinate in the quantum well, and the second term in parentheses describes the
screening effect by the charge carriers.
After differentiation of Eq. (1) and taking into account Eq. (3), we get
⎛
⎞
2 2/3
z
2
3
·
3π
4π
·
e
d
−1/3
(n(z))
=−
(n(z)) = e · ⎝−Ei +
· n(ζ) · dζ ⎠ .
(4)
2
m dz
χ
0
Charge carrier concentration, cm3
This equation describes the equilibrium of attractive forces of electrons to the ions’ layer and the repelling forces of electrons from the wall by the Fermi pressure in each interval dz in the potential well.
8 1017
41011
6 1017
2.51011
21011
4 1017
1.61011
2 1017
0
0
5. 107
1. 106
1.5 106
2. 106
2.5 106
3. 106
Coordinate, cm
Fig. 1 Charge carriers concentration profiles for structures with various doped levels (shown on the figure
legend).
We denote
E0 =
2 · e · m · Ei
,
3(3π)2/3 · 2
K0 =
8π · e2 · m
.
3(3π)2/3 · 2 · χ
Then the Eq. (4) takes on a simpler form:
z
−E0 + K0
n(ζ) · dζ = [n(z)]−1/3 ·
0
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
dn
dz
.
(5)
www.ann-phys.org
Ann. Phys. (Berlin) 18, No. 12 (2009)
931
We solved Eq. (5) with the boundary condition dn
dz |z=L = 0, which means that on the right border the
field of ions is fully offset by the field of charge carriers.
In addition, we use the condition n(z = L) = nL is the charge carrier concentration at z = L.
Solution of Eq. (5) is given below in the Appendix.
We have obtained the exact solution of Eq. (5), which expresses in terms of hypergeometric functions.
From this solution follows
3
5 · (3π)2/3 · 2 · χ −5/6 1
2/3
z = L− ·
· nL
· (n)
4
8π · e2 · m
2
2 1 7
5/3
, , , (n/nL ) − 1
· Hypergeometric
(6)
5 2 5
(and n(z) is given by the corresponding inverse function).
Equation (6) is valid for any ratio between the screening length and width of the well. This formula is
suitable to describe the electron density distribution with any degree of inhomogeneity.
Based on formula (6), we carried out numerical computations of the profile of carriers’ concentrations
in the modulation doped Si/SiGe/Si, for which there are experimental data [2,6]. Figure 1 shows the results
of our calculations for structures with 30 nm potential well and various doping levels: 1.6 × 1011 , 2 × 1011 ,
2.5 × 1011 , 4 × 1011 cm−2 .
Note that partial penetration of particles in the barrier, which is not taken into account in the above
model would distort the concentration profile. However, according to our estimates, this does not affect
appreciably in the integral characteristics of the charge carriers, such as conductivity and resistance (magnetoresistance).
Large positive magnetoresistance in a strong parallel magnetic field observed in the modulation doped
Si/SiGe/Si structures with wide quantum well [2, 6]. It is important that magnetoresistance in a parallel
magnetic field is substantially different character for the samples with various carrier concentrations. The
effect of magnetic field is dramatic for the structures with lowest carrier density, nD = 8.2 × 1010cm−2 . In
this structure at low temperatures (T = 0.3 − 2K) magnetoresistance increased by more than three orders
of magnitude with increasing magnetic field up to 18 Tesla. While for the sample with a greater carrier
density, nD = 2 × 1011 cm−2 , magnetoresistance increased only in a few times. In [6] for structures with
the even greater concentration of charge carriers, nD = 2.5×1011cm−2 , magnetoresistance experimentally
observed in a parallel magnetic field was only tens of percent.
Indeed, in the lowest doped structures, nD = 8.2 × 1010 cm−2 , level L ≤ λ states of discrete spectrum
are formed. Holes in the fist quantum confinement level are mostly in the center of the quantum well. Giant
magnetoresistance in this structure due to the orbital motion of carriers in a strong magnetic field (r < L/2,
r is the radius of cyclotron orbit).
In the higher doped structures holes are pressed to the interface (Fig. 1). For these holes, their average
distance to the interface is less than their cyclotron radius (for magnetic fields in the experiment [2,6]). The
effect of orbital motion is weak for these holes. This effect weakens with increasing degree of heterogeneity
in the structures with more doping. Our estimates of the magnetoresistance in zero magnetic field ρ =
6 × 10−3 Ω for the structure with nD = 2 × 1011 cm−2 and ρ = 10−4 Ω for the structure with nD =
2 × 1011 cm−2 is consistent with experimental data [2, 6].
Appendix
We describe a method of solving Eq. (5) on the interval [0, L]
z
−E0 + K0
0
www.ann-phys.org
n(ζ) · dζ = [n(z)]−1/3 ·
dn
dz
.
(A1)
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
932
L. Yu. Shchurova: Calculation of profile of charge carrier concentration
with the boundary conditions
dn
|z=L = 0,
dz
(A2)
n(z = L) = nL .
(A3)
By differentiating Eq. (A1), we obtain
2
1
K0 · n = −
n−4/3 · n + n−1/3 · n
3
Multiplying (A4) to n−1 provides
2
1
K0 = −
n−7/3 · n + n−4/3 · n
3
(A4)
(A5)
Since Eq. (A5) does not include an independent variable z, the standard replacement n = dn
dz = p(n)
reduces the degree of this equation.
dn
= dp(n)
By differentiating with respect to z, we obtain n = dp(n)
dz
dn dz = p p. Then Eq. (A4) takes the
form
1
K0 = −
(A6)
n−7/3 · p2 + p · p.
3
Another substitution p2 = q in Eq. (A6) gives the equation
1
n−4/3 ·q
K0 = −
n−7/3 · q +
3
2
(A7)
dq
.
Here q = dn
Eq. (A7) is equivalent to the original equation (A1), but is much easier. Eq. (A7) is a first order differential equation for the unknown function q. The general solution of inhomogeneous equation (A7) includes
the general solution of homogeneous differential equation
n−4/3 1
·q =0
(A8)
−
n−7/3 · q +
3
2
and a particular solution of Eq. (A7).
The general solution of Eq. (A8) is obtained by a separation of variables. Multiplying (A8) by n and
dq
making the change of q = dn
, we obtain the equation
dq
2 dn
=
.
3 n
q
(A9)
Following the integration of (A9), we obtain
q = C · n2/3 .
(A10)
We find the general solution of inhomogeneous equation (A7) by using the method of variation of
constants: q = C(n) · n2/3 , which gives
C(n) = K0 ·
6 5/3
· n + CL .
5
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.ann-phys.org
Ann. Phys. (Berlin) 18, No. 12 (2009)
933
The equation for the constant CL is shown below.
Then,
6
q = K0 · · n7/3 + CL · n2/3 ,
5
6
√
K0 · n7/3 + CL · n2/3 .
p= q=
5
We recall the replacement of n = p(n); then
6
dn
=
K0 · n7/3 + CL · n2/3 .
dz
5
Eq. (A11) represents the equation with separable variables, from which directly follows
nL
n
dζ
6
5 K0
· n7/3 + CL · n2/3
(A11)
= z + C1 .
From the boundary condition (A2) follows C1 = −L. In addition, from the conditions (A2) and (A3)
follows
6
5/3
CL = − K0 · nL .
5
We have the equation
n
−
nL
dζ
6
5 K0
· ζ 7/3 − 65 K0 · nL 5/3 · ζ 2/3
or
n
nL
dζ
=
ζ 7/3 − nL 5/3 · ζ 2/3
= z − L,
6
K0 · (L − z).
5
Then, we transform the integral:
n
nL
dζ
−1/6 −1/6
nL
= nL
ζ 7/3 − nL 5/3 · ζ 2/3
= 3 · nL −1/6
n/n
L
ζ
nL
−1/3
ζ
nL
1
1/3
(n/n
L)
1
·d
5/3
ζ
nL
−1
y · dy
.
(y 5 − 1)
The last integral is expressed in terms of hypergeometric functions:
1/3
(n/n
L)
1
y · dy
1
2/3
= (n/nL ) · Hypergeometric
5
2
(y − 1)
2 1 7
, , ,
5 2 5
5/3
(n/nL ) − 1 .
Thus, we have an expression in which n and z are on the opposite sides of the equality sign:
2 1 7
6
1
2/3
5/3
1/6 1
(n/nL ) · Hypergeometric
, , , (n/nL ) − 1 = nL
K0 · (L − z) .
· ·
2
5 2 5
3
5
Therefore, z is evidently expressed in terms of n.
www.ann-phys.org
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
934
L. Yu. Shchurova: Calculation of profile of charge carrier concentration
References
[1]
[2]
[3]
[4]
T. Okamoto, K. Ooya, K. Hosoya, and S. Kawaji, Phys. Rev. B 69, 041202 (2004).
I. L. Drichko, I. Yu. Smirnov, A. V. Suslov, O. A. Mironov, and D. R. Leadley, Phys. Rev. B 79, 205310 (2009).
I.-H. Tan, G. S. Snider, L. D. Chang, and E. L. Hu, J. Appl. Phys. 68, 4071 (1990).
F. Neumann, Y. A. Genenko, C. Melzer, S. V. Yampolskii, and H. von Seggern, Phys. Rev. B 75, 205–322
(2007).
[5] S. Das Sarma and E. H. Hwang, Phys. Rev. Lett. 84, 5596 (2000).
[6] S. I. Dorozhkin, C. J. Emeleus, T. E. Whall, and G. Landwehr, Phys. Rev. B 52, R11638 (1995).
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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