Interaction Effects in Conductivity of a Two-Valley Electron System in High-Mobility Si Inversion Layers

Abstract

We have measured the conductivity of high-mobility (001) Si
metal-oxide-semiconductor field effect transistors (MOSFETs) over wide ranges
of electron densities n=(1.8-15)x10^11cm^2, temperatures T=30mK-4.2K, and
in-plane magnetic fields B=0-5T. The experimental data have been analyzed using
the theory of interaction effects in the conductivity of disordered 2D systems.
The parameters essential for comparison with the theory, such as the
intervalley scattering time and valley splitting, have been measured or
evaluated in independent experiments. The observed behavior of the
conductivity, including its quasi-linear increase with decreasing T down to
~0.4K and its downturn at lower temperatures, is in agreement with the theory.
The values of the Fermi- liquid parameter obtained from the comparison agree
with the corresponding values extracted from the analysis of Shubnikov-de Haas
oscillations based on the theory of magnetooscillations in interacting 2D
systems.

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arXiv:0809.2750v1 [cond-mat.dis-nn] 16 Sep 2008
Interaction Effects in Conductivity of a Two-Valley Electron
System in High-Mobility Si Inversion Layers
N. N. Klimov1,3, D. A. Knyazev2, O. E. Omel’yanovskii2,
V. M. Pudalov2, H. Kojima1, and M. E. Gershenson1
1Department of Physics and Astronomy,
Rutgers University,
New Jersey 08854, USA.
2P. N. Lebedev Physical Institute,
Moscow 119991, Russia.
3P. N. Lebedev Physics Research Center,
Moscow 119991, Russia.
(Dated: September 16, 2008)
Abstract
We have measured the conductivity of high-mobility (001) Si metal-oxide-semiconductor field
effect transistors (MOSFETs) over wide ranges of electron densities n = (1.8 − 15) × 1011cm−2,
temperatures T = 30mK – 4.2K, and in-plane magnetic fields B?= (0 − 5)T. The experimental
data have been analyzed using the theory of interaction effects in the conductivity σ of disordered
2D systems. The parameters essential for comparison with the theory, such as the intervalley
scattering time and valley splitting, have been measured or evaluated in independent experiments.
The observed behavior of σ, including its quasi-linear increase with decreasing T down to ∼ 0.4K
and its downturn at lower temperatures, is in agreement with the theory. The values of the Fermi-
liquid parameter obtained from the comparison agree with the corresponding values extracted
from the analysis of Shubnikov-de Haas oscillations based on the theory of magnetooscillations in
interacting 2D systems.
PACS numbers: PACS:
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I. INTRODUCTION
The beginning of the 80’s witnessed a triumph of the one-parameter scaling theory of
localization1,2and the theory of electron-electron interactions in disordered conductors (for a
review, see Ref. 3). As a result, the peculiar low-temperature behavior of the conductivity of
numerous low-dimensional systems has been successfully attributed to quantum interference
effects (see, e.g., Ref. 4). Curiously, application of these ideas to the two-dimensional (2D)
electron liquid in Si MOSFETs - one of the most ubiquitous 2D systems - remained a chal-
lenge for more than 25 years. In the early experiments5,6with low-mobility (µ ∼ 0.1m2/Vs)
structures in the regime of relatively high electron densities (n > 1012cm−2), a decrease of
the conductivity with cooling has been observed, apparently in qualitative agreement with
the ideas of weak localization1,7and electron-electron interactions3. However, the quantita-
tive description of this behavior remained a problem8. The disagreement with the theory
became qualitative with the advent of high-mobility (µ ≥ 2m2/Vs) structures: the low−T
conductivity of high-µ Si MOSFETs increased with decreasing temperature9, in striking
contrast to the behavior of many other 2D systems. This “metallic” behavior of the con-
ductivity is especially pronounced at low electron densities n ∼ (1−2)×1011cm−2, where a
five-fold increase of σ(T) with cooling was observed10,11,12,13. Later, the quasi-linear increase
of σ(T) was observed in many high-mobility 2D systems in the “dilute” regime, including
GaAs heterostructures (both p-type14,15,16,17and n-type18,19), Si/SiGe20, and AlAs21quan-
tum wells.
The subsequent development of theory and experiment led to significant progress in our
understanding of the low-temperature transport in high-mobility systems in the regime of
low electron densities. In the mid-80’s, the quasi-linear “metallic” dependence observed in
the ballistic interaction regime Tτ ≫ 1 (τ is the transport mean free time, here and below we
set ? = kB= e = 1) was attributed22to weakening of the screening of the scattering potential
with increasing T. Observation of a strong negative magnetoconductance induced by the
in-plane magnetic field23and renormalization of the effective electron mass and g−factor in
these structures24,25also suggested that the electron-electron interactions play an important
role in this phenomenon. More recently, Zala, Narozhny, and Aleiner (ZNA)26developed a
theory that took into account all interaction contributions to the conductivity, including the
exchange ones. This theory offers a unified approach to both ballistic (Tτ ≫ 1) and diffusive
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(Tτ ≪ 1) interaction regimes by considering the quantum interference between electron
waves scattered off a short-range random potential “dressed” by Friedel oscillations of the
electron density. The theory was extended for the case of a long-range scattering potential
by Gornyi and Mirlin (GM)27. The theories26,27naturally incorporate the Altshuler-Aronov
results for the interaction corrections to the conductivity in the diffusive regime3.
For the diffusive regime, a more general approach to interacting systems based on the
nonlinear σ-model has been developed by Finkel’stein28. Recently, the renormalization group
(RG) equations of this theory29,30(obtained in the first order in
1
πσand in all orders in
interaction) have been compared with the conductivity of Si MOSFETs at low electron
densities29,31,32,33.
The RG equations28,29,30,34,35describe the length scale (temperature) evolution of the
resistivity and interaction parameters for a 2D electron system in the diffusive regime28.
However, at high electron densities, the temperature range corresponding to the diffusive
regime shrinks. In contrast, the theory of interaction corrections26,27is applicable over a
wider T range (that includes both ballistic and diffusive regimes) provided σ ≫ 1 and
∆σ/σ ≪ 1; these assumptions are well justified at high densities.
The theories26,27predict that the magnitude and sign of the the interaction correction
∆σ(T,B) is determined by the value of the Fermi-liquid parameter Fσ
0(which can be found
by measuring the Shubnikov-de Haas (SdH) oscillations in weak magnetic fields perpendic-
ular to the plane of a 2D structure24,25or the magnetoresistance in strong in-plane magnetic
fields36). In particular, it is expected that the σ(T) dependence becomes “metallic” when
Fσ
0is negative and its absolute value exceeds a certain threshold (see Sec. II).
The experimental studies of the conductivity in various low-carrier-density 2D systems
in the ballistic (high-temperature) regime are in agreement with the ZNA and GM theories.
The “metallicity” in all these systems is enhanced at low n due to an increase of the absolute
value of Fσ
0(see, e.g., Refs. 37,38,39). In Si MOSFETs, the interaction effects are especially
strong due to the presence of two nearly degenerate valleys in the electron spectrum29.
This enhancement, however, diminishes if the temperature T becomes smaller than the
valley splitting ∆V and intervalley scattering rate τ−1
v. As a result, with lowering T, the
“metallic” dependence of σ(T) is expected to become weaker or even to be replaced with
an “insulating” one. To the best of our knowledge, this behavior has not been observed for
Si MOSFETs prior to our work.
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This paper aims to study in detail the conductivity of 2D electron liquid in high-mobility
(001) Si MOSFETs over a wide temperature range (T = 0.03−4.2K) that includes both the
diffusive and ballistic regimes. In particular, we observed for the first time that the “metallic”
increase of σ with cooling is followed by the downturn of σ(T) at lower temperatures. For the
purpose of comparison with the ZNA theory26, we studied the range of not-too-low densities,
n = (1.8−15)×1011cm2, where the temperature and magnetic field dependences ∆σ(T,B)
can still be treated as small corrections to the Drude conductivity σD. In principle, no fitting
parameters are required for comparison with the theory, because we have measured Fσ
0, ∆V,
and τvin independent experiments. However, below we take a slightly different approach:
we obtain the Fσ
0(n) values from fitting the ∆σ(T,B?) dependences with the ZNA theory26,
and show that these values are consistent with the corresponding values extracted from the
analysis of SdH oscillations25. We have also revealed shortcomings of earlier analysis of
∆σ(T), reanalyzed the available data, and compared the extracted values of Fσ
0(n) with
corresponding values from other measurements. We conclude that the experimental data
are well described by the theory of interaction corrections26at intermediate temperatures
T ≈ (0.3 − 4.2)K. For a quantitative analysis at ultra-low temperatures (T ≤ 0.3K), the
interaction correction theory should be modified by taking into account finite intervalley
scattering rates.
The paper is organized as follows. In Section II we briefly summarize the theoretical
results26for the interaction corrections to the conductivity of a two-valley system. The
experimental data are presented in Section III, along with the data analysis and discussion.
The summary is given in Section IV.
II.INTERACTION CORRECTIONS TO THE CONDUCTIVITY
A.Temperature dependence of the conductivity in zero magnetic field
In the ZNA theory26, the corrections ∆σeeto the Drude conductivity σD= nτ/mb(mb≈
0.205meis the electron band mass in (001) Si MOSFETs; for more detail, see discussion in
section IIIA3 and references therein) were calculated in both ballistic and diffusive regimes
for all orders of the interaction strength and the leading order in 1/(EFτ) and T/EF. In
particular, the theory reproduces the Altshuler-Aronov correction3to the conductivity in the
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diffusive regime. To adapt the theoretical results26to the case of (001) Si MOSFETs, one
should take into account that the electron spectrum in this system has two almost degenerate
valleys40. In zero magnetic field, ∆σee(T) for a system with two degenerate valleys in the
absence of intervalley scattering can be written as follows38:
∆σee(T) = δσC(T) + 15δσT(T).(1)
Here δσC is the so-called “charge” contribution which combines Fock correction and the
singlet part of Hartree correction, and δσT is the “triplet” contribution due to the triplet
part of Hartree term. The valley index can be considered as a pseudo-spin in multi-valley
systems29, and the valley degeneracy determines the number of triplet terms due to the spin
exchange processes between electrons in different valleys. For a system with two degenerate
valleys, the total number of interaction channels is 4 × 4 = 16, among them 1 singlet and
15 triplet terms (for comparison, there are 1 singlet and 3 triplet terms for a single-valley
system).
Below we assume that the scattering potential is short-ranged which is relevant to Si
MOSFETs. According to Ref. 26, the charge term does not depend on the details of inter-
actions:
δσC=1
π
?
(Tτ)
?
1 −3
8f(Tτ)
?
−
1
2πlnEF
T
?
, (2)
whereas the magnitude and sign of the triplet term is controlled by the Fermi-liquid
parameter26Fσ
0:
δσT=1
π
?
(Tτ)
Fσ
0
1 + Fσ
0
?
1 −3
8t(Tτ;Fσ
0)
?
−
?
1 −ln(1 + Fσ
0)
Fσ
0
?
1
2πlnEF
T
?
.(3)
The functions f and t in Eqs. (2,3) describe the crossover between the diffusive (∆σee∝ lnT)
and ballistic (∆σee∝ T ) regimes; outside the crossover region, they change the value of
∆σeeby only a few percent. The explicit expressions for these functions can be found in
Ref. 26. The diffusive-ballistic crossover is expected over some temperature range near
T∗=1 + Fσ
0
2πτ
.(4)
Equations (2,3) describe the quantum corrections in a system with the conductance σ ≫
1/2π at temperatures well below the Fermi energy?T ≪ (1 + Fσ
The sign and the magnitude of ∆σeeis controlled by the Fermi-liquid parameter Fσ
0)2EF
?.
0. For
a rough estimate, deeply in the ballistic regime the lnT terms and the crossover functions
5