Sharp increase of the effective mass near the critical density
in a metallic two-dimensional electron system
A. A. Shashkin
*
and S. V. Kravchenko
Physics Department, Northeastern University, Boston, Massachusetts 02115
V. T. Dolgopolov
Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia
T. M. Klapwijk
Department of Applied Physics, Delft University of Technology, 2628 CJ Delft, The Netherlands
共Received 13 May 2002; published 6 August 2002兲
We find that at intermediate temperatures, the metallic temperature dependence of the conductivity
(T)of
two-dimensional electrons in silicon is described well by a recent interaction-based theory of Zala et al. 关Phys.
Rev. B 64, 214204 共2001兲兴. The tendency of the slope
⫺ 1
d
/dT to diverge near the critical electron density
is in agreement with the previously suggested ferromagnetic instability in this electron system. Comparing
theory and experiment, we arrive at a conclusion that the instability, unexpectedly, originates from the sharp
enhancement of the effective mass, while the effective Lande
´
g factor remains nearly constant and close to its
value in bulk silicon.
DOI: 10.1103/PhysRevB.66.073303 PACS number共s兲: 71.30.⫹h, 73.40.Qv
In dilute two-dimensional 共2D兲 electron systems, the en-
ergy of electron-electron interactions dominates the kinetic
energy making the system strongly correlated.
1
The interac-
tion strength is characterized by the Wigner-Seitz radius r
s
⫽ 1/(
n
s
)
1/2
a
B
共where n
s
is the electron density and a
B
is
the effective Bohr radius in semiconductor兲, which is equal
in the single-valley case to the ratio, r
s
⬘
, of the Coulomb and
the Fermi energies assuming that the effective electron mass
m is equal to the band mass m
b
. According to the Fermi
liquid theory
2
共whose applicability to dilute 2D electron sys-
tems is discussed in Ref. 3兲, the electron-electron interac-
tions should give rise to a renormalization of the system
parameters including the effective mass and the effective g
factor. A sharp enhancement of the product gm with decreas-
ing electron density— possibly, a precursor of the ferromag-
netic instability—has been observed in recent studies of the
parallel field magnetoresistance of a metallic 2D electron
system in high-mobility silicon metal-oxide-semiconductor
field-effect transistors 共MOSFET’s兲.
4
This agrees well with
the gm value obtained from the beating pattern of
Shubnikov–de Haas oscillations in tilted magnetic fields.
5–8
Although physics behind the metal-insulator transition ob-
served in strongly interacting 2D systems still remains
illusive,
1
significant progress has been recently made in un-
derstanding of the nature of the metallic conductivity deeper
in the metallic state. Zala et al.
9
have calculated temperature-
dependent corrections to conductivity due to electron-
electron interactions based on the Fermi liquid approach. In
contrast to pre-existing theories 共see, e.g., Refs. 10,11兲, the
new theory incorporates strongly interacting 2D electron sys-
tems with electron densities down to the vicinity of the
metal-insulator transition 共provided that the conductivity
Ⰷ e
2
/h). For sufficiently strong interactions, it predicts a me-
tallic temperature dependence of conductivity in the entire
temperature range. At very low temperatures, in the ‘‘diffu-
sive’’ regime (TⰆ ប/k
B
, where
is the elastic relaxation
time兲, this is Finkelstein’s weakly metallic 共logarithmic兲
conductivity.
12,13
At intermediate temperatures, in the ‘‘bal-
listic’’ regime (Tⲏ ប/k
B
; T⬎ 0.2⫺ 0.5 K under the condi-
tions of our experiments兲, the predicted
(T) is similar to
the Gold-Dolgopolov linear dependence
10
共
T
兲
0
⫽ 1⫺ Ak
B
T, 共1兲
where the slope A is determined by the interaction-related
parameters: the Fermi liquid constants F
0
a
and F
1
s
. These
parameters are responsible for the renormalization of the g
factor and the effective mass
3
g
g
0
⫽
1
1⫹ F
0
a
,
m
m
b
⫽ 1⫹ F
1
s
共2兲
and can be determined experimentally.
14
The slope A is pre-
dicted to rise as the interaction strength increases and the 2D
electron system is driven toward the ferromagnetic instabil-
ity. The last is expected to occur, in the simplest case, at
F
0
a
⫽⫺1, which corresponds to the diverging effective g
factor.
In this paper, we perform precision measurements of the
temperature-dependent conductivity in a metallic 2D elec-
tron system in silicon over a wide range of electron densities
above the critical electron density n
c
for the metal-insulator
transition. The theory of Zala et al.
9
is found to be successful
in interpreting the experimental data in the ballistic regime.
Knowing the product gm from independent measurements,
we determine both g and m as a function of n
s
from the slope
of the temperature dependence of the conductivity. The ten-
dency of the slope to diverge near the critical density is con-
sistent with the suggested ferromagnetic instability in this
electron system.
4,15
However, unlike in the simplest scenario
PHYSICAL REVIEW B 66, 073303 共2002兲
0163-1829/2002/66共7兲/073303共4兲/$20.00 ©2002 The American Physical Society66 073303-1
for the ferromagnetic instability, comparison between experi-
ment and theory shows that it is the value of the effective
mass that becomes strongly enhanced with decreasing elec-
tron density, while the g factor remains nearly constant g
⬇g
0
⫽ 2 in bulk silicon.
Measurements were made in an Oxford dilution refrigera-
tor with a base temperature of ⬇30 mK on high-mobility
共100兲-silicon samples similar to those previously used in Ref.
16. The resistance was measured by a standard four-terminal
low-frequency technique. Excitation current was kept low
enough to ensure that measurements were taken in the linear
regime of response. Contact resistances in our samples were
minimized by using a split-gate technique that allows one to
maintain a high electron density in the vicinity of the con-
tacts regardless of its value in the main part of the sample. In
this paper we show results obtained on a sample with a peak
mobility close to 3 m
2
/Vs at 0.1 K.
Typical dependences of the normalized conductivity on
temperature
(T)/
0
are displayed in Fig. 1 at different
electron densities above the critical electron density for the
metal-insulator transition which in this sample occurs at n
c
⫽ 8⫻ 10
10
cm
⫺ 2
;
17
the value
0
, which has been used to
normalize
, was obtained by extrapolating the linear inter-
val of the
(T) dependence to T⫽ 0. As long as the devia-
tion
兩
/
0
⫺ 1
兩
is sufficiently small, the conductivity
in-
creases linearly with decreasing T in agreement with Eq. 共1兲,
until it saturates at the lowest temperatures.
18
As seen from
the figure, the linear interval of the dependence is wide
enough to make a reliable fit.
In Fig. 2, we show the n
s
dependence of the inverse slope
1/A extracted from the
(T) data. Also shown for compari-
son is the magnetic energy
B
B
c
⫽
ប
2
n
s
/gm 共where
B
is
the Bohr magneton and B
c
is the parallel field of the magne-
toresistance saturation兲, corresponding to the onset of full
spin polarization in this electron system, which is governed
by the 共enhanced兲 product gm.
4
Over a wide range of elec-
tron densities, the values 1/A and
B
B
c
turn out to be close
to each other. The low density data for 1/A are approximated
well by a linear dependence which extrapolates to the critical
electron density n
c
in a similar way to the behavior of the
polarization field B
c
. We emphasize that the density of the
delocalized electrons in the metallic phase is practically co-
incident with n
s
in our samples, as inferred from the above
agreement between the gm data obtained by different mea-
surement methods.
7
As has already been mentioned, the coefficient A in the
linear-in-T correction to conductivity of Eq. 共1兲 is deter-
mined by the Fermi liquid parameters
9,19
A⫽⫺
共
1⫹
␣
F
0
a
兲
gm
ប
2
n
s
. 共3兲
The factor
␣
is equal to 8 in our case.
20
This theoretical
relation allows us to determine the many-body enhanced g
factor and mass m separately using the data for the slope A
and the product gm as a function of n
s
共the latter is known
from independent measurements similar to those described in
Ref. 4兲.
Before presenting data for g and m, we would like to
emphasize that possible uncertainties in the coefficients of
Eq. 共3兲 do not affect the main result of the paper: a sharp
increase of m and approximate constancy of g as n
s
→ n
c
.
This result directly follows from the fact that the experimen-
tally observed dependence of 1/A on n
s
is linear and, there-
fore, F
0
a
is n
s
independent, according to the functional form
of Eq. 共3兲. In addition, the extracted g and m values are only
weakly sensitive to the coefficients in Eq. 共3兲 because g turns
out to be close to g
0
.
In Fig. 3, we show the so-determined values g/g
0
and
m/m
b
as a function of the electron density 共the band mass m
b
is equal to 0.19m
e
, where m
e
is the free electron mass兲. Note
that in the range of n
s
studied here, the low-temperature
conductivity
⬎ 8e
2
/h. The behavior of g and m at electron
densities below n
s
⫽ 3⫻ 10
11
cm
⫺ 2
共corresponding to r
s
⬇4.8) turns out to be very different from that at electron
densities above this value. In the high n
s
region 共lower r
s
),
the enhancement of both g and m is relatively small, both
values slightly increasing with decreasing electron density in
agreement with earlier data.
21
Also, the renormalization of
the g factor is dominant compared to that of the effective
FIG. 1. The temperature dependence of the normalized conduc-
tivity at different electron densities 共indicated in units of
10
11
cm
⫺ 2
) above the critical electron density for the metal-
insulator transition. The dashed lines are fits of the linear interval of
the dependence.
FIG. 2. Comparison of the inverse slope 1/A 共dots兲 and the data
for the polarization field B
c
共diamonds兲 as a function of electron
density. The dashed lines are linear fits which extrapolate to the
critical electron density for the metal-insulator transition.
BRIEF REPORTS PHYSICAL REVIEW B 66, 073303 共2002兲
073303-2
mass, which is consistent with theoretical studies.
22
The de-
pendence g(n
s
) is described reasonably well by the theory:
the inset of Fig. 3 compares the theoretical renormalization
parameter F
0
a
⫽⫺r
s
/2(2r
s
⫹
冑
2) 共Ref. 9兲 to that calculated
using Eq. 共2兲 and the data for g(n
s
).
In contrast, the renormalization in the low n
s
共critical兲
region, where r
s
Ⰷ 1, is much more striking. As the electron
density is decreased, Fig. 3 shows that the renormalization of
the effective mass overshoots abruptly while that of the g
factor remains relatively small g⬇g
0
, without tending to in-
crease. Hence, the current analysis indicates that it is the
effective mass that is responsible for the drastically enhanced
gm value near the metal-insulator transition. The present re-
sults for the effective mass and g factor in the critical region
are consistent with both the evaluation of m(n
s
) and g(n
s
)
obtained by analysis of the Shubnikov–de Haas oscillations
in high-mobility Si MOSFET’s 共Ref. 8兲 and the data for the
spin and the cyclotron gaps obtained by magnetocapacitance
spectroscopy.
23
This gives support to our procedure and con-
clusions.
One can already see from Fig. 2 that the data points span
close enough to the critical electron density. In the inset to
Fig. 4, this is additionally checked for the m data plotted
against r
s
⬘
⫽ 2(m/m
b
)r
s
共the factor of 2 comes from the val-
ley degeneracy兲. As the electron density is decreased, the
dependence of m on r
s
⬘
approaches the linear dependence
with the slope 1/2r
s
(n
c
) determined by n
c
. At the lowest n
s
,
the effective mass increases approximately proportionally
with r
s
⬘
and, therefore, m changes sharply with r
s
共and n
s
)in
the reached vicinity of the critical electron density.
It is important to discuss another consequence of the
theory:
9
the slope A of the temperature dependence of the
conductivity should increase as the ferromagnetic instability
in a dilute 2D electron system is approached. Since renor-
malization parameters have not been theoretically calculated
in the limit r
s
Ⰷ 1, the simplest scenario of the ferromagnetic
instability is prompted by Eq. 共2兲: F
0
a
→ ⫺ 1 causes the effec-
tive g factor 共and the slope A) to diverge. Experimentally,
the slope A tends to diverge near the critical electron density
in a way similar to the behavior of the product gm seen in
Fig. 2. This is consistent with the conclusion of Refs. 4,15
about the possibility of ferromagnetic instability in this elec-
tron system. At the same time, the simplest scenario of a
diverging g factor is not the case; instead, it is the growing
effective mass which controls the anomalous behavior of the
dilute 2D electron system near the metal-insulator transition.
In principle, g or m or both might be diverging at the
occurrence of a ferromagnetic instability. It is important that
rather than r
s
, the ratio r
s
⬘
of the interaction and the kinetic
energies is the relevant parameter for ferromagnetic instabil-
ity. Spontaneous spin polarization is expected to occur at the
large r
s
⬘
at which the increase in the kinetic energy of a fully
spin-polarized electron system is excelled by the decreasing
energy of electron-electron interactions. In our case, both the
kinetic energy drop and the increase of r
s
⬘
with decreasing
electron density are controlled by the sharply growing effec-
tive mass.
The effective mass enhancement was traditionally consid-
ered to be small and, therefore, the value m⬇m
b
was used to
calculate some of the important system parameters, e.g., the
elastic relaxation time
extracted from mobility.
21
In Fig. 4,
we compare the so-determined
共circles兲 with that calcu-
lated taking into account the enhancement of the effective
mass 共squares兲. As seen from the figure, in the range of elec-
tron densities studied, the corrected
keeps increasing down
to the lowest n
s
; note that such a behavior is representative
of surface roughness scattering.
21
Therefore, the mobility
drop at low n
s
in high-mobility Si MOSFET’s 共see, e.g., Ref.
24兲 turns out to originate from the m enhancement rather
than from the decrease in
, although the value
is still
expected to vanish in the insulating phase. The observed be-
havior of
is consistent with that of the temperature range
corresponding to the ballistic regime 共see Fig. 1兲, which
gives additional confidence in our analysis down to the vi-
cinity of the metal-insulator transition. Finally, values of
much larger than those previously estimated yield apprecia-
bly smaller quantum level widths in perpendicular magnetic
FIG. 3. Renormalization of the effective mass 共squares兲 and g
factor 共dots兲 as a function of electron density. The dashed lines are
guides to the eye. The inset compares the theoretical dependence of
the renormalization parameter F
0
a
on n
s
共solid line兲 with the data
共dots兲 calculated using Eq. 共2兲 from our g values.
FIG. 4. The elastic relaxation time versus electron density at a
temperature of 0.1 K, assuming m⫽ m
b
共dots兲, and taking into ac-
count the renormalization of m 共squares兲.The dashed lines are
guides to the eye. The inset shows the effective mass data of Fig. 3
as a function of the parameter r
s
⬘
. A linear fit 共dashed line兲 of the
low n
s
data has a slope which is equal to 1/2r
s
(n
c
).
BRIEF REPORTS PHYSICAL REVIEW B 66, 073303 共2002兲
073303-3
fields, which helps to understand why the Shubnikov–de
Haas oscillations survive near the metal-insulator transition,
as well as the origin of the oscillations of the metal-insulator
phase boundary as a function of 共perpendicular兲 magnetic
field.
25
In summary, we have studied the temperature-dependent
conductivity in a wide range of electron densities above the
critical electron density for the metal-insulator transition. Us-
ing the recent theory of interaction-driven corrections to
conductivity,
9
we extract Fermi-liquid parameters from the
experimental data and determine the many-body enhanced g
factor and the effective mass. The tendency of the slope A of
the temperature dependence of the conductivity to diverge
near the critical density is in agreement with the suggested
ferromagnetic instability in this electron system.
4,15
Unex-
pectedly, it is found to originate from the growing effective
mass rather than the g factor. In addition, the mass enhance-
ment is found to be responsible for the previously underes-
timated values of elastic scattering time near the metal-
insulator transition.
We gratefully acknowledge discussions with I. L. Aleiner,
P. T. Coleridge, L. I. Glazman, D. Heiman, J. P. Kotthaus, B.
N. Narozhny, and A. Punnoose. This work was supported by
NSF Grants No. DMR-9803440 and DMR-9988283, RFBR
Grant No. 01-02-16424, Forschungspreis of A. von Hum-
boldt Foundation, and the Sloan Foundation. T.M.K. ac-
knowledges support through NSF Grant No. PHY99-07949.
*
Permanent address: Institute of Solid State Physics, Cher-
nogolovka, Moscow District 142432, Russia.
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The formula for A has been obtained from Eqs. 共2.16b兲 and
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the resulting underestimation 共overestimation兲 of the g factor
共effective mass兲 does not exceed 10%.
20
I.L. Aleiner, private communication. For low intervalley scatter-
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␣
⫽ 8ifTⱗ⌬
v
and
␣
⫽ 16 if TⰇ ⌬
v
, where ⌬
v
is the valley
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timate for ⌬
v
⬇1.5 K.
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