Metal-Insulator%20Transition%20in%202D%20Electron%20Systems:%20Recent%20Progress

1
Metal-Insulator Transition in 2D Electron
Systems Recent Progress
P.N. Lebedev Physical Institute, Moscow
L.D. Landau Institute, Chernogolovka
Experiment Dima Knyazev, Oleg
Omelyanovskii Vladimir Pudalov
Theory Igor Burmistrov, Nickolai
Chtchelkatchev
Schegolev memorial conference. Oct. 11-16, 2009
2
Major question to be addressed
Groundstate(s) of the 2D electron liquid (T ? 0)

• Outline
• Historical intro classical, semiclassical,
  quantum transport and 1-parameter scaling
• MIT in high mobility 2D systems
• The puzzle of the metallic-like conduction
• Quantifying e-e interaction in 2D
• Transport in the critical regime 2 parameter RG
  theory
• Data analysis in the vicinity of the fixed point
• Data analysis in the vicinity of the fixed point

3
1.1. Classical charge transport

• 1. T gtgthwD. Phonon scattering s
  ?1/T
• 2. T ltlt hwD. Phonon scattering s
  ?1/T 5
• T ltlt TF. e-e scattering s ?1/T 2
• 4. T ltlt TF. Impurity scattering s ?Const

Umklapp
Note (a) There is no s(T) dependence in the T0
limit ! (within the classical approximation,
for non-interacting electrons )
4
1.2.Semiclassical concept of transport (1960)
Ioffe-Regel criterion A.F. Ioffe and A.R. Regel,
Prog. Semicond. 4, 237 (1960).
Abram F. Ioffe
minimum metallic conductivity
Anatoly R. Regel
Nevil Mott (1905-96)
5
Semiclassical picture MIT at T 0 (1970s)
Possible behavior of resistivity (dimensionality
is irrelevant)
6
1.3. Quantum concept of transport (1979)
Competition between dimensionality and
interefrence
Interference of electron waves causes localization
B
E.Abrahams
A
Note (b)
T.V. Ramakrishnan
All electrons in 2D become localized at T ? 0
D.Khmelnitskii
for ln(1/T?) ? ?
P.W. Anderson
L.P.Gorkov
7
1.4. Scaling ideas in the quantum transport
picture Thouless (1974, 77) Abrahams,
Anderson, Licciardello, Ramakrishnan (79)
Wegner (79).
Renormalization Group transformation The block
size is increased from ltr to L
g(L) dimensionless conductance for a sample
(size L) in units of e2/h
1-Parameter scaling equation
At the MIT
For 2D system ß is always lt0 there is no
metallic state and no MIT
8
One-parameter scaling and experiment
Low-mobility sample (µ1.5?103cm2/Vs)
n
Note (c) The sign of d?/dT at finite T is not
indicative of the metallic or insulating state
9
2.Metal-insulator transition in high mobility 2D
system
?4,5m2/Vs
density
N 1011cm-2
S.Kravchenko, VP, et al., PRB 50, 8039 (1994)
10
Similar r(T) behavior was found in many other 2D
systems p-GaAs, n-GaAs, p-Si/SiGe,
n-Si/SiGe, n-SOI, p-AlAs/GaAs, etc.
n-AlAs-GaAs
p-GaAs/AlAs
? (?/?)
? (?/?)
Y.Hanein et al. PRL (1998)
Papadakis, Shayegan, PRB (1998)
11
There is no metallic state and no MIT - in the
noninteracting 2D systems

• Spin-orbit interaction ?

Not renormalized

• Electron-phonon interaction ?

Too low temperature and too weak e-ph coupling
Electron-electron interaction
12
High mobility
?4,5m2/Vs
density
Eee/EF rs10
13
e-e interaction in Si-MOS structures
Note1 Within the concept of the e-e
correlations, the role of high mobility in the 2D
MIT becomes transparent

• The high mobility
• Increases t and, hence, the amplitude of
  interaction corrections (? Tt)
• Translates down the critical density range
  (decreases the density of impurities ni)
• Increases the magnitude of interaction effects
  (? F0s(n) Tt).

14
2.1. Signatures of the critical phenomenon - QPT
Mirror reflection symmetry ?(Dn,T)/?c
?c/?(-Dn,T) data scaling
r/rc f T/T0(n) Critical behavior T0 ?
n-nc-z?
Symmetry holds here and is missing outside
S.V.Kravchenko, W.E.Mason, G.E.Bowker,
J.E.Furneaux, V.M.Pudalov, M.D'Iorio, PRB 1995
15
MIT in 2D system
(1994)
?35,000cm2/Vs
16
MIT in 2D system
(1994)
?35,000cm2/Vs
17
2.2. Problems of the data (mis)interpretation
In analogy with the 1-parameter scaling

• If MIT is a QPT, it is expected
• rc to be universal,
• scaling persists to the lowest T
• horizontal separatrix rc ? f(T)
• z, ? are universal

• Experimentally, however,
• rc0.5?5 is sample dependent,
• z? 0.9 ? 2 is sample dependent,
• reflection symmetry fails at low T
• and at high Tgt2K
• rins ?cexp(T0/T)p1 (p10.5 ?1)
• rmet ?cexp(-T0/T)p2r0 (p20.5 ?1)
• separatrix is T-dependent

The failure of the OPST approach is not
surprising interactions How to proceed in
the 2-parameter problem ? Which parameters
should be universal ? Definitions of the
critical density, critical resistivity etc. ?
18
3. Solving the puzzle of the metallic-like
conduction at g gtgte2/h (2000-2004)
Ballistic interaction regime Ttgtgt1
19
Quantifying e-e interaction in 2D (2000-2004)
Fi a,s FL-constants (harmonics) of the e-e
interaction
20

• Strong growth in ? ? mg, m and
  g
• as n decreases

V.M.Pudalov, M.E.Gershenson, H.Kojima,
Phys.Rev.Lett. 88, 196404 (2002)
21
Fermi-liquid parameter F0s
N.Klimov, M.Gershenson, VP, et al. PRB 78, 195308
(2008)
22
No parameter comparison of the data and theory in
the ballistic regime T? gtgt1 (2002-2004)
Theory Zala, Narozhny, Aleiner, PRB (2001-2002)
Exper. VP, Gershenson, Kojima, et al. PRL 93
(2004)
23
4. Transport in the critical regime
motivated us to apply the same ideas to the
regime of low density/strong disorder (r 1)
Successful description of the transport in terms
of e-e interaction effects in the high
density/low disorder (r ltlt1) regime
VP et al. JETP Lett. (1998)
24
Theory Two- parameter renorm. group equations
s is in units of e2/h
Interplay of disorder and interaction
25
Exact RG results for B0
nv2
One-loop,
rmax
A.A.Finkelstein, Punnoose, Phys.Rev.Lett. (2005)
26
Transport data in the critical regime
27
Magnetotransport in the critical regime
Quantitative agreement of the data with theory
Knyazev, Omelyanovskii, Burmistrov, Pudalov,
JETP Lett. (2006)
Anissimova, Kravchenko, Punnoose, Finkel'stein,
Klapwijk, Nature Phys. 3, 707 (2007)
RG equation in B field Burmistrov,
Chtchelkatchev, JETP Lett. (2006)
28
g2(T) comparison with theory
Quantitative agreement with theory for both,
r(T) and g2(T)
29
Anissimova, Kravchenko, Punnoose, Finkel'stein,
Klapwijk, Nature Phys. 3, 707 (2007)
30
Interplay of disorder and interaction
RG-result in the two-loop approximation
Finkelstein, Punnoose, Science (2005)
No crossover 2D metal localized state
31
6. Fixed point (QCP)
Two-loop approximation, nv?
r/rc
g2
32
Data analysis in the vicinity of the fixed point
Linearising RG equations close to the fixed
point bs bg2 0
k p/(2n)
z -py/2
p for heat capacity, n for correlation length
Knyazev, Omelyanovskii, Pudalov, Burmistrov, PRL
100, 046405 (2008)
33
Scaling of the r/rc(T) data
separatrix
Note The quality of the data scaling relative
the tilted separatrix rc(T)
Separatrix is a power low function, with no
maxima and inflection. Exponent z must be lt 1.
34
R(T) data scaling in a wide range of (X,Y gt1)
Fits 64000 data points to within 4 over the
range Xlt5, Ylt3
f1 -X0.07X20.01X3
(1-Y1.48Y2) (11.9Y21.7Y3)
f2
Reflection symmetry holds within (0.8) for
Xlt0.5, Ylt0.7
separatrix
35
Empiric scaling function R(X,Y) data spline for
5 samples
Knyazev, Omelyanovskii, Pudalov, Burmistrov,
PRL100, 046405 (2008)
36
Current understanding of the 2D systems
Summary

• Metallic conduction in 2D systems for ? gtgt
  e2/h - the result of e-e interactions
• Interplay of disorder and e-e interaction
  radically changes the common believe that the
  metallic state can not exist in 2D
• Agreement of the data with RG theory and the
  2-parameter data scaling

• In RG theory, the 2D metal always exist for nv2
  (or at large g2 for nv1), whereas M-I-T is a
  quantum phase transition

More realistic RG calculations are needed (finite
nv, two-loop)
37
Thank you for attention!
Theory Sasha Finkelstein - Texas U. Boris
Altshuler - Columbia U. Igor Aleiner
- Columbia U. Dmitrii Maslov
- U.of Florida Valentin Kachorovskii -
Ioffe Inst. Nikita Averkiev - Ioffe
Inst. Alex Punnoose - Lucent
Experiment Dima Rinberg - Harvard
Univ. Sergei Kravchenko - SEU,
Boston, Mary DIorio - NRC,
Canada John Campbell - NRC,
Canada Robert Fletcher - Queens Univ.
Gerhard Brunthaler - JKU, Linz Adrian
Prinz - JKU, Linz Misha
Reznikov - Technion, Haifa Kolya Klimov
- Rutgers Univ. Misha Gershenson
- Rutgers Univ. Harry Kojima -
Rutgers Univ. Nick Busch -
Rutgers Univ. Sasha Kuntsevich -Lebedev Inst.