Metal-Insulator Transition Metal-Insulator Transition in 2D Electron Systems: in 2D Electron Systems: Recent Progress Recent Progress Experiment: Dima Knyazev, Oleg Omel’yanovskii Vladimir Pudalov Theory: Igor Burmistrov, Nickolai Chtchelkatchev chegolev memorial conference. Oct. 11-16, 2009 P.N. Lebedev Physical Institute, Moscow L.D. Landau Institute, Chernogolovka
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Metal-Insulator Transition in 2D Electron Systems: Recent Progress Experiment: Dima Knyazev, Oleg Omel’yanovskii Vladimir Pudalov Theory: Igor Burmistrov,
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Metal-Insulator Transition in 2D Metal-Insulator Transition in 2D Electron Systems: Recent Electron Systems: Recent
ProgressProgress
Experiment:
Dima Knyazev,Oleg Omel’yanovskiiVladimir Pudalov
Theory:
Igor Burmistrov, Nickolai Chtchelkatchev
Schegolev memorial conference. Oct. 11-16, 2009
P.N. Lebedev Physical Institute, Moscow
L.D. Landau Institute, Chernogolovka
Groundstate(s) of the 2D electron liquid (T 0)
Major question to be addressed:
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
1.1.1.1. ClassicalClassical charge transportcharge transport
1. T >>hD. Phonon scattering 1/T
2. T << hD. Phonon scattering 1/T 5
3. T << TF. e-e scattering 1/T 2
4. T << TF. Impurity scattering ConstNote (a): Note (a): There is no σ(T) dependence in the T=0 limit !
(within the classical approximation, for non-interacting electrons )
+ Umklapp
1.2.Semiclassical concept of 1.2.Semiclassical concept of transport (1960)transport (1960)
Ioffe-Regel criterion
A.F. Ioffe and A.R. Regel, Prog. Semicond. 4, 237 (1960).
Abram F. Ioffe
Anatoly R. Regel
“minimum metallic conductivity”
2
2
1
25.82kΩD
e
h
Fkl
1~min
h
e
k
lek
m
ne
F
F22222 )2/(
Nevil Mott (1905-96)
min
cn
cn
min
Possible behavior of resistivity (dimensionality is irrelevant):
metalic
0 T
insulating
0 T
insulating
metalic
Semiclassical picture: MIT at T =
0 (1970’s)
All electrons in 2D become localized at T 0
1.3. Quantum concept of transport (1979):
E.Abrahams
T.V. Ramakrishnan
A
B
Competition between dimensionality and Competition between dimensionality and interefrenceinterefrence
Interference of electron waves causes localization
2
ln( )D
eT
h
for ln(1/T)
Note (b)Note (b)
P.W. Anderson
D.Khmelnitskii
L.P.Gorkov
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
1-Parameter scaling equation
( ) ; ln ( / ).tr
dgg L l
d
( ) 0critg g At the MIT:
g(L) – dimensionless conductance for a sample (size L) in units of e2/h
For 2D system: β is always <0; there is no metallic state and no MIT
TlL
1~
One-parameter scaling and experiment
0,1 10,1
1
10
Si39
(
h/e2 )
Temperature (K)1 2 3
0,1
1
10Si39
(
h/e2 )
Temperature (K)
Note (c)Note (c): The sign of dρ/dT at finite T is not indicative of the metallic or insulating state
Low-mobility sample (μ=1.5103cm2/Vs)
n
2.Metal-insulator transition in2.Metal-insulator transition in high high mobility 2D systemmobility 2D system
0 1 2 3 40.1
1
10
100
Si-62
(
h/e2 )
T (K)
S.Kravchenko, VP, et al., PRB 50, 8039 (1994)
N ~1011cm-2
dens
ity =4,5m2/Vs
Similar (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.
Y.Hanein et al. PRL (1998)Papadakis, Shayegan, PRB (1998)
n-AlAs-GaAs p-GaAs/AlAs
(
/)
(
/)
There is no metallic state and no MIT - There is no metallic state and no MIT - in the in the noninteractingnoninteracting 2 2D systemsD systems
Spin-orbit interaction ?
Electron-phonon interaction ?
Too low temperature and too weak e-ph coupling
Not renormalized
Electron-electron interaction
0 1 2 3 40.1
1
10
100
Si-62
(
h/e2 )
T (K)
High mobility
Eee/EF= rs~10
dens
ity
=4,5m2/Vs
13
e-e interaction in Si-MOS structurese-e interaction in Si-MOS structures
Note1:Note1: Within the concept of the e-e correlations, the role of high high
mobilitymobility in the 2D MIT becomes transparent
The high mobility:
• Increases and, hence, the amplitude of interaction corrections ( T);
• Translates down the critical density range (decreases the density of impurities ni)
• Increases the magnitude of interaction effects ( F0n).
2.2. Problems of the data (mis)interpretation2.2. Problems of the data (mis)interpretation
If “MIT” is a QPT, it is expected:
• c to be universal,
•scaling persists to the lowest T
• horizontal “separatrix” c f(T)
• z, are universal
Experimentally, however,• c=0.55 is sample dependent,• z =0.9 2 is sample dependent, • reflection symmetry fails at low Tand at high T>2Kins =cexp(T0/T)p1 (p1=0.5 1)met =cexp(-T0/T)p2+0 (p2=0.5 1)• separatrix is T-dependent
The failure of the OPST approach is not surprising: interactionsHow to proceed in the 2-parameter problem ?
Which parameters should be universal ?
Definitions of the critical density, critical resistivity etc. ?
In analogy with the 1-parameter scaling:
0 1 2 3 40.1
1
10
100
Si-62
(
h/e2 )
T (K)
3. Solving the puzzle of the metallic-like conduction at g >>e2/h (2000-2004)
Ballistic interaction regime T>>1
QuantifyingQuantifying e-ee-e interaction ininteraction in 2D (2000-2004)2D (2000-2004)
Fi a,s – FL-constants (harmonics) of the e-e interaction
N.Klimov, M.Gershenson, VP, et al. PRB 78, 195308 (2008)
1 2 3 4 5 6 7 8-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
F0
rS
No parameter comparisonNo parameter comparison of the data and theory in the ballistic of the data and theory in the ballistic regimeregime T >>1 (2002-2004):
0 2 4 6
50
60
70
80
90
100
110
120
(e2 /h
)
T (K)
Exper.: VP, Gershenson, Kojima, et al. PRL 93 (2004)
Theory: Zala, Narozhny, Aleiner, PRB (2001-2002)
0 1 2 3 40.1
1
10
100
Si-62
(
h/e2 )
T (K)VP et al. JETP Lett. (1998)
Successful description of the transport in terms of e-e interaction effects in the “high density/low disorder ( <<1) regime
motivated us to apply the same ideas to the regime of low density/strong disorder ( ~1)
4. Transport in the critical regime4. Transport in the critical regime
Theory: Two- parameter renorm. group equations
02
01
ln
F
F
L
l
1
LT
is in units of e2/h
Interplay of disorder and interaction
nv=2
Exact RG results forExact RG results for BB=0=0
One-loop,
A.A.Finkelstein, Punnoose, Phys.Rev.Lett. (2005)
max
Transport data Transport data in the critical regimein the critical regime
Magnetotransport in the critical regime
1 2 3 4
0.8
0.9
1.0
1.1
1.2
Si2 , n = 1.075
B|| = 0
(h/
e2 )
T (K)
B|| = 2.5T
Quantitative agreement of the
data with theory
Knyazev, Omelyanovskii, Burmistrov, Pudalov, JETP Lett. (2006)
Current understanding of the 2D systemsCurrent understanding of the 2D systems “Metallic” conduction in 2D systems for >> 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 nv=2 (or at large 2 for nv=1), whereas M-I-T is a quantum M-I-T is a quantum phase transitionphase transition
Summary
More realistic RG calculations are needed (finite nv, two-loop)