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).

2.1. Signatures of the critical phenomenon - QPT

•Mirror reflection symmetry: (n,T)/c = c/(-n,T)

•data scaling /c= f [T/T0(n)]

•Critical behavior T0 |n-nc|-z

S.V.Kravchenko, W.E.Mason, G.E.Bowker, J.E.Furneaux, V.M.Pudalov, M.D'Iorio, PRB 1995

Symmetry: holds hereand is missing outside

0 1 2 3 40.1

1

10

100

Si-62

(

h/e2 )

T (K)

=35,000cm2/Vs

MIT in 2D system

(1994)

0 1 2 3 40.1

1

10

100

Si-62

(

h/e2 )

T (K)

=35,000cm2/Vs

MIT in 2D system

(1994)

17

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

Strong growth in * m*g*, m* and g* as n decreases

V.M.Pudalov, M.E.Gershenson, H.Kojima, Phys.Rev.Lett. 88, 196404 (2002)

Fermi-liquid parameter F0

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)

Anissimova, Kravchenko, Punnoose, Finkel'stein, Klapwijk, Nature Phys. 3, 707 (2007)RG equation in B|| field:

Burmistrov, Chtchelkatchev, JETP Lett. (2006)

2(T) – comparison with theory

Quantitative agreement with theory for both, (T) and 2(T)

-2 -1 0 1

0.4

0.5

0.6

0.7

Finkelstein's theory Si2 Si6-14

max

ln(T/Tmax

)

2

X=

Anissimova, Kravchenko, Punnoose, Finkel'stein, Klapwijk, Nature Phys. 3, 707 (2007)

InterplayInterplay of disorderof disorder andand interactioninteraction

No crossoverNo crossover “2D metal”“2D metal” –– localized statelocalized state

RG-resultRG-result inin the twothe two--looploop approximationapproximation

Finkelstein, Punnoose, Science (2005)

6. Fixed6. Fixed point (QCP)point (QCP)

Two-loop approximation, nv=

c

Data analysis in the vicinity of the fixed pointData analysis in the vicinity of the fixed point

( , ) ( , )T n X Y

0

1

( )c

c

n n TX

n T

TY

T

0d

dl

Linearising RG equations close to the fixed point= 2 = 0:

= p/(2)

= -py/2

p – for heat capacity, – for correlation length

( , ) [1 ]XT n e Y

Knyazev, Omelyanovskii, Pudalov, Burmistrov, PRL 100, 046405 (2008)

Scaling of the c(T) data

( , ) ( , )cXT n T n e

0

1

( , ) 1c c

TT n

T

Note: The quality of the data scaling relative the tilted separatrix rc(T)

0.10 0.15 0.20 0.25 0.30

0.5

0.6

0.10 0.12

0.62

0.64

(

T)e

xp(X

)/ c0

Si2

0.896

0.941

0.963

0.874

0.918

Separatrix – is a power low function, with no maxima and inflection.

Exponent must be < 1.

separatrix

R(T) data scaling in a wide range of (X,Y >1)

Reflection symmetry holds within (0.8%) for |X|<0.5, Y<0.7

1 2( , ) exp ( ) ( )X Y f X f Y

Fits 64000 data points to within 4%

over the range|X|<5, Y<3

separatrix

f1= -X+0.07X2+0.01X3

(1-Y+1.48Y2)

(1+1.9Y2+1.7Y3)f2=

Empiric scaling function R(X,Y) – data spline for 5 samples

Knyazev, Omelyanovskii, Pudalov, Burmistrov, PRL100, 046405 (2008)

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)

Thank you for attention! Thank you for attention!

Theory:

Sasha Finkelstein - Texas U.Boris Al’tshuler - Columbia U.Igor Aleiner - Columbia U.Dmitrii Maslov - U.of FloridaValentin Kachorovskii - Ioffe Inst.Nikita Averkiev - Ioffe Inst.Alex Punnoose - Lucent

Experiment

Dima Rinberg - Harvard Univ.Sergei Kravchenko - SEU, Boston,Mary D’Iorio - NRC, CanadaJohn Campbell - NRC, CanadaRobert Fletcher - Queens Univ. Gerhard Brunthaler - JKU, LinzAdrian Prinz - JKU, Linz Misha Reznikov - Technion, HaifaKolya Klimov - Rutgers Univ.Misha Gershenson - Rutgers Univ.Harry Kojima - Rutgers Univ.Nick Busch - Rutgers Univ.Sasha Kuntsevich-Lebedev Inst.