Transcript 2D Metal-insulator transition revisited: Experimental

Metal-Insulator Transition in 2D
Electron Systems: Recent
Progress
P.N. Lebedev Physical Institute,
Moscow
Experiment:
Dima Knyazev,
Oleg Omel’yanovskii
Vladimir Pudalov
Schegolev memorial conference.
L.D. Landau Institute,
Chernogolovka
Theory:
Igor Burmistrov,
Nickolai Chtchelkatchev
Oct. 11-16, 2009
Major question to be addressed:
Groundstate(s) of the 2D electron liquid (T  0)
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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. Classical charge transport
1. T >>hwD. Phonon scattering
s 1/T
2. T << hwD. Phonon scattering
s 1/T 5
3. T << TF. e-e scattering + Umklapp
s 1/T 2
4. T << TF. Impurity scattering
s Const
Note (a): There is no σ(T) dependence in the T=0 limit !
(within the classical approximation, for non-interacting electrons )
1.2.Semiclassical concept of transport
(1960)
lmin
1
~
kF
Ioffe-Regel criterion
A.F. Ioffe and A.R. Regel, Prog.
Semicond. 4, 237 (1960).
ne 
(k
s 

m
2
2
F
/ 2 )e l
e

k F
h
2
2
2
Abram F. Ioffe
“minimum metallic
conductivity”
s 2D
Nevil Mott (1905-96)
e2
1
 
h 25.82kΩ
Anatoly R. Regel
Semiclassical picture: s
MIT at T = 0
s
(1970’s)
 s min
nc

 s min
Possible behavior of resistivity
(dimensionality is irrelevant):
insulating
metalic
0T

nc
insulating
metalic
0T
1.3. Quantum concept of transport (1979):
E.Abrahams
Competition between dimensionality and
interefrence
B Interference of electron waves
causes localization
A
s sD 
T.V. Ramakrishnan
e
2
h
Note (b)
ln(T )
All electrons in 2D
become localized at
T0
D.Khmelnitskii
for ln(1/T)  s
P.W. Anderson
L.P.Gorkov
1.4. Scaling ideas in the quantum transport picture:
Thouless (1974, 77);
Wegner (’79).
Abrahams, Anderson, Licciardello, Ramakrishnan (’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
dg
  ( g ) ;   ln ( L / ltr ).
d
At the MIT:
For 2D system:
1
L ~ l 
T
 ( g  gcrit )  0
β is always <0;
there is no metallic state and no MIT
One-parameter scaling and experiment
Low-mobility sample (μ=1.5103cm2/Vs)
10
10
2
 (h/e )
Si39
2
 (h/e )
Si39
1
0,1
n
1
2
Temperature (K)
3
1
0,1
0,1
1
Temperature (K)
Note (c): The sign of dρ/dT at finite T is not indicative of the
metallic or insulating state
density
2.Metal-insulator transition in high mobility
2D system
=4,5m2/Vs
2
 (h/e )
100
10
N ~1011cm-2
1
Si-62
0.1
0
1
2
T (K)
3
4
S.Kravchenko, VP, et al.,
PRB 50, 8039 (1994)
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.
Papadakis, Shayegan, PRB (1998)
p-GaAs/AlAs
 (/)
 (/)
n-AlAs-GaAs
Y.Hanein et al. PRL (1998)
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
density
High mobility
=4,5m2/Vs
2
 (h/e )
100
10
Eee/EF= rs~10
1
Si-62
0.1
0
1
2
T (K)
3
4
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  and, hence, the amplitude of interaction
corrections ( T);
• Translates down the critical density range (decreases the
density of impurities ni)
s
• Increases the magnitude of interaction effects ( F0 (n) T).
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2.1. Signatures of the critical phenomenon - QPT
•Mirror reflection
symmetry:
(Dn,T)/c = c/(-Dn,T)
•data scaling
/c= 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
MIT in 2D system
(1994)
100
2
 (h/e )
=35,000cm2/Vs
10
1
Si-62
0.1
0
1
2
T (K)
3
4
MIT in 2D system
(1994)
100
2
 (h/e )
=35,000cm2/Vs
10
1
Si-62
0.1
0
1
2
T (K)
3
4
2.2. Problems of the data (mis)interpretation
In analogy with the 1-parameter scaling:
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=0.55 is sample dependent,
• z =0.9  2 is sample dependent,
• reflection symmetry fails at low T
and at high T>2K
ins =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: interactions
How to proceed in the 2-parameter problem ?
Which parameters should be universal ?
Definitions of the critical density, critical resistivity etc. ?
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3. Solving the puzzle of the metallic-like conduction at
g >>e2/h (2000-2004)
2
 (h/e )
100
Ballistic interaction regime
T>>1
10
1
Si-62
0.1
0
1
2
T (K)
3
4
Quantifying e-e interaction in 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
s
-0,1
-0,3
F0
s
-0,2
-0,4
-0,5
-0,6
1
2
3
N.Klimov, M.Gershenson, VP, et al.
PRB 78, 195308 (2008)
4
rS
5
6
7
8
No parameter comparison of the data and theory in the ballistic
regime T >>1
(2002-2004):
Theory: Zala, Narozhny, Aleiner,
PRB (2001-2002)
120
110
Exper.: VP, Gershenson, Kojima, et
al. PRL 93 (2004)
90
2
s (e /h)
100
80
70
60
50
0
2
4
T (K)
6
4. Transport in the critical regime
motivated us to apply the same
ideas to the regime of low
density/strong disorder ( ~1)
2
 (h/e )
100
10
1
Successful description of the
transport in terms of e-e
interaction effects in the “high
density/low disorder ( <<1)
regime
Si-62
0.1
0
1
2
T (K)
3
VP et al. JETP Lett. (1998)
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Theory: Two- parameter renorm. group equations
 F0s
2 
1  F0s
L
  ln
l
s is in units of e2/h
1
L
T
Interplay of disorder and interaction
Exact RG results for B=0
One-loop, nv=2
max
A.A.Finkelstein, Punnoose, Phys.Rev.Lett. (2005)
Transport data in the critical regime
3
2
 (h/e )
1000
Si 62
2
 (h/e )
100
2
1
10
0
1
2
3
T (K)
1
0.1
0
1
2
3
4
T (K)
5
6
7
4
Magnetotransport in the critical regime
2
 (h/e )
1.2
B|| = 2.5T
1.1
Quantitative
agreement of the
data with theory
1.0
B|| = 0
0.9
Knyazev, Omelyanovskii,
Burmistrov, Pudalov,
JETP Lett. (2006)
Si2 , n = 1.075
0.8
1
2
3
4
T (K)
RG equation in B|| field:
Burmistrov, Chtchelkatchev, JETP Lett. (2006)
Anissimova, Kravchenko,
Punnoose, Finkel'stein,
Klapwijk, Nature Phys. 3,
707 (2007)
2(T) – comparison with theory
Quantitative
agreement with
theory for both,
(T) and 2(T)
0.7
Finkelstein's theory
theory
Si2
experiment
0.7
Si6-14
0.6
2
2
0.6
0.5
0.5
0.4
0.4
2
-2
3
-1
4
T (K)
0
X= maxln(T/Tmax)
1
Anissimova, Kravchenko, Punnoose, Finkel'stein,
Klapwijk, Nature Phys. 3, 707 (2007)
Interplay of disorder and interaction
RG-result in the two-loop approximation
Finkelstein, Punnoose, Science (2005)
No crossover “2D metal” – localized state
6. Fixed point (QCP)
/c
Two-loop approximation, nv=

Data analysis in the vicinity of the fixed point
Linearising RG equations close to the fixed point
s = 2 = 0:
d
0
dl
(n  nc )  T 
X
 
nc  T0 

T 
Y  
 T1 

 (T , n)  ( X , Y )
 = p/(2)
 = -py/2
p – for heat capacity,
 – for correlation length
 (T , n)  e X [1  Y ]
Knyazev, Omelyanovskii, Pudalov, Burmistrov, PRL 100, 046405 (2008)
Scaling of the /c(T) data
(T)exp(X)/c
0
Si2
 (T , n)  c (T , n)e X
0.6
0.10
0.896
0.5
  
T
0 
c (T , n)  c 1   
  T1 

0.12
0.64
0.941
0.963
0.874
0.62
0.918
separatrix
0.10
0.15
0.20
T/T
0.25
0.30
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  must be < 1.




R(T) data scaling in a wide range of (X,Y >1)
( X , Y )  exp  f1 ( X ) f2 (Y )
Fits 64000 data
points to within 4%
over the range
|X|<5, Y<3
f1= -X+0.07X2+0.01X3
f2 =
(1-Y+1.48Y2)
(1+1.9Y2+1.7Y3)
Reflection symmetry holds within
(0.8%) for |X|<0.5, Y<0.7
separatrix
Empiric scaling function R(X,Y) – data spline for 5 samples
Knyazev, Omelyanovskii, Pudalov, Burmistrov, PRL100, 046405 (2008)
Summary
Current understanding of the 2D systems
 “Metallic” conduction in 2D systems for s >> 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 2parameter 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 phase
transition
More realistic RG calculations are needed (finite nv, two-loop)
Thank you for attention!
Theory:
Sasha Finkelstein
Boris Al’tshuler
Igor Aleiner
Dmitrii Maslov
Valentin Kachorovskii
Nikita Averkiev
Alex Punnoose
Experiment
- Texas U.
- Columbia U.
- Columbia U.
- U.of Florida
- Ioffe Inst.
- Ioffe Inst.
- Lucent
Dima Rinberg
Sergei Kravchenko
Mary D’Iorio
John Campbell
Robert Fletcher
Gerhard Brunthaler
Adrian Prinz
Misha Reznikov
Kolya Klimov
Misha Gershenson
Harry Kojima
Nick Busch
Sasha Kuntsevich
- Harvard Univ.
- SEU, Boston,
- NRC, Canada
- NRC, Canada
- Queens Univ.
- JKU, Linz
- JKU, Linz
- Technion, Haifa
- Rutgers Univ.
- Rutgers Univ.
- Rutgers Univ.
- Rutgers Univ.
-Lebedev Inst.