Transcript No Slide Title

Happy Birthday Carlo!
Interplay between interactions and disorder
in two dimensions
Sergey Kravchenko
in collaboration with:
S. Anissimova, V.T. Dolgopolov, A. M. Finkelstein, T.M. Klapwijk,
A. Punnoose, A.A. Shashkin
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Outline
• Metal-insulator transition in Si MOSFETs and other 2D
systems
• Spin susceptibility, effective mass and g-factor
• What do theorists have to say? (more on this at 11:50)
• Comparison between experiments and recent RG theory
(more on this at 11:50)
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Why Si MOSFETs?
• large m*= 0.19 m0
• two valleys
• low average dielectric constant e=7.7
As a result, at low densities, Coulomb energy strongly exceeds Fermi
energy: EC >> EF
rs = EC / EF >10 can easily be reached in clean samples
EF
EF, EC
EC
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electron density
Strongly disordered Si MOSFET
(Pudalov et al.)
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Clean sample, much lower electron densities
Kravchenko, Mason, Bowker,
Furneaux, Pudalov, and
D’Iorio, PRB 1995
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Similar transition is also observed
in other 2D structures:
•p-Si:Ge (Coleridge’s group; Ensslin’s
group)
•p-GaAs/AlGaAs (Tsui’s group,
Boebinger’s group)
•n-GaAs/AlGaAs (Tsui’s group,
Stormer’s group, Eisenstein’s group)
•n-Si:Ge (Okamoto’s group, Tsui’s
group)
•p-AlAs (Shayegan’s group)
Hanein, Shahar, Tsui et al., PRL 1998
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In very clean samples, the transition is practically universal:
Klapwijk’s sample:
Pudalov’s sample:
6
resistivity r (Ohm)
10
5
10
11
4
(Note: samples from
different sources,
measured in different labs)
3
10
0
0.5
1
temperature T (K)
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-2
0.86x10 cm
0.88
0.90
0.93
0.95
0.99
1.10
10
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1.5
2
Magnetoresistance in a parallel magnetic field
5
10
15
1.01x10
-2
m
15
1.20x10
T = 30 mK
r (Ohm)
Bc
4
10
15
1.68x10
Bc
Bc 2.40x1015
Shashkin, Kravchenko,
Dolgopolov, and
Klapwijk, PRL 2001
15
3.18x10
3
10
0
2
4
6
8
B (Tesla)
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10
12
Spins become fully polarized
(Okamoto et al., PRL 1999;
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Vitkalov et al., PRL 2000)
Extrapolated polarization field, Bc,
vanishes at a finite electron density, nc
 B (meV)
5
0.2
B c
3
B c
 B (meV)
4
0.4
0
0.8
1.2
15
n (10
s
2
2
-2
Shashkin, Kravchenko,
Dolgopolov, and
Klapwijk, PRL 2001
m )
nnc
1
0
1.6
c
0
2
4
15
ns (10
6
-2
8
10
m )
Spontaneous spin polarization at nc?
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Comparison to other groups’ data
6
Shashkin et al, 2001
5
Vitkalov, Sarachik et al, 2001
4
3
B c
 B (meV)
Pudalov et al, 2002
2
nncc
1
0
0
2
4
6
8
10
12
n (1015 m-2)
s
nc @ 7.5.1010 cm-2 is sample-independent
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Measurements of thermodynamic magnetization
suggested by B. I. Halperin (1998); first implemented by O. Prus, M. Reznikov, U. Sivan et al. (2002)
1010 Ohm
+
Gate
Vg
Current amplifier
SiO2
Si
2D electron gas
Modulated magnetic field
B + dB
Ohmic contact
i ~ d/dB = - dM/dns
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Raw magnetization data:
current vs. gate voltage
d/dBinduced
= - dM/dn
1
2
0.5
-15
i (10 A)
B
1
0
0
-1
-0.5
B|| = 5 tesla
-2
-1
0
1
2
3
4
11
n (10
s
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d/dB ( )
1 fA!!
5
6
-2
cm )
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7
Raw magnetization
Integral of the
data:
previous
induced
slide
current
givesvs.
M gate
(ns): voltage
complete spin polarization
at ns=1.5x1011 cm-2
2
M (10  /cm )
1.5
B|| = 5 tesla
0.5
metal
insulator
11
B
1
0
0
2
n (10
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s
4
11
6
-2
cm )
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Spin susceptibility exhibits critical behavior near the
metal-insulator transition:
c ~ ns/(ns – nc)
7
6
magnetization data
magnetocapacitance data
integral of the master curve
transport data
c/c
0
5
4
3
insulator
2
n
c
T-dependent 1
regime
0.5 1 1.5 2 2.5 3 3.5
n (10
s
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11
-2
cm )
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Effective mass vs. g-factor
4
m/mb , g/g0
3
m/m
b
2
Shashkin, Kravchenko,
Dolgopolov, and Klapwijk,
PRB 66, 073303 (2002)
1
g/g
0
0
0
2
4
11
n (10
s
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6
-2
8
cm )
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10
Effective mass as a function of rs-2 in Si(111) and Si(100)
Si (111)
Si(111): peak mobility 2.5x103 cm2/Vs
Si(100): peak mobility 3x104 cm2/Vs
Si (100)
Shashkin, Kapustin, Deviatov, Dolgopolov, and Kvon, in preparation
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Partial summary:
• In strongly correlated 2D electron and hole systems, there appears to
exists a transition from an insulating to a metallic state
• The metallic state is wiped out by a parallel magnetic field strong
enough to polarize carriers’ spins
• Spin susceptibility tends to diverge at electron density close to (but
slightly below) the MIT point
• This effect is due to the strong enhancement of the effective mass
while the g-factor remains practically constant
• The relative enhancement of the effective mass seems to be function of
rs only
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What do theorists have to say?
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Corrections to conductivity due to electron-electron interactions
in the diffusive regime (Tt < 1)
 always insulating behavior
However, later this prediction was shown to be incorrect
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Sov. Phys. – JETP
Volume 57
1983
Influence of Coulomb interaction on the properties of disordered metals
A. M. Finkel'stein
L.D. Landau Inst. for Theoretical Phys., Acad. of Sci., Moscow, USSR




ln 1  F0
e
d  2  lnTt   1  3  1 

2 
F
0


2
 


behavior when interactions are weak (0.45  F0  0)
(1  F0  0.45)
 Altshuler-AronovMetallic
behavior
when
interactions
are
strong
Lee’s result
Finkelstein’s & CastellaniEffective strength of interactions grows as the temperature
decreases
DiCastro-Lee-Ma’s
term
 Insulating
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Same mechanism persists to ballistic regime (Tt > 1),
but corrections become linear in temperature
3F0
e 2 k BTt 
 T  
1
   1  F0




This is reminiscent of earlier Stern-Das Sarma’s result
e 2 k BTt
 T 
C (ns ) where C(ns)
 
 
<0
(However, Das Sarma’s calculations are not applicable to strongly interacting regime because
at rs>1, the screening length becomes smaller than the separation between electrons.)
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Recent development: two-loop RG theory
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disorder takes over
disorder

QCP

interactions
Punnoose and Finkelstein, Science
310, 289 (2005)
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metallic phase stabilized
by e-e interaction
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Experimental test
First, one needs to ensure that the system is in the diffusive regime (Tt < 1).
One can distinguish between diffusive and ballistic regimes by studying
magnetoconductance:
B
 B, T    
T 
2
- diffusive: low temperatures, higher disorder (Tt < 1).
2
B
- ballistic: low disorder, higher temperatures (Tt > 1).
 B, T  
T
The exact formula for magnetoconductance (Lee and Ramakrishnan, 1982):
 0.091e 2
 B, T   4

  h
2
2

 g B   B 

   2  2  1  
 k
 T 

 B
  

2
 g B B 

  1
 k T 
 B

Low-field magnetoconductance in the diffusive regime a
yields
strength of electron-electron interactionsF0
In standard Fermi-liquid notations,    
a
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1  F0
Experimental results (low-disordered Si MOSFETs;
“just metallic” regime; ns= 9.14x1010 cm-2):
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Temperature dependences of the
resistance (a) and strength of interactions (b)
This is the first time effective strength of interactions
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has been
seen to depend on T
Experimental disorder-interaction flow diagram of the 2D electron liquid
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Experimental vs. theoretical flow diagram
(qualitative comparison b/c the 2-loop theory was developed for multi-valley systems)
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Quantitative predictions of the one-loop RG for 2-valley systems
(Punnoose and Finkelstein, Phys. Rev. Lett. 2002)
rmax
r(T)
Solutions of the RG-equations for r << h/e2:
a series of non-monotonic curves r(T). After
rescaling, the solutions are described by a single
universal curve:
ρ(T) = ρmax R(η)
Tmax
For a 2-valley system (like Si MOSFET),
metallic r(T) sets in when 2 > 0.45
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(T)
η = ρmax ln(Tmax /T)
2 = 0.45
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rmax ln(T/Tmax)
Resistance and interactions vs. T
Note that the metallic behavior sets in when 2 ~ 0.45,
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Di Castro feastby
(Rome
2007)
exactly asCarlo
predicted
the
RG theory
Comparison between theory (lines) and experiment (symbols)
(no adjustable parameters used!)
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Similar experiments were also performed by Pudalov’s group
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2 vs. temperature extracted by Pudalov et al.
Knyazev et al., JETP Lett. (2006)
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g-factor grows as T decreases
4.5
g*  2(1    )
g*
4
ns = 9.9 x 1010 cm-2
3.5
3
2.5
0
1
2
T (K)
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3
4
“ballistic” value
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SUMMARY:
•
In the clean (ballistic) regime, Pauli spin susceptibility critically grows with
a tendency to diverge near a certain electron density nc suggesting a
quantum phase transition at ns=nc
•
However, upon approaching to nc, one leaves the clean regime and enters
the diffusive regime, where the physics is governed by the interplay
between interactions and disorder. In this regime, both interactions and
disorder become temperature-dependent
•
RG theory gives quantitatively correct description of the metallic state. In
particular, in excellent agreement with theory, the metallic behavior sets in
once 2 > 0.45!
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