Transcript No Slide Title

Effects of electron-electron interactions
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
• Do interactions modify “all states are localized in 2D” paradigm?
(or: what happens to the Anderson localization in the presence of interactions?)
• Samples
• What do experiments show?
• “Clean” regime: diverging spin susceptibility
• “Dirty” regime: interplay between disorder and interactions
• Summary
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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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Zeitschrift fur Physik B (Condensed Matter) -- 1984 -- vol.56, no.3, pp. 189-96
Weak localization and Coulomb interaction in disordered systems
Finkel'stein, A.M.
L.D. Landau Inst. for Theoretical Phys., Acad. of Sci., Moscow, USSR




ln 1  F0
e
  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 r s>1, the screening length becomes smaller than the separation between electrons.)
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• Do interactions modify “all states are localized in 2D” paradigm?
(or: what happens to the Anderson localization in the presence of interactions?)
• Samples
• What do experiments show?
• “Clean” regime: diverging spin susceptibility
• “Dirty” regime: interplay between disorder and interactions
• Summary
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silicon MOSFET
Al
SiO2
p-Si
energy
2D electrons
conductance band
chemical potential
valence band
+ _
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distance
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into the sample (perpendicular
to the surface)
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
• Do interactions modify “all states are localized in 2D” paradigm?
(or: what happens to the Anderson localization in the presence of interactions?)
• Samples
• What do experiments show?
• “Clean” regime: diverging spin susceptibility
• “Dirty” regime: interplay between disorder and interactions
• Summary
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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
… in contrast to strongly disordered samples:
clean sample:
disordered sample:
6
5
10
resistivity
r
(Ohm)
10
11
4
3
10
-2
0.86x10 cm
0.88
0.90
0.93
0.95
0.99
1.10
10
0
0.5
1
1.5
2
temperature T (K)
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The effect of magnetic field
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5
10
15
1.01x10
-2
m
15
r (Ohm)
1.20x10
T = 30 mK
4
10
15
1.68x10
15
2.40x10
15
3.18x10
3
10
0
2
4
6
8
10
B (Tesla)
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12
Shashkin, Kravchenko,
Dolgopolov, and
Klapwijk, PRL 2001
Magnetic field, by aligning spins, changes metallic R(T) to insulating:
Shashkin et al., 2000
6
6
r (W)
10
10
0.765
0.780
0.795
0.810
0.825
1.095
1.125
1.155
1.185
1.215
5
10
5
4
10
10
10
4
0
0.3
0.6
0.9
1.2
0
T (K)
0.3
0.6
0.9
T (K)
=0
> Bsat
Such aBdramatic
reaction on parallelBmagnetic
field suggests unusual magnetic properties
(spins aligned)
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1.2
• Do interactions modify “all states are localized in 2D” paradigm?
(or: what happens to the Anderson localization in the presence of interactions?)
• Samples
• What do experiments show?
• “Clean” regime: diverging spin susceptibility
• “Dirty” regime: interplay between disorder and interactions
• Summary
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How to study magnetic properties of
2D electrons?
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Method 1: 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
10
B (Tesla)
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12
Spins become fully polarized
(Okamoto et al., PRL 1999;
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 @ 8.1010 cm-2 is sample-independent
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Method 2: 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
Modulated magnetic field
B + B
Ohmic contact
Si
2D electron gas
i ~ d/dB = - dM/dns
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Magnetization of non-interacting electrons
spin-up
spin-down
gBB
dM
dns
M
B
ns
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ns
Magnetic field of the full spin polarization vs. ns
spontaneous
spin polarization
non-interacting
system at nc:
Bc = h2ns/
/2
g*m*
mb
BB
M = Bx ns =
0
B
nc
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Bns for B > Bc
dM
dns B > B
c
Bc
0
Bns B/Bc for B < Bc
ns
B < Bc
ns
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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
-2
cm )
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6
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
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4
11
6
-2
cm )
Summary of the results obtained by four
independent methods (including transport)
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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
cm )
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8
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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disorder
Anderson insulator
Disorder increases at low density
due to reduced screening
paramagnetic Fermi-liquid
Wigner crystal?
Liquid ferromagnet?
Density-independent disorder
electron density
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• Do interactions modify “all states are localized in 2D” paradigm?
(or: what happens to the Anderson localization in the presence of interactions?)
• Samples
• What do experiments show?
• “Clean” regime: diverging spin susceptibility
• “Dirty” regime: interplay between disorder and interactions
• Summary
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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)
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2 = 0.45
rmax ln(T/Tmax)
Resistance and interactions vs. T
Note that the metallic behavior sets in when 2 ~ 0.45,
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exactly as predicted
by the RG theory
Comparison between theory (lines) and experiment (symbols)
(no adjustable parameters used!)
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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
3
T (K)
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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 the
existence of a magnetic phase transition.
•
However, upon approaching to nc, one leaves the clean regime and enters
the “Punnoose-Finkelstein” regime where the physics is governed by
interplay between interactions and disorder. In this regime, both
interactions and disorder become temperature-dependent.
•
Punnoose-Finkelstein theory gives quantitatively correct description of the
metal-insulator transition. In particular, in excellent agreement with
theory, the metallic behavior sets in once 2 > 0.45!
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