Download Coulomb Drag to Measure Electron-Electron Interaction in Bilayer

Coulomb drag : An experimenter's handle to
the electron- electron interaction problem
Kantimay Das Gupta
Dept. of Physics, IIT Bombay
Semiconductor Physics Group
Cavendish Laboratory
Cambridge
United Kingdom
Low Dimensional Quantum Systems. HRI, Oct 2011 (Wed 12/10 @ 10.00)
Q. What is the interaction energy between two electrons?
2
vacuum
slightly polarisable
lattice but with no
other free electrons
slightly polarisable
lattice with more
free electrons
1 e
4  0 r
2
1
e
4  0 r r
Explicit expression for V(r)
is not possible in general
V(r) is not very important, we need its Fourier transform V(q) in
most cases. Why?
Any quantum mechanical problem would require matrix elements like
1
~
τ(q)
=
2
∣〈 k f ∣V (r )∣k i 〉∣ × DOS
∣∫ dV
V (r )e
i(k i −k f ).r 2
∣
×DOS
between free
electron states
q=kf - ki
2
= ∣V (q)∣ × DOS
The simplest “textbook case” of one static charge and all other
moving ones goes quite far
ext  
2
∇ V=

0 r 0 r
EF
CB
place for
more
electrons
less
electrons
2
 =e.eV  D E F =qTF V
How is the bare potential of a charge modified due to the presence of
other electrons?
Fourier transforming the equation leads to the solution:
(
V (q)=
In general
)
1 ρext (q)
1
ϵ0 ϵ r q2
1+q2TF /q2
V ext q
V  q=
q
Assumption is that the
source of the extra
potential is static
What if the source of the potential also varies in time ?
V ext q , 
V  q , =
q , 
Q: Can the denominator be zero?
Density waves, Plasmon modes....
How is the electron-impurity scattering different from electronelectron scattering as far as “screening” is concerned?
Impurites don't move about . So we use :
q , 0
Electrons move about, So we must use:
q , 
This is also why electronimpurity scatt rate will not
tell a lot about ε(q,ω)
How does the experimenter try to measure this (in clean metals) ?
 T = 0eP T Aee T
residual resistance
due to impurities
(Mathiesen & Vogt 1964)
electron-phonon
Bloch 1930
Gruniessen 1933
2
electron-electron
Landau & Pomeranchuk (1936)
McDonald et al (1981)
Resistivity of ultra-pure Silver. What is the power law for ρ(T)?
l=v F ≈500  m
In very “clean” samples e-e and
electron-impurity interactions
might become comparable.
BUT
Power law with N = 2 to 4 have
been reported.
Comparison of data from different
groups. no agreement !!
Besides we can't change density
of electrons in a metal. They
cannot be “gated”.
Khoshnevisan et al,
Jl of Phys F: Metal Phys. 9, L1 (1979)
A new idea: Instead of trying to measure the momentum lost by
the particle try to measure the momentum gained by them.
M.B Pogrebinsky, Sov Phys Semicond. 11,372 (1977)
P. Price , Physica 117B, 750 (1983)
V
Consider 2 parallel layers. Moving electrons in one layer transfers
momentum to the other layer. Try to measure that. There was no way
to do that in 1977. Two films of metal won't work!
Quantum well and modulation doping had not yet come.
How to analyse the amount of momentum transferred and what
can be inferred about the interactions from that?

F
∂ v . ∇  
.∇v
r
∂t
m

∂ f 1 r , v , t 
f 1 r , v , t =
∂t
∣
−e E1
∂ f1
0
. ∇ k f 1=
ℏ
∂t
Small deviation from
Interlayer
equilibrium due to
scattering
current flow in layer 2
 2 ,  2' ∝ I 2
⃗' ⃗'
⃗
k⃗1 + k 2 → k 1 + k 2
∂ f1
=
∂t
d
k2
collision
2
1
V
'
d
k1
∑ ∫ 2  2 ∫ 2  2 W 1,2  1' , 2'  1 2− 1 ' − 2 '  ×

f 10 f 02 1− f 01' 1− f 02 '  12 −1 ' −2 ' 
Jauho & Smith PRB 47,4420 ( 1993) Zheng & MacDonald PRB 48, 8203 (1993)
Yurtsever et al. Solid State Comm . 125,575 (2003) Hwang & Das Sarma PRB 78, 075430 (2008)
The deviation functions:
momentum conservation gives
'
1
 1= =0
eE 2  2 v 2x
 2 =−
kT
This is a shift in the
Fermi circle
m2 v 2x−m 2 v 2' x =m 1 v1 ' x −m1 v1x
We want to isolate the electric field in layer 1, so multiply both sides of the
equation by k1x and integrate over all k-space.
0
0
e E1
d k1
∂f1
e E1
∂ f 1
n1e E1
LHS = −
2∫
k
=−
D
− d =
∫
2 1x
ℏ
∂ k 1x
ℏ
∂
ℏ
2 
d k1
∂ f 01
RHS = ∑ ∫
k
=
2 1x
∂t
2 
ℏ 2 e E 2
d
k 2 d k '1 d 
k1
W 1,2 1 ' , 2 ' k 1x k 1x−k 1 ' x 
∑ ∫ 2
2
2
m2 kT
2  2   2 
0 0
0
0
× f 1 f 2 1− f 1 ' 1− f 2'   1 2− 1 ' − 2 ' 
Symmetries of this expression
allow simplification
The symmetry allows replacing
The current and voltages in
layer 2 are easily related
1
1 2
2
k 1x (k 1x −k 1 ' x )→ (k 1x−k 1 ' x ) → q
2
4
E2=
I 2 m2
n2 e 2 τ 2
n1 e E 1 ℏ τ 2 e E 2
d ⃗k 2 d ⃗
k '1 d ⃗k 1
=
W (1,2 → 1 ' , 2 ' )k 1x (k 1x −k 1 ' x )
∑
2
2
2
σ∫
ℏ
m2 kT
(2 π) ( 2 π) (2 π)
× f 01 f 02 (1− f 01' )(1− f 02' )δ(ϵ1 +ϵ 2−ϵ 1 '−ϵ2 ' )
Notice that individual layer scattering times are going to disappear from the ratio
between E1 and I2. This is immensely important - because we have now related a
transport measurement to electron-electron scattering .
The effect of disorder has somehow disappeared - at least within the relaxation
time approximation. Usually the disorder scattering is 100-1000 times stronger
than e-e even in very “clean” samples.
Expression for Coulomb drag including dynamic screening:
 DRAG =
ℏ
2
∞
2
2 k B Te
2
q
∫
np
0
∞
3
2
dq∫ d ∣V  q , ∣
I m e q ,  I m h  q , 
0
2
sinh ℏ  /2k B T 
V bare
det q , 
det ε (q,ω) can be zero or very small and these collective modes of the
2- component plasma can contribute very significantly to Coulomb drag
2
me mh  3
k B T 
 DRAG =
npe 2 16 ℏ k Fe d  k Fh d  qTFe d  qTFh d 
holds for high densities and
large interlayer separations
k F d ≫1
T /T F ≪1
What difficulties has been swept under the carpet? Why did
everyone not start doing this?
A. The conditions under which the effect can be appreciable are not trivial. Also
there are possible sources of errors.
What difficulties has been swept under the carpet?
1. Independent contacts to two layers spaced by about 10 nm
ohmic
ohmic
EF
depletion
gate
AlGaAs
AlGaAs
GaAs
depletion
gate
AlGaAs
CB
Eisenstein et al. APL (1990)
Gramila et al PRL, 66, 1216 (1991)
Linfield et al Semicond. Sci & Tech. 8, 415 (1993)
NPR Hill et al PRL, 78, 2204 (1997)
2 x 2DEG
GaAs
Requires two-sided lithography OR Focussed Ion Beam patterning
Q. What would be realistic gate voltages needed? How can we see both sides of a GaAs wafer?
Top side of chip
with scribes on glass
(a)
500 µm
2. Aligning the gates on the top and bottom side is absolutely
necessary, with better than 5 micron accuracy
Acetone wash
(b)
Thinned to ~60µm
Back side of chip
Devices
GaAs chip
Crystalbond-509
Cover slip
~5µm wide
scribes on glass
100µm
Alignment marks for
backside lithography
Croxall et al JAP 104, 113715 (2008)
3. Even a small leakage between the two layers would produce a
spurious signal, masking the real one.
Shift the bias point and check if the signal changes.......
Less than 100pA over 100x100 microns is generally necessary
Often less than 1 in 50 devices would meet this requirement
Let's summarise the three key requirements
1. Independent contacts
2. Very low leakage barrier (1.5V, 10nm, 100x100 micron)
3. Gating from both top and bottom
1 cm
The relation between Coulomb drag and Onsager reciprocity
relation for four-terminal linear response.
I-
I+
V
V+
V-
OR
V-
V+
V
I+
I-
H.B.G. Cassimir Rev. Mod. Phy. (1945)
The first measurement of drag effect in an electron-electron bilayer
ρdrag
<0.01
ρlayer
11
-2
N=1.5 x 10 cm
µ=3.5x106 cm2V-1s-1
Gramilla, Eisenstein et al PRL 66, 1216 (1991)
But that is no longer
a problem.
The many significances of ε(q,ω): relation to density-density
response and local field corrections
Total potential = External potential + potential due to induced charges
0
ϵ(q , ω)=1 − v q χ (q , ω)
0
δ n=χ V
tot
e
Density-density response or
charge susceptibility
Fourier transform of the Coulomb
potential. 1/q in 2D, 1/q2 in 3D
The Thomas-Fermi form predicts a constant χ(q) for all q. This cannot be
correct. Implication is that the system responds equally well to all
frequencies. In fact it leads to some problems...
A theory of charge susceptibility is also a theory of pair correlation
Singwi-Tosi-Land-Sjolander (1968)
1
d
k k . 
q
G q=− ∫
[ S  k −
q −1]
2
2
n  2  q
0
χ
χ=
1−v q (1−G (q))χ0
ℏ
S  q=− I m

Need to fix this
K.S. Singwi & M. Tosi:
“Correlations in electron liquids”
Generalisation to bilayers :
Liu, Swierkowski. Neilson , Szymanski
PRB 53, 7923 (1996)
Zheng & Macdonald PRB 49, 5522 (1994)
2X2DEG (10nm barrier)
Similar data exists for holehole bilayers
Vig
n
Sin ale
gw i
STLS
Verifying the local field correction in a bilayer
ic
m
na
y
D
A
P
R
RPA
c
i
t
Sta
V
Data from M. Kellog et al
Solid State Comm 123, 515 (2003)
Calculated curves:
Yurtsever. Moldaveanu, Tanatar
Solid State Comm 125, 575 (2003)
Measure drag at low densities – large rs
Does a fermi liquid at large r_s continue to be a “fermi liquid” ?
This question can be asked
using Coulomb drag as a
probe.
Can local field corrections
explain the huge
enhancement in hole-drag?
We can ask these questions
without worrying about
disorder.
R. Pillarisetty at al.
PRL 89, 016805 (2002)
Why interaction effects can appear in a bilayer more easily than
they do in a single layer?
l
E ee
e2  N
=
4   0
EF
ℏ N
=
meff
2
If n=1×1011 cm-2 then
E ee
1
rs =
=
EF
aB  N
l ~ 30 nm rs ≈ 1.8 in GaAs
Q. Effective mass in Si is higher, but not ideal for these, why?
Why interaction effects can appear in a bilayer more easily than
they do in a single layer....
Total potential = External potential +
contribution due to polarization charges in same layer +
contribution due to polarization charges in other layer
tot
ext
−qd
V e =V e +v q δ n e +v q e δ n h
ext
−qd
V tot
=V
+v
δ
n
+v
e
δ ne
h
h
q
h
q
0
e
0
h
δ ne =χ V
δ n h=χ V
tot
e
tot
h
e2
v q=
2 ϵ0 ϵr q
[
1−v q χ
−qd
−v q e
0
e
−qd
−v q e
0
χh
χ
0
1−v q χ h
0
h
][ ] [ ]
V
V
tot
e
tot
h
=
V
V
ext
e
ext
h
Q. What if the matrix has
det = 0 ?
A. Spontaneous density
modulations possible, with
well behaved single layers
What are the other possible phases in a bilayer?
d ~ aB
l
l
d
l
=
=

1
2 n
kFd
~ 1
assuming equal densities
in both layers...
Such low densities and high
mobilities are possible only in
GaAs/AlGaAs
10-20 nm separation needed.
Bound states are not the only possibility.....density waves may occur.....
Coulomb drag measurements in an electron-hole bilayer
The strength of the
interlayer to
intralayer
interaction.....
1
 2 n
d / l≈1.8
l=
Croxall et al. PRL 101, 246801 (2008)
Coulomb drag measurements in an electron-hole bilayer
20nm barrier EHBL
Data from Seamons et al
PRL 102, 026804 (2009)
No phase space is expected for scattering at T=0
Between two Fermi liquids.
How does the electron-hole bilayer seem to have a finite scattering rate at T=0?
What kind of questions can be addressed using the
Coulomb drag measurements?
Electron-electron scattering rates are not masked by electron-impurity scattering rates, we
can use this in many situations. Like..
Fermi-liq / non-Fermi liq at low densities?
Emergence of density wave modes/bound states/bilayer Quantum Hall states in a bilayer.
Correctness/verification of local field corrections.
The idea can be extended to experiments on narrow channels, where the experimental
evidence of Luttinger liquid phases is sketchy...
Acknowledgements.
The work was done in the semiconductor Physics group of Prof David Ritchie and
Michael Pepper.
●The MBE growth was done by Christine Nicoll, Harvey Beere & Ian Farrer.
●The devices were fabricated and measured by KDG with Andy Croxall, James Keogh,
Mamta Thangaraj & Joanna Waldie at various times.
●The work was funded by EPSRC, UK.
●