Quantum transport
and its classical limit
Piet Brouwer
Laboratory of Atomic and
Solid State Physics
Cornell University
Lecture 1
Capri spring school on Transport in Nanostructures,
March 25-31, 2007
Quantum Transport
About the manifestations of quantum
mechanics on the electrical transport
properties of conductors
sample
These lectures: signatures of quantum interference
Quantum effects not covered here:
•Interaction effects
•Shot noise
•Mesoscopic superconductivity
Quantum Transport
dG
G (e2/h)
G1+2 (e2/h)
These lectures: signatures of quantum interference
What to expect?
B (mT)
B (10-4T)
Magnetofingerprint
B
Nonlocality
G1=G2=2e2/h
R1+2=R1+R2
Figures adapted from: Mailly and Sanquer (1991)
Webb, Washburn, Umbach, and Laibowitz (1985)
Marcus (2005)
Landauer-Buettiker formalism
Ideal leads
sample
y
W
x
N: number of propagating transverse modes or “channels”
N depends on energy e, width W
an: electrons moving towards sample
bn: electrons moving away from sample
Note: |an|2 and |bn|2 determine flux in each channel, not density
Scattering Matrix: Definition
sample
More than one lead:
• Nj is number of channels in lead j
• Use amplitudes anj, bnj for incoming,
outgoing electrons, n = 1, …, Nj.
Linear relationship between anj, bnj:
S: “scattering matrix”
• |Smj;nk|2 describes what fraction of the flux of electrons entering in lead k,
channel n, leaves sample through lead j, channel m.
• Probability that an electron entering in lead k, channel n, leaves sample
through lead j, channel m is |Smj;nk|2 vnk/vmj.
Scattering matrix: Properties
Linear relationship between anj, bnj:
sample
S: “scattering matrix”
• Current conservation: S is unitary
• Time-reversal symmetry:
If y is a solution of the Schroedinger
equation at magnetic field B, then y*
is a solution at magnetic field –B.
Landauer-Buettiker formalism
Reservoirs
Each lead j is connected to an
electron reservoir at temperature
T and chemical potential mj.
sample
mj, T
Distribution function for electrons originating from reservoir j
is f(e-mj).
Landauer-Buettiker formalism
Current in leads
sample
Ij,in
In one dimension:
Ij,out
mj, T
= (nnkh)-1
Buettiker (1985)
Landauer-Buettiker formalism
Linear response
mj = m – eVj
sample
Expand to first order in Vj:
Ij,in
Zero temperature
Ij,out
mj, T
Conductance coefficients
sample
• Current conservation
and gauge invariance
Ij
mj=m-eVj
• Time-reversal
Note:
Otherwise
only if B=0 or if there are only two leads.
and
in general.
Multiterminal measurements
In four-terminal measurement, one measures a combination of the 16
coefficients Gjk. Different ways to perform the measurement correspond
to different combinations of the Gjk, so they give different results!
V
I
V
I
I
Benoit, Washburn, Umbach,
Laibowitz, Webb (1986)
V
Landauer formula: spin
• Without spin-dependent scattering: Factor two for spin
degeneracy
• With spin-dependent scattering: Use separate sets of
channels for each spin direction. Dimension of scattering
matrix is doubled.
Conductance measured in units of 2e2/h:
“Dimensionless conductance”.
Two-terminal geometry
r’
t
r
r, r’: “reflection matrices”
t, t’: “transmission matrices”
t’
f(e)
e
e
in
f(e)
Anthore, Pierre, Pothier,
Devoret (2003)
out
e
e
e (meV)
eV
|t’|2 |r|2
f(e)
|t|2 |r’|2
Quantum transport
Landauer formula
t
r
r’
t’
sample
What is the “sample”?
• Point contact
• Quantum dot
• Disordered metal wire
• Metal ring
• Molecule
• Graphene sheet
Example: adiabatic point contact
N(x)
Nmin
10
8
6
4
2
0
g
x
Vgate (V)
-2.0 -1.8 -1.6 -1.4 -1.2 -1.0
Van Wees et al. (1988)
Quantum interference
In general: dg small,
random sign
a
b
tnm,a , tnm,b : amplitude for
transmission along paths a, b
Quantum interference
Three prototypical examples:
• Disordered wire
• Disordered quantum dot
• Ballistic quantum dot
Scattering matrix and Green function
Recall: retarded Green function is solution of
In one dimension:
Green function in channel basis:
ek = e and v = h-1dek/dk
r in lead j; r’ in lead k
Substitute 1d form of Green function
If j = k:
Quantum transport
and its classical limit
Piet Brouwer
Laboratory of Atomic and
Solid State Physics
Cornell University
Lecture 2
Capri spring school on Transport in Nanostructures,
March 25-31, 2007
Characteristic time scales
L
h/eF
t
terg
tD
Ballistic quantum dot: t ~ terg ~ L/vF, l ~ L
Diffusive conductor: terg ~ L2/D
tH
Inverse level spacing:
relevant for closed samples
l
Elastic mean free time
lF
Characteristic conductances
Conductances of the contacts: g1, g2
Conductance of sample without contacts: gsample
if g >> 1
• ‘Bulk measurement’: g1,2 >> gsample
• Quantum dot: g1,2 << gsample
general relationships:
g dominated by sample
g dominated by contacts
Assumptions and restrictions
Always: lF << l.
Well-defined momentum between scattering events
Diagrammatic perturbation theory: g >> 1.
This implies tD << tH
Only ‘nonperturbative’ methods can describe the regime g ~ 1 or,
equivalently, times up to tH.
Examples are certain field theories, random matrix theory.
Quantum interference corrections
G
Weak localization
Small negative correction to the ensembleaveraged conductance at zero magnetic field
Conductance fluctuations
Reproducible fluctuations of the samplespecific conductance as a function of
magnetic field or Fermi energy
B
G
B
Anderson, Abrahams, Ramakrishnan (1979)
Gorkov, Larkin, Khmelnitskii (1979)
Altshuler (1985)
Lee and Stone (1985)
Weak localization (1)
Nonzero (negative) ensemble average
d g at zero magnetic field
g
dg
B
a
b
=
+
+ permutations
‘Cooperon’
‘Hikami box’
Interfering trajectories propagating
in opposite directions
Weak localization (2)
Nonzero (negative) ensemble average
d g at zero magnetic field
g
dg
B
Sign of effect follows directly from quantum correction
to reflection.
a
b
Trajectories propagating at the same angle in the
leads contribute to the same element of the
reflection matrix r. Such trajectories can interfere.
Weak localization (3)
G (e2/h)
Disordered wire:
(no derivation here)
Disordered quantum dot:
(derivation later)
a
b
F
N1 channels
N2 channels
B (10-4T)
Mailly and Sanquer (1991)
Weak localization (4)
G (e2/h)
dg
B (10-4T)
a
Magnetic field
suppresses WL.
b
F
10-3
0
DR/R
H(kOe)
-10-3
Chentsov (1948)
Weak localization (5)
a
b
F
Typical dwell time for transmitted electrons: terg
Typical area enclosed in that time: sample area A.
WL suppressed at flux F ~ hc/e through sample.
Typical area enclosed in time terg: sample area A.
Typical area enclosed in timetD: A(tD/terg)1/2.
WL suppressed at F ~ (hc/e)(terg/tD)1/2 << hc/e.
Weak localization (6)
b
F
a
In a ring, all trajectories enclose
multiples of the same area.
If F is a multiple of hc/2e, all phase
differences are multiples of 2p :
d g oscillates with period hc/2e.
‘hc/2e Aharonov-Bohm effect’
Altshuler, Aronov, Spivak (1981)
Note: phases picked up by
individual trajectories are
multiples of p, not 2p!
Sharvin and Sharvin (1981)
Conductance fluctuations (1)
Fluctuations of d g with
applied magnetic field
dg
a
b’
a
b
a’
a
b
b’
a
b
b’
a’
a’
b
b’
a’
Umbach, Washburn,
Laibowitz, Webb (1984)
“diffuson”
interfering trajectories
in the same direction
“cooperon”
interfering trajectories
in the opposite direction
Conductance fluctuations (2)
Fluctuations of d g with
applied magnetic field
dg
G (e2/h)
Disordered wire:
Disordered quantum dot:
Jalabert, Pichard, Beenakker (1994)
Baranger and Mello (1994)
N1 channels
N2 channels
B (mT)
Marcus (2005)
Conductance fluctuations (3)
dg
b
F
a
In a ring: sample-specific
conductance g is periodic
funtion of F with period hc/e.
‘hc/e Aharonov-Bohm effect’
Webb, Washburn, Umbach, and Laibowitz (1985)
Random Matrix Theory
Quantum dot
Ideal contacts: every electron that
reaches the contact is transmitted.
For ideal contacts: all elements of
S have random phase.
N1 channels
N2 channels
Ansatz: S is as random as possible,
with constraints of unitarity and
time-reversal symmetry,
“Dyson’s circular ensemble”
Dimension of S is N1+N2.
Assign channels m=1, …, N1 to lead 1,
channels m=N1+1, …, N1+N2 to lead 2
Bluemel and Smilansky (1988)
RMT: Without time-reversal symmetry
Quantum dot
N1 channels
N2 channels
Ansatz: S is as random as possible,
with constraint of unitarity
Probability to find certain S does not change if
• We permute rows or columns
• We multiply a row or column by eif
Average conductance:
No interference correction to average conductance
RMT: with time-reversal symmetry
Quantum dot
Additional constraint:
Probability to find certain S does not change if
• We permute rows and columns,
• We multiply a row and columns by eif,
while keeping S symmetric
N1 channels
N2 channels
Average conductance:
Interference correction to average conductance
RMT: with time-reversal symmetry
Quantum dot
Weak localization correction is
difference with classical conductance
N1 channels
N2 channels
For N1, N2 >> 1:
Jalabert, Pichard, Beenakker (1994)
Baranger and Mello (1994)
Same as diagrammatic perturbation theory
RMT: conductance fluctuations
Quantum dot
Without time-reversal symmetry:
With time-reversal symmetry:
N1 channels
N2 channels
Jalabert, Pichard, Beenakker (1994)
Baranger and Mello (1994)
Same as diagrammatic perturbation theory
There exist extensions of RMT to deal
with contacts that contain tunnel barriers,
magnetic-field dependence, etc.
Quantum transport
and its classical limit
Piet Brouwer
Laboratory of Atomic and
Solid State Physics
Cornell University
Lecture 3
Capri spring school on Transport in Nanostructures,
March 25-31, 2007
Ballistic quantum dots
Past lectures:
• Qualitative microscopic picture of interference
corrections in disordered conductors;
Quantitative calculations can be done using
diagrammatic perturbation theory
• Quantitative non-microscopic theory of
interference corrections in quantum dots (RMT).
This lecture:
Microscopic theory of interference
corrections in ballistic quantum dots
Assumptions and restrictions: lF << l, g >> 1
Method: semiclassics, quantum properties are obtained
from the classical dynamics
Semiclassical Green function
Relation between transmission matrix and Green function
Semiclassical Green function (two dimensions)
a: classical trajectory connection r’ and r
S: classical action of a
ma: Maslov index
Aa: stability amplitude
a
r’
r
q’
r
’
Comparison to exact Green function
Semiclassical Green function (two dimensions)
Exact Green function (two dimensions)
Asymptotic behavior for k|r-r’| >> 1
equals semiclassical Green function
Semiclassical scattering matrix
Insert semiclassical Green function
and Fourier transform to y, y’. This replaces y, y’ by the conjugate
momenta py, py’ and fixes these to
Result:
Jalabert, Baranger, Stone (1990)
Legendre transformed action
q
y
a
Semiclassical scattering matrix
Transmission matrix
Legendre transformed action
Stability amplitude
transverse momenta of a fixed at
q
y
a
Reflection matrix
Diagonal approximation
Reflection probability
a
b
Dominant contribution from terms a = b.
probability
to return to
contact 1
Enhanced diagonal reflection
Reflection probability
b=a
a
If m=n: also contribution if b = a timereversed of a:
Without magnetic field: a and a have equal actions, hence
Factor-two enhancement of diagonal reflection
Doron, Smilansky, Frenkel (1991)
Lewenkopf, Weidenmueller (1991)
diagonal approximation: limitations
We found
One expects a corresponding reduction
of the transmission. Where is it?
The diagonal approximation gives
b=a
a
Note: Time-reversed of transmitting trajectories
contribute to t’, not t. No interference!
Compare to RMT:
captured by diagonal approximation
missed by diagonal approximation
Lesson from disordered metals
a
Weak localization correction to
reflection:
Do not need Hikami box.
b
Weak localization correction to
transmission:
Need Hikami box.
a
b
=
+
+ permutations
‘Hikami box’
Ballistic Hikami box?
In a quantum dot with smooth
boundaries: Wavepackets follow
classical trajectories.
Ballistic Hikami box?
But… quantum interference
corrections d g and var g exist in
ballistic quantum dots!
Marcus group
Ballistic Hikami box?
l: Lyapunov exponent
Initial uncertainty is
magnified by chaotic
boundary scattering.
Time until initial uncertainty
~lF has reached dot size ~L:
L=lF exp(l t)
t=
“Ehrenfest time”
Aleiner and Larkin (1996)
Richter and Sieber (2002)
Interference corrections in
ballistic quantum dot same
as in disordered quantum
dot if tE << tD
Ballistic weak localization
Probability to remain in dot:
special for
ballistic dot
also for disordered
quantum dot:
included in RMT
tE
t loop
Aleiner and Larkin (1996)
Adagideli (2003)
Rahav and Brouwer (2005)
Semiclassical theory
Landauer formula
Jalabert, Baranger, Stone (1990)
• Sa, Sb: classical action
• angles of a, b consistent with
transverse momentum in lead,
• Aa, Ab: stability amplitudes
Semiclassical theory
tenc
Landauer formula
(0,0)
s, u: distances along stable,
unstable phase space directions
encounter region: |s|,|u| < c
c: classical cut-off scale
Action difference Sa-Sb = su
(s,u)
s a e-lt
u a elt
Richter and Sieber (2002)
Spehner (2003)
Turek and Richter (2003)
Müller et al. (2004)
Heusler et al. (2006)
Semiclassical theory
tenc
Landauer formula
(0,0)
s, u: distances along stable,
unstable phase space directions
(s,u)
t’
P1, P2: probabilities to enter,
exit through contacts 1,2
Aleiner and Larkin (1996)
Adagideli (2003)
Rahav and Brouwer (2005)
c: classical cut-off scale
Classical Limit
Take limit lF/L 0 without changing the
classical dynamics of the dot, including its
contacts
L
diverges in this limit!
0
Aleiner and Larkin (1996)
but… var g remains finite!
Brouwer and Rahav (2006)
THE END