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### Quantum transport and its classical limit

### Quantum transport and its classical limit

### 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

These lectures: signatures of quantum interference

What to expect?

G1+2 (e2/h)

G(e2/h)

dG

B (mT)

G1=G2=2e2/h

B (10-4T)

B

R1+2=R1+R2

Magnetofingerprint

Nonlocality

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

- More than one lead:
- Nj is number of channels in lead j
- Use amplitudes anj, bnj for incoming,
- outgoing electrons, n = 1, …, Nj.

sample

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, theny* 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

Ij,out

mj, T

In one dimension:

= (nnkh)-1

Buettiker (1985)

Landauer-Buettiker formalism

Linear response

mj = m – eVj

sample

Expand to first order in Vj:

Ij,in

Ij,out

mj, T

Zero temperature

Conductance coefficients

sample

- Current conservation
- and gauge invariance

Ij

mj=m-eVj

- Time-reversal

Note:

only if B=0 or if there are only two leads.

Otherwise

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

V

Benoit, Washburn, Umbach, Laibowitz, Webb (1986)

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’: “reflection matrices”

t, t’: “transmission matrices”

r

t’

in

e

e

f(e)

f(e)

eV

out

e

e

e (meV)

f(e)

Anthore, Pierre, Pothier, Devoret (2003)

|t’|2

|r|2

|t|2

|r’|2

Quantum transport

Landauer formula

r’

t

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

g

10

x

8

6

4

2

Vgate (V)

0

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

Van Wees et al. (1988)

Quantum interference

a

In general: dg small, random sign

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:

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

lF

l

L

t

tD

tH

h/eF

terg

Elastic mean free time

Inverse level spacing: relevant for closed samples

Ballistic quantum dot: t ~ terg ~ L/vF, l ~ L

Diffusive conductor: terg ~ L2/D

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

g dominated by sample

g dominated by contacts

general relationships:

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

Weak localization

Small negative correction to the ensemble-averaged conductance at zero magnetic field

Conductance fluctuations

Reproducible fluctuations of the sample-specific 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

dg at zero magnetic field

g

dg

B

a

=

+

b

+ permutations

‘Hikami box’

‘Cooperon’

Interfering trajectories propagating in opposite directions

Weak localization (2)

Nonzero (negative) ensemble average

dg 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)

Disordered wire:

(no derivation here)

G(e2/h)

Disordered quantum dot:

B (10-4T)

N1 channels

N2 channels

(derivation later)

Mailly and Sanquer (1991)

a

b

F

Weak localization (4)

dg

G(e2/h)

B (10-4T)

a

Magnetic field suppresses WL.

10-3

DR/R

b

0

H(kOe)

F

-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

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 :

dg oscillates with period hc/2e.

a

F

‘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 dg with applied magnetic field

dg

Umbach, Washburn, Laibowitz, Webb (1984)

a

b

a

b

“diffuson”

interfering trajectories in the same direction

b’

a’

a’

b’

a

a

“cooperon”

interfering trajectories in the opposite direction

b

b

a’

a’

b’

b’

Conductance fluctuations (2)

Fluctuations of dg with applied magnetic field

dg

Disordered wire:

G(e2/h)

Disordered quantum dot:

B (mT)

N1 channels

N2 channels

Jalabert, Pichard, Beenakker (1994)

Baranger and Mello (1994)

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

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

N1 channels

N2 channels

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.

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 ofa

ma: Maslov index

Aa: stability amplitude

a

q’

r

r

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

a

y

Semiclassical scattering matrix

Transmission matrix

transverse momenta of a fixed at

Legendre transformed action

q

a

Stability amplitude

y

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 time-reversed 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

b=a

a

One expects a corresponding reduction of the transmission. Where is it?

The diagonal approximation gives

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 dg and varg 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:

Aleiner and Larkin (1996)

Richter and Sieber (2002)

Interference corrections in ballistic quantum dot same as in disordered quantum dot if tE << tD

L=lF exp(l t)

t =

“Ehrenfest time”

Ballistic weak localization

Probability to remain in dot:

tE

special for ballistic dot

tloop

also for disordered quantum dot: included in RMT

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)

sae-lt

u a elt

s, u: distances along stable, unstable phase space directions

encounter region: |s|,|u| < c

c: classical cut-off scale

Richter and Sieber (2002)

Spehner (2003)

Turek and Richter (2003)

Müller et al. (2004)

Heusler et al. (2006)

Action difference Sa-Sb = su

Semiclassical theory

tenc

Landauer formula

(0,0)

(s,u)

s, u: distances along stable, unstable phase space directions

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

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