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Quantum transport and its classical limit

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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

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

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)

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

- 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.

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.

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).

Current in leads

sample

Ij,in

Ij,out

mj, T

In one dimension:

= (nnkh)-1

Buettiker (1985)

Linear response

mj = m – eVj

sample

Expand to first order in Vj:

Ij,in

Ij,out

mj, T

Zero temperature

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.

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)

- 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”.

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

Landauer formula

r’

t

r

t’

sample

What is the “sample”?

- Point contact
- Quantum dot
- Disordered metal wire
- Metal ring
- Molecule
- Graphene sheet

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)

a

In general: dg small, random sign

b

tnm,a , tnm,b : amplitude for transmission along paths a, b

- Three prototypical examples:
- Disordered wire
- Disordered quantum dot
- Ballistic quantum dot

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

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

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:

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.

G

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)

Nonzero (negative) ensemble average

dg at zero magnetic field

g

dg

B

a

=

+

b

+ permutations

‘Hikami box’

‘Cooperon’

Interfering trajectories propagating in opposite directions

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.

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

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)

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.

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)

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’

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)

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)

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)

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

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

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

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

- 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

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’

’

Semiclassical Green function (two dimensions)

Exact Green function (two dimensions)

Asymptotic behavior for k|r-r’| >> 1

equals semiclassical Green function

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

Transmission matrix

transverse momenta of a fixed at

Legendre transformed action

q

a

Stability amplitude

y

Reflection matrix

Reflection probability

a

b

Dominant contribution from terms a = b.

probability to return to contact 1

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)

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

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’

In a quantum dot with smooth boundaries: Wavepackets follow classical trajectories.

But… quantum interference corrections dg and varg exist in ballistic quantum dots!

Marcus group

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”

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)

Landauer formula

Jalabert, Baranger, Stone (1990)

- Sa, Sb: classical action
- angles of a, b consistent with
- transverse momentum in lead,
- Aa, Ab: stability amplitudes

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

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

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