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The story so far:

Landauer formula + scattering matrix approach: generalway of treating (noninteracting, small bias) two-terminal

conductance of quantum coherent system attached to classical

reservoirs at absolute zero, independent of details of thequantum system.

Subtle issues about conductance: ballistic system has finite

conductance.

Energy relaxation processes typically modeled as takingplace in leads or contacts, resulting in very nonthermal /

nonequilibrium electronic distributions in active region of

device.

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On the plate today:

Zeroth order effect of interactions

Multiterminal generalization of Landauer formula: the

Buttiker formula.

Reciprocity relations

Finite temperature and larger biases

Combining scattering matrices

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Resistivity dipole - Coulomb interactions

Tcontact 1 contact 2

lead 1 lead 2

+kstates-kstatesT

1-T

x

When computing chemical

potential changes, we showed

abrupt changes (a) at interfacesbetween contacts and leads;

and (b) across a scatterer of

transmittance T.

While may change abruptly,we know electrostatic potential

cannot, because ofscreening.

Quick accounting for averaged

electron-electron interactions:

Poisson equation and screening

length.

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Resistivity dipole - Coulomb interactions

-+

electron density

electric field

screening length

Discontinuity in chemical

potential leads to

smeared discontinuity inelectrostatic potential.

Charge builds up

microscopically like a

dipole around thescatterer.

Whole system is solved

self-consistently.

In systems with poor

screening, effects of

interfaces can be very big!

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Buttiker formula

Using this formula, potential of terminal n is determined

by potentials of other terminals weighted by transmission

functions:

=

nq

nq

nq

qnq

nG

VG

V

Note that, in general,

though

pqqp GG

)()( BGBG pqqp =+

reciprocity -- not easy to show in general.

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Buttiker formula: 4-terminal example

++

++

++

++

=

4

3

2

1

434241434241

343432313231

242324232121

141312141312

4

3

2

1

V

V

V

V

GGGGGG

GGGGGG

GGGGGG

GGGGGG

I

I

I

I

Can set V4 = 0 without loss of generality.

++

++

++

=

3

2

1

3432313231

2324232121

1312141312

3

2

1

V

V

V

GGGGG

GGGGG

GGGGG

I

I

I