Quantum Mechanics Course Buttikerformula

The story so far: • Landauer formula + scattering matrix approach: general way of treating (noninteracting, small bias) two-terminal conductance of quantum coherent system attached to classical reservoirs at absolute zero, independent of details of the quantum system. • Subtle issues about conductance: ballistic system has finite conductance. • Energy relaxation processes typically modeled as taking place in leads or contacts, resulting in very nonthermal / nonequilibrium electronic distributions in “active” region of device.

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

“Resistivity dipole” - Coulomb interactions When computing chemical potential changes, we showed abrupt changes (a) at interfaces between contacts and leads; and (b) across a scatterer of transmittance T. While µ may change abruptly, we know electrostatic potential cannot, because of screening. Quick accounting for averaged electron-electron interactions: Poisson equation and screening length.

contact 1

contact 2

T lead 1

lead 2

µ 1-T -k states

+k states T x

“Resistivity dipole” - Coulomb interactions Discontinuity in chemical potential leads to smeared discontinuity in electrostatic potential. Charge builds up microscopically like a dipole around the scatterer. Whole system is solved self-consistently. In systems with poor screening, effects of interfaces can be very big!

µ φ screening length electron density -

electric field

+

Buttiker formula (1988) Treats multiple probe measurements such that all probes are on equal footing:

2e I p = ∑ (Tq ← p µ p − Tp ←q µ q ) h q Contributions from scattering with to/from terminals q.

Net current out of terminal p Rewriting

I p = ∑ (GqpV p −G pqVq )

2e 2 G pq ≡ Tp←q h

q

Sum rule (guarantees I = 0 when all V are same):

∑G

qp

q

= ∑ G pq q

I p = ∑ G pq (V p − Vq ) equivalent to Kirchhoff’s law q

Buttiker formula Using this formula, potential of terminal n is determined by potentials of other terminals weighted by transmission functions:

Vn

∑G V = ∑G q≠n

nq q

q≠n

nq

Note that, in general, Gqp ≠ G pq though Gqp (+ B ) = G pq (− B )

“reciprocity” -- not easy to show in general.

Buttiker formula: 4-terminal example ⎛ I1 ⎞ ⎡G12 + G13 + G14 ⎜ ⎟ ⎢ − G21 ⎜ I2 ⎟ ⎢ ⎜I ⎟ = ⎢ − G31 ⎜ 3⎟ ⎢ ⎜I ⎟ − G41 ⎝ 4⎠ ⎣

− G12

− G13

G21 + G23 + G24 − G32

− G23

− G42

− G14

G31 + G32 + G34 − G43

⎤⎛ V1 ⎞ ⎥⎜V ⎟ − G24 ⎥⎜ 2 ⎟ ⎥⎜ V3 ⎟ − G34 ⎥⎜⎜ ⎟⎟ G41 + G42 + G43 ⎦⎝V4 ⎠

Can set V4 = 0 without loss of generality…. ⎛ I1 ⎞ ⎡G12 + G13 + G14 ⎜ ⎟ ⎢ − G21 ⎜ I2 ⎟ = ⎢ ⎜I ⎟ ⎢ − G31 ⎝ 3⎠ ⎣

− G12 G21 + G23 + G24 − G32

− G13 − G23

⎤⎛ V1 ⎞ ⎥⎜V ⎟ ⎥⎜ 2 ⎟ G31 + G32 + G34 ⎥⎦⎜⎝ V3 ⎟⎠