8.513 Lecture 3. Microscopic approach; Nonlocal transport; Voltage probes; Resonances
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1 8.513 Lecture 3 Microscopic approach; Nonlocal transport; Voltage probes; Resonances
2 Derivation of Landauer formulas (multi-lead, multi-channel)
3 from Nazarov & Blanter Second quantization summary
4 from Nazarov & Blanter Second quantization summary
5 Second-quantized scattering states Left lead Right lead
6 Operators for scattering states A complete set: scattering states An alternative complete set: unscat t er ing st at es from Nazarov&Blanter
7 General properties Anticommutation relations Occupation numbers in reservoirs: NB: operators a and b do not obey such relations! Field operators that annihilate or create particle in the left lead (similar for the right lead) Current in the left lead
8 Electric current in the left lead Sum over microscopic states in the leads Handle the continuum of states Interpretation: current = # right-moving particles - # left-moving particles (express b's through a's) NB: this expression is still true for an arbitrary number of leads
9 Averaging over states in reservoirs For two leads, using unitarity t^+t+r^+r=1, find True for arbitrary energy distribution is reservoirs (T>0 etc), and for energy-dependent transmission T( ) Taking V to zero get the Landauer formula
10 9 Multiterminal devices: rings, interferometers, etc Webb et al Heiblum et al. 1996
11 10 Multi-probe conductors Buttiker, PRL 57, 1761 (1986); IBM J. Res. Developm. 32, 317 (1988) Quantum Kirchhoff law; current conservation; gauge invariance
12 Multi-probe conductors: scattering matrix magnetic field symmetry 11
13 12 Four-probe resistances G has eigenvalue zero! Current contacts Voltage probes
14 Voltage probes Voltage probes essential for electric circuits. What about quantum circuits? An ideal probe: infinite resistance, thus non-invasive, does not perturb the distribution of currents in the tested circuit Possible for nanostructructures, e.g. Scanning Tunneling Microscopy (STM) However, interpreting results my be tricky b/c the laws of classical circuitry fail Landauer-Buttiker three-terminal description of a voltage probe: In a classical circuit: Measurement is noninvasive if
15 Scatterer with voltage probes Apply to a single-lead scatterer (1D conductor) Probe A, left lead: Contribution of reflected current: Voltage on Probe A: Compare to classical relation Series resistor representation
16 T or T/(1-T)? Unexpected but no paradox, both formulas describe realistic but different situations Two-probe conductance T vs. four-probe conductance T/(1-T) Cannot apply voltage to a scatterer so that G=T/(1-T) b/c voltage in a quantum scatterer is applied to reservoirs, BUT can measure using voltage probes; IIlustrates nonlocality of quantum transport
17 Experiment Four-terminal resistance of a ballistic quantum wire, de Picciotto et al. Nature 411, 51 (2001) four-probe two-probe
18 Multi-probe conductors: scattering matrix magnetic field symmetry 11
19 Reciprocity From and 13
20 Reciprocity: Benoit et al. Benoit, Washburn, Umbach, Laibowitz, Webb, PRL 57, 1765 (1986) 14
21 15 Reciprocity: van Houten et al. skipping orbit van Houten et al., Phys. Rev. B39, 8556 (1989) electron focusing
22 Microscopic derivation
23 Conductance from transmission conductance quantum resistance quantum dissipation and irreversibility boundary conditions 17
24 Conductance: finite temperature 18 current of left movers current of right movers net current linear response conductance Transmission probability evaluated in the equilibrium potential
25 Multi-probe conductance: leads asymptotic perfect translation invariant potential separable wave function channel threshold energy of transverse motion energy for transverse and longitudinal motion scattering channel 19
26 Occupation number and current amplitudes Buttiker, PRB 46, (1992) Incident current at kt = 0 Incident current at kt > 0 Occupation number < > = statistical average Creation and annihilation operators «Incident current» «Current amplitude» 20
27 Current operator 21 Buttiker, PRL 65, 2901 (1990) Current in contact current amplitude: Current in contact single channel result (incoming) multi-channel channel result (outgoing)
28 Current operator Buttiker, PRL 65, 2901 (1990) 22 component vectors quantum statistical average average current, conductance
29 Conductance: finite temperature quantum statistical average 23
30 Advanced Quantum Mechanics: Resonance Scattering
31
32
33 ?
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