Semiclassical formulation

Transcription

1 The story so far: Transport coefficients relate current densities and electric fields (currents and voltages). Can define differential transport coefficients + mobility. Drude picture: treat electrons like classical, independent pointlike particles; motion is ballistic or diffusive, depending on sample size compared to elastic mean free path. Classical size effects: boundary scattering, cyclotron motion & magnetoresistance. Semiclassical picture: Bloch electrons; still ignores quantum interference effects.

2 Semiclassical formulation Quantum numbers: n, k, s. Energy: Velocity: Wave function: E n (k) = E n (k +K ), given by band structure v ψ 1 E ( k) k) = k ( r) = exp( ik r) u ( r) n n( nk nk Velocity expression in analogy to plane wave case. Pauli exclusion holds, with states counted in k space using density of states. Ignore interference effects. Allow a specific r as well as a k.

3 Semiclassical equations of motion [ ] ), ( ) ( ), ( ) ( 1 ) ( t t e E n n n r B k v r E k k k k v r + = = = Assume band index n is constant of the motion. Electrons not near extrema of E n (k) are always in motion! No scattering or forces from ion cores here - all that is taken into account by our definition of Bloch waves.

4 Crystal momentum vs. momentum k is the crystal momentum. W/ o external fields, crystal momentum is a constant of the motion for electrons in a periodic potential. As said on previous page, is unaffected by forces from periodic ion cores. Rate of change of crystal momentum given by total external forces on the electron. When periodicity / symmetry are broken, k does not have to be conserved (reflection from a boundary).

5 Regime of validity When can we safely use semiclassical model? ( Ec( k) Ev ( k)) low electric fields e E a << E low magnetic fields low disorder ( Ec( k) Ev ( k)) ωc << E k F l >>1 F F 2 2 low frequencies long length scales large samples ω << E ( k) E ( k) λ >> a λ c F <<, L, w t v Can be violated in some directions but not others.

6 Relaxation time approximation Way of incorporating elastic scattering off defects. Same as in classical picture. Define some time τ for relaxation of (crystal) momentum through elastic scattering processes. These processes allow drift velocity to develop, and allow Fermi surface to maintain its shape. k y k y v d = k m * k x k x E x

7 Results of semiclassical picture Reproduces Drude results, but justifies lack of scattering off ions. Properly handles Pauli principle. Semiclassical equations of motion allow mapping of Fermi surface through effects at high magnetic fields (DeHaas-van Alphen oscillations; Shubnikov-DeHaas oscillations) -- beyond scope of this course. One validity criterion we didn t mention: incoherent samples - all length scales must be >> L φ, a coherence length.

8 Quantum coherence A system is said to be quantum coherent if, to calculate probabilities of processes, one must include interference terms. When worrying about an electron in a hydrogen atom, one must use quantum mechanics all the time. However, one doesn t need to use quantum probabillity rules when discussing electrons in macroscopic objects at room temperature. Consider dropping an electron into a solid. Quantum coherent corrections to the electron s motion are only needed over a finite length scale.

9 Quantum interference Prototypical quantum coherent system: two-slit interference experiment. Amplitude for path A = φ A Amplitude for path B = φ B Probability = φ A + φ B 2 = φ A 2 + φ B Re φ A* φ B Classical result Electron beam traveling in isolation, in vacuum. Elastic scattering (diffraction) of electrons off slits. Coherence seen over large length scale.

10 Decoherence Detector wavefunction: χ(η, t) Initial wavefunction: [φ A (x, t = 0) + φ B (x, t = 0)] χ(η, t=0) The wavefunction at time τ (after electron goes through slit) : φ A (x,τ) χ A (η,τ) + φ B (x,τ) χ B (η,τ) So, the interference term is now: 2 Re [ φ A(x,τ) φ B (x,τ) χ A (η,τ) χ B (η,τ) ] The state of the detector multiplies the interference term! We see apparent decoherence because inelastic interaction entangles electrons with detector, and we trace over detector final states.

11 Coherence time / length Equivalent formulation: interaction with detector perturbs relative phases of electron paths in effectively random way; decoherence sometimes known as dephasing. Consider dropping an electron into a solid, where at some rate it undergoes inelastic interactions with the environment (other electrons, phonons, magnetic impurities), On some time scale τ φ, the phase of the electron becomes essentially uncorrelated with its initial phase. Result: washing out of interference effects. Distance the electron moves in this time in a diffusive system: L φ = (D τ φ ) 1/2 = coherence length. Note: elastic scattering off disorder or edges does not cause decoherence no change in state of environment.

12 Why this matters: conductance as transmission If we have to worry about interference effects, every conductance measurement really becomes an interference experiment! Should sum amplitudes for all possible trajectories through disordered system, and then square to find probabilities. Many cross terms! Since L usually > L φ, we can treat often treat these corrections perturbatively.

13 Quantum coherence: Aharonov-Bohm effect In magnetic field, k k + qa the vector potential A A dr = B, = Φ Presence of A leads to particles moving along a trajectory picking up an additional quantum mechanical phase: ψ ψ exp i ea dr Interference fringes shift as current through solenoid is varied. Complete shift of 2π when flux enclosed = 1 flux quantum, h/e.

14 Quantum coherence: Aharonov-Bohm effect can see Aharonov-Bohm effect in solids if ring circumference is <~ coherence length! erference/interf.html

15 Quantum coherence: Aharonov-Bohm effect Power Spectrum [arb. unit] h/e Resistance [Ω] h/2e B-field [T] /B [1/T] Ag sample, ~ 0.5 micron diameter ring, T < 1 K

16 Quantum coherence: conductance fluctuations In fully coherent region, conductance involves adding amplitudes from all possible trajectories, where phases are distributed randomly. Analogous to diffraction off array of randomly placed slits. Interference pattern on a screen would be random bright and dark regions - speckles. Shifting the relative phases of the waves would move the speckles randomly yet deterministically.

17 Quantum coherence: conductance fluctuations One way of shifting relative trajectory phases: Aharonov-Bohm. Result: applying a magnetic field leads to fluctuations in sample conductance δg(b) that depend on exact configuration of scatterers in that sample. In coherent volume, rms δg ~ e 2 /h Field scale B c ~ 1 flux quantum through a typical coherent area, L φ2. Skocpol et al., PRL 56, 2865 (1986).

18 Quantum coherence: conductance fluctuations Another way of altering relative phases of trajectories: consider moving a scatterer. Within coherent volume, if typical scatterer moves distance d such that k F d > ~ 1, can change conductance by ~ e 2 /h. Result: Noise! d = 5nm, L = 750nm I = 10nA, τ = 3 s T = 81K T = 7.7K Resistance fluctuates as a function of time. R/R time [min]

19 Conductance fluctuations and ensemble averaging Why don t we see quantum CF all the time? Fluctuations are not correlated from coherent volume to coherent volume - they average away to zero when L>>L φ. For N coherent volumes in series, the size of fluctuations, δg is down by factor of 1/N 1/2. A correlation energy scale also exists for coherence phenomena: D Eϕ = 2 Lφ Thermal smearing (k B T > E φ ) suppresses fluctuations by similar factor. Large voltage drops can also suppress fluctuations (ev > E φ ) Noise in real devices at room temp. may still have a contribution from these effects.

20 Conductance fluctuations and ensemble averaging I R (Ω) K 8 K B [T] 14 K Ag wire, 140 nm wide

21 Conductance fluctuations and ensemble averaging II conductance noise power K, 0 T 4 K, 0 T 6 K, 0 T 8 K, 0 T 10 K, 0 T 8 K, 2 T I ac [µa]

22 Quantum coherence: weak localization A correction not suppressed by ensemble averaging! Time-reversed paths shorter than L φ interfere constructively, affecting the conductance. One flux quantum through typical trajectory suppresses this effect due to Aharonov-Bohm phase. Result: a magnetoresistance with a size and field scale set by τ φ and conductor properties (dimensionality, D, etc.). Brunthaler et al., cond-mat/

23 Quantum coherence: weak localization Not surpressed by ensemble averaging because each coherent volume s contribution adds in series! Field scale of magnetoresistance allows one to infer the coherence length as a function of temperature. In 1d, must thread h/e worth of flux through a typical loop trajectory, w x L φ.

24 Do we understand quantum coherence in the solid state? As inelastic processes freeze out, τ φ is expected to diverge, with power law exponent set by dimensionality and inelastic mechanism: electron-3d-phonon dominates: τ φ ~ T -3 electron-electron dominates: τ φ ~ T 2/(d-4) Still some experimental concerns. In many experiments at very low temperatures, these power laws don t seem to hold. Subtle experimental problems to debug.

25 Demonstration: weak coupling to a detector Aharonov-Bohm ring weakly coupled to a quantum point contact detector. Using the detector suppresses amplitude of AB oscillations as expected by theory. Buks et al., Nature 391, (1998)

26 How relevant is quantum coherence to technology? Right now, not very, except in specialized cases like the noise described earlier. In the future, however: Quantum interference effects essential to understanding conductance of very small systems, like single molecules. Coherence times for spins are much longer than for orbital wavefunctions: possibility of using spin-based properties for novel devices. Coherence lengths at room temperature are often very short (few atoms), but not inaccessible any more.

27 Can we use quantum coherence for useful purposes? Highly likely that we ll be able to in the future. Interferometric-based switches Quantum cryptography Quantum information processing.

28 Summary: Semiclassical picture with Bloch waves does reasonable job reproducing observed electronic transport properties of solids. This picture explains why the free-electron Drude model works as well as it does. Quantum coherence corrections (interference effects) include: Aharonov-Bohm oscillations, conductance fluctuations, and weak localization. Quantum coherence is suppressed on some time (and length) scale by inelastic interactions between the system and dynamic environmental degrees of freedom. Understanding coherence is an active area of research, likely to be of technological relevance as devices get very small and / or as spin becomes an important degree of freedom.

29 Next time: Tunneling - a primer.