The story so far: Charge transport

Transcription

1 The story so far: Reasonable understanding of (simplified) ground state of many-electron system in bulk solids, with sense of where approximations fail at small size scales. Single-particle states fall into bands of allowed energies, with atomic and crystal structure determining details. Excitations are electrons and holes. Can label single-particle states with k, analogous to a plane wave prop. vector for reduced symmetry (compared to free particle) system. Altering band structure, confinement geometry, or confining potential can lead to engineered states with interesting properties. Charge transport System properties often studied through experiments that add, remove, or rearrange the charge distribution in the material. Have already seen one example (Coulomb blockade in quantum dots) of using electronic transport to study spectrum of singleparticle states. Idea: measure some transport-related quantity and turn some knob, such as T,, B, etc. Plan: review terminology, classical transport discuss semiclassical transport quantum corrections 1

2 Low frequency measurements: -terminal what most people think of when discussing electrical conduction measurements. R dr 0 0 I conductance G I 0 contact effects contribute directly. resistance R 0 Rdr for R<<R dr Low frequency measurements: 4-terminal R dr 0 0 I I G R 0 Rdr for R<<R dr Mitigates lead effects b/c no current flows through ideal voltage probes. Truly ideal voltage probe: capacitive coupling allows electrostatic potentials to equalize w/o any charge exchange.

3 Transport terminology Can define multiterminal resistances and conductances: Usual 4-terminal resistance: R 14,3 I 3 14 Non-local resistances: R 3,14 I 14 3 Seems like should R 14,3 R 1,34 I 34 1 Seems like should 0 I- characteristics I Red curve is Ohmic: I ~. Blue curve is nonlinear & subohmic. di/d Differential conductance: G( ) di d 3

4 Differential resistance and NDR I d/di I R( ) d di d/di can be < 0! Useful for active circuit elements gain. Transport coefficients Intensive quantities: J σ E J J J x y z J current density σ conductivity ρ resistivity σ xx σ yx σ zx σ xy σ yy σ zy σ xz σ yz σ zz Ex E y E z E ρ J E E E x y z ρ xx ρ yx ρ zx ρ ρ ρ xy yy zy ρ xz J x ρ yz J y zz J ρ z Onsager relation: σ xy ( B) σ yx ( B) Comes from basic symmetries of equations of motion Reciprocity relation: R ab B) R ( ), cd ( cd, ab B 4

5 Kinetic concepts Treat electrons like classical gas of noninteracting particles (Drude), with density n 3d and charge q. Elastic mean free path l: typical distance traveled before scattering event randomizes direction of momentum, while conserving energy. l Scattering is off static defects and disorder. Assume typical speed v F. Collision time (momentum relaxation time): τ l / vf On scales short compared to l, motion is ballistic. On scales long compared to l, motion is diffusive. Drude conductivity and mobility Consider what happens when such a gas is put into an electric field: Particles accelerate in response to E, but experience collisions in time τ that randomize direction of momentum. Result: a drift velocity. qe v d τ m Resulting current density: n3d q τ J n3 dqvd E m Drude conductivity: n3 d q τ σ m Mobility: v d qτ µ E m σ n 3d qµ 5

6 More about mobility v d qτ µ E m Units: cm /s May be extracted from low temperature conductivity measurements. Effective mobility limited at high temperatures by inelastic processes (e.g. electron-phonon scattering). Directly related to elastic scattering time. Serves as a measure of material cleanliness. Typical mobility in single-crystal Si at 300 K: ~ 1000 cm /s. τ ~ x s. For v F ~ 10 5 m/s, l ~ 0 nm. In GaAs deg at low temperatures, µ ~ 10 7 cm /s. τ ~ 4 x s. For v F ~ 10 5 m/s, l ~ 40 µm (!) Diffusive regime At scales longer than l transport looks diffusive random walk of step size l. l Diffusion constant: 1 1 D v l d d F v F Dimensionality d is that for diffusive process. That is, for sample of length L, width w, thickness t, 3d L,w,t >> l d L,w >> l >> t 1d L >> l >> w,t τ 6

7 Einstein relation Slight departure here from classical treatment. We know only electrons near Fermi surface contribute to conduction. Rather than using full electron density: σ e ν 3 d ( EF ) D Einstein relation Knowing ν(e F ) from, e.g., heat capacity measurements, one can then find D by measuring σ. Why do we care about D? Gives us l and dimensionality of diffusion. Can then find characteristic lengths associated with time scales. Example: given some time τ φ, one can find an associated length, L φ (D τ φ ) 1/ Typical values in a metal Resistivity of clean gold ~ µω-cm. σ 5 x 10 7 S/m. Density of states ν 3d (E F ) 1 x J -1 m -3. So, D 0.0 m /s. For Au, v F 1.5 x 10 6 m/s: l 40 nm τ.7 x s. 7

8 Boundary scattering An example of a purely classical finite size effect. w Specular scattering. smooth surfaces does not randomize momentum does not affect l. Diffuse scattering. rough surfaces randomizes momentum reduces l l eff l w σ eff w σ l Effects of a magnetic field: Hall effect B B z I / I longitudinal resistance 5 / I Hall resistance Hall voltage, H Transverse voltage develops across sample. In equilibrium, average transverse electric force + average Lorenz force + scattering must all balance. 8

9 9 Hall effect d case [ B] v E v + d d e m τ Writing out components, and remembering definition of J, y x d d d d y x J J n e m n e B m n e B m n e m E E τ τ µ τ µ τ / / / / So, d xy yx xx e n B 1 /, ρ ρ σ ρ Remember, 0., /, /, / y x H y x x J w I J w E L E Hall effect d case Putting these together, we see why Hall measurements are so useful. L w n e I db d e I n x d H d / /, / / µ Measuring Hall voltage vs. B tells us d carrier density. With that and sample dimensions, longitudinal resistance lets us find mobility. Lower carrier densities larger Hall voltages. Key to some very sensitive magnetic field sensors.

10 Magnetic fields: two more length scales Magnetic field curves classical electron trajectories. Cyclotron orbits have radius: r c mvf eb A classical length scale Archetypal case: varying B tunes this scale, and interesting things happen as r c crosses other lengths in the problem. Consider effect of B on resistance of sample with diffuse boundary scattering. As B is increased, more trajectories hit the boundaries than at B 0. resistance increases. Eventually, if r c << w, particles move in such tight spirals that boundary scattering is reduced. resistance decreases. Classical conductivity not monotonic in B in this case. Magnetic fields: a quantum length scale Can combine fundamental constants to get a quantity with units of magnetic flux: h/e Given some magnetic field, can define a length scale, the magnetic length: 1/ h l B eb At 1 Tesla, this is ~ 45 nm. What does this length scale mean? At very high fields, this is roughly the size of a singleparticle state wave function for a free particle in d. 10

11 Magnetic fields: cyclotron orbits & Landau levels Consider a free particle in d in a large magnetic field in a box of side L. For an electron with Fermi velocity v F, what is the frequency of the circular motion? vf eb ωc π πr m Energy of a single particle state can then be written as ε ν c ( ν +1/ ) h ω c Each Landau level, labeled by ν, is highly degenerate. Degeneracy: ebl That is, divide sample up hc into chunks of size l B on a side. Summary of classical transport ideas: Transport coefficients relate currents to fields. Can do multiterminal measurements, + define differential transport coefficients. Treat electrons like independent pointlike particles; motion is ballistic or diffusive, depending on sample size compared to elastic mean free path. Can define mobility, and find it using Hall effect. Classical size effects: boundary scattering, cyclotron motion & magnetoresistance. 11

12 What does semiclassical mean Remember that real single-particle states are Bloch waves, and that we fill those states from the bottom up. Microscopic equations of motion now relate to evolution of quantum number k rather than to some classical momentum p. Why classical at all? Neglect quantum interference effects! Act like position and momentum can both be known. Relevant picture here: with no lattice, Sommerfeld; with lattice + band structure, Bloch. Next time: Semiclassical equations of motion Quantum coherence and mesoscopic effects 1

13 Appendix: other ways of measuring charge motion High frequency method: impedance spectroscopy High frequency method: resonant cavity High frequency method: optical spectroscopy High frequency method: surface acoustic waves Chemical method: electrochemistry High frequency method: impedance spectroscopy rf Basic idea: matched impedance no reflected rf power. Monitor changes in sample impedance by watching variations in reflected power coming back. Example: Rimberg group rf-set. 13

14 Noncontact methods: resonant cavity Take a microwave resonating cavity. Insert a sample of the material of interest (must be a large quantity). Conductivity of sample damping of cavity resonance + shift of resonant frequency. Can vary some parameter (T) and watch cavity resonance behavior. Related nano-technique: scanning microwave microscopy Noncontact methods: optical measurements Charge motion can lead to optical absorption and re-emission. Good example: plasmon resonance. Different modes of charge sloshing can give absorption at different frequencies. (a) (b) Extinction (Arb. Units) 0 nm 10 nm 7 nm 5 nm Wavelength (nm) 0 nm 60 nm Core Radius 60 nm Core Radius 0 nm Shell 5 nm Shell Example: Halas group nanoshells 14

15 Noncontact methods: surface acoustic waves Some materials are piezoelectric - because of asymmetry of crystal structure, strain fields produce electric fields. Can use electrodes to launch surface acoustic waves at ~ GHz frequencies across GaAs surface. Charges on surface can feel (piezo)electric fields & move in response, damping & shifting the frequency of the surface acoustic wave. Example: R.L. Willett - probing fractional quantum Hall state. Chemical methods: electrochemistry One common technique: cyclic voltammetry, to try to find oxidation and reduction potentials of molecules. 15

16 Chemical methods: cyclic voltammetry A number of assumptions needed to interpret data quantitatively. 16

17 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. Semiclassical formulation Quantum numbers: n, k, s. Energy: elocity: Wave function: E n (k) E n (k +K ), given by band structure 1 En ( k) v n( k) h k ψ ( r) exp( ik r) u ( r) nk nk elocity 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. 1

18 Semiclassical equations of motion 1 En ( k) r& v n ( k) h k h k & e [ E( r, t) + v ( k) B( r, t) ] n 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. Crystal momentum vs. momentum hk 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).

19 Regime of validity When can we safely use semiclassical model? low electric fields low magnetic fields low disorder ( Ec ( k) Ev ( k)) e E a << E ( Ec ( k) Ev ( k)) hω c << E k F l >>1 F F low frequencies long length scales large samples hω << E ( k) E ( k) λ >> a λ << L, w t c F, v Can be violated in some directions but not others. 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 hk m * k x k x E x 3

20 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. 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. 4

21 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 φ A + 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. Decoherence Detector wavefunction: χ(η, t) Initial wavefunction: [φ A (x, t 0) + φ B (x, t 0)] χ(η, t0) 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: 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. 5

22 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/ coherence length. Note: elastic scattering off disorder or edges does not cause decoherence no change in state of environment. 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. 6

23 Quantum coherence: Aharonov-Bohm effect In magnetic field, hk hk + qa the vector potential A B, A dr Φ Presence of A leads to particles moving along a trajectory picking up an additional quantum mechanical phase: ψ ψ expi ea dr h Interference fringes shift as current through solenoid is varied. Complete shift of π when flux enclosed 1 flux quantum, h/e. Quantum coherence: Aharonov-Bohm effect can see Aharonov-Bohm effect in solids if ring circumference is <~ coherence length! erference/interf.html 7

24 Quantum coherence: Aharonov-Bohm effect Power Spectrum [arb. unit] h/e Resistance [Ω] h/e B-field [T] /B [1/T] Ag sample, ~ 0.5 micron diameter ring, T < 1 K 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. 8

25 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 /h Field scale B c ~ 1 flux quantum through a typical coherent area, L φ. Skocpol et al., PRL 56, 865 (1986). 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 /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] 9

26 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/. A correlation energy scale also exists for coherence phenomena: hd Eϕ Lφ Thermal smearing (k B T > E φ ) suppresses fluctuations by similar factor. Large voltage drops can also suppress fluctuations (e > E φ ) Noise in real devices at room temp. may still have a contribution from these effects. Conductance fluctuations and ensemble averaging I R (Ω) B [T] K 8 K 14 K Ag wire, 140 nm wide 10

27 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, T I ac [µa] 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/

28 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. 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 /(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. 1

29 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) 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. 13

30 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. 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. 14

31 Next time: Tunneling - a primer. 15

32 Tunneling - a primer Nano often appears in real technology in the form of thin layers or barriers. We re going to look at several ways electrons can transport over or through these barriers under various conditions. Thermionic emission (classical over-barrier) Poole-Frenkel behavior (hopping out of tilted potential wells) Tunneling, rectangular barrier (simple quantum problem) Tunneling, generic barrier Fowler-Nordheim tunneling (triangular barrier - field emission) Resonant tunneling (Semi)Classical thermionic emission E metal, voltage φ b e Distribution of incident particle energies. Those with high enough energy can make it over classical barrier. Net current density J s-m J m-s semiconductor, voltage 0. e bi Thermionic emission: applications Thermionic emission often governs: Electron injection into semiconductors from metals. Emission of electrons from hot materials into vacuum (the filament in every cathode ray tube, ion gauge, etc.) Partly responsible for emission in thermal field emission sources (electron microscopy). Thermionic emission: calculation semiconductor to metal: dn( E) ν 3d ( E) f ( E, T ) de Assume high T (nondegenerate) 1/ 4 (m ) π * E Ec exp( ( E Ec en ) / kbt ) de 3 h + Rewrite in terms of velocities (semiclassical picture): 3 m* e m* v n exp exp (4πv dv) h k BT kbt Add up current heading toward barrier, knowing that minimum velocity to get over barrier at bias is: 1 m* v0, x e( bi ) Thermionic emission: calculation Result is, skipping some algebra, J s m πem k 3 h T 4 * B φ A *, Richardson constant. Heading the other direction, one finds eφ B J m s A* T exp kbt Net result: J tot e B e exp exp kbt kbt eφ exp B e A * T exp 1 kbt kbt Thermionic emission: details J tot eφ exp B e A* T exp 1 kbt kbt Temperature dependence is strong. Exponential voltage dependence at high bias. Other perturbing effects (some tunneling contribution, for example) can cause measured A * value to deviate from theory prediction. Basically just a classical statistical mechanics effect. 1

33 Strongly disordered systems In strongly disordered systems, the Bloch wave picture breaks down. Wavefunctions are localized, decaying exponentially away from trap sites (potential wells). Conduction is due to hopping from localized site to localized site. Hopping can be thermally activated (though it can also be due to tunneling). Applying a large electric field can affect thermal hopping rates: Poole-Frenkel behavior Lowering of hopping barriers in large electric fields is called Poole-Frenkel behavior. e Assume potential well is electrostatic: ( r) 4πεε 0r Potential maximum occurs when external field and trap field balance: e 0 4πεε r max ef r max Amount barrier potential is lowered: e 4 πεε0f 1/ efr e max 4 πεε0f e 1/ Poole-Frenkel behavior If hopping rate is activated, Tunneling through a rectangular barrier σ ~ exp[ U 0 / kbt] Lowering of barrier means A B F G 3 σ ~ exp[ ( e F / 4πεε ) 0 1/ / k T] Scaling of conductance with T and F like this is the signature of Poole-Frenkel behavior. Prevalent in organic semiconductors, amorphous Si, etc. B h + m* ( z) z eff -a +a ( z) φ ( z) Eφ( z) Ae ikz φ( z) Ce Fe γz ikz + Be + De + Ge ikz γz ikz 1 φ φ( z), m* ( z) z z < a a < z < a z > a continuous. Do case where E < 0. Tunneling through a rectangular barrier Applying boundary conditions, Tunneling through a rectangular barrier Combining, one gets Ae ika ik[ Ae γ [ Ce γa + Be ika γa Ce + De ika Ce γa + De ika γa γa Be ] γ [ Ce De ] γa Fe ika + Ge γa De ] ik[ Fe ika γa ik + γ ( e A ik B ik γ e ik ika Ge ika ik + γ ( ik e C γ D ik γ ( e γ ] ik γ ) a ( ik γ ) a γ ) a ik+ γ ) a ik γ ( e ik ik + γ e ik ik+ γ ) a ( ik γ ) a ik γ e γ ik + γ ( e γ C D ( ik+ γ ) a ik γ ) a F G Incident flux: A M11 M1 F where elements come from B M 1 M G multiplying matrices on prev. slide. f inc Transmission coefficient: Limit of wide barrier: hk A transmitted flux: m * f T( E) f tran inc 1+ 1 f tran k +γ ( ) sinh (γa) kγ 4kγ T( E) e k + γ F 1 A M 4γa e 4a 11 m* ( 0 E ) / h hk F m *

34 Tunneling through a rectangular barrier Reflection coefficient 1 - T, unsurprisingly. Can also do case where E > 0 : 1 T ( E) 1+ where k' m ( ) * E 0 k ( ) k ' sin (k' a) kk' Combined picture: Tunneling through a generic barrier: the WKB approximation classically forbidden x 1 x define a local effective wave vector - imaginary in classically forbidden region. x T ( E) exp h x1 1/ { m ( ( x) E } dx * ) valid when turning points are far apart (many wavelengths). valid when (x) varies slowly (compared to wavelength). image from Ferry Realistic tunneling probabilities As we saw from rectangular barrier case, tunneling only occurs with significant probability at very short length scales (very narrow barriers). Tunneling is exponentially suppressed with barrier width. Factoid: metal-metal tunneling current in vacuum (or air) decays with distance roughly like 1 order of magnitude per Angstrom (!). One of your homework problems: what is the probability of a olkswagon Beetle tunneling through a speed bump? A triangular barrier: Fowler-Nordheim tunneling Can pull barrier down by applying electric field. Fowler-Nordheim limit: barrier gets thin enough to allow tunneling. E F φ b e Analytical expression can be obtained using WKB, as long as barrier doesn t get too thin: ( ( x ) E ) e φ T ( E) exp h 0 exp φb 4 ( 3 / ef B { m ( eφ efx) } * 1/ m* e) φ hf efx B 3/ B 1/ dx -efx Double barriers Double barriers B A F G A B F G B A F G A B F G We can easily consider the case of two barriers in a row. This case gets particularly interesting if the barriers are sufficiently close that there are bound states in the space between the barriers that have energies close to those of incident electrons. Matrix-wise, we know we can combine the two single-barrier problems. -a z0 b Need to relate phases of A, B to F, G: Define transfer matrix: F e G 0 ikb A ikb ' Fe, B' Ge 0 A' A' M ikb W e B' B' Combined system: A F' M LMW M R B G' where the M L and M R are matrices for each barrier individually. ikb 3

35 Resonant tunneling Write the combined matrix as M tot M L M W M R, and write the matrix elements as a magnitude and a phase. Final result: Resonant tunneling What about asymmetric case? Relevant at finite bias: T tot 1 T1 ( E) M T1 + 4R1 cos ( kb ) θ tot11 k γ θ arctan tanh( a) + ka k γ γ Transmission through the whole structure has resonances! These correspond to cases when the energy of the incident particle coincides with bound state energies of the well. Incident + reflected waves interfere constructively in well! At resonance, T tot 1. Minimum transmission: T tot ~ T 1 /4. Expect to see peak in transmission when bias tilt aligns bound level with source or drain energy levels. At higher bias, expect transmission to drop again as resonance condition vanishes. Predicts NDR! Resonant tunneling Algebra gets a bit messy. Resonant tunneling diode Final result: T tot ( E) (1 R R ) + 4 R R cos ( Φ) 1 TT 1 1 θ Φ k b + Again, at resonance Off resonance 1 L1 + θ R1 θ L11 θ R11 4TT 1 T res 1 for T 1 T. ( T + T ) 1 T off TT 1 4 image from Ferry Resonant tunneling diode Does this actually work? A very subtle question A philosophy question worth pondering, even though it s not directly germane to the course: How long does the tunneling process take? That is, for particles that successfully traverse barrier, how long are they in the classically forbidden region? Consider incident Gaussian wavepacket + transmitted Gaussian wavepacket. Surprising answer: measuring positions of peaks of wavepacket, tunneling velocity can greatly exceed c! GaAs RTD, AlGaAs barriers, a b 5 nm. For more information, read Landauer and Martin, RMP 66, 17 (1994). image from Ferry 4

36 To summarize: Thermionic emission classical thermal over-barrier Poole-Frenkel field-assisted classical thermal hopping Single-barrier tunneling is straightforward. Generic barriers: WKB approximation Fowler-Nordheim field-assisted tunneling Double barriers: must account for interference Result: resonant tunneling diodes w/ NDR Nontrivial interpretation issues associated with tunneling! Next time: Scattering matrices Landauer-Buttiker formalism - conductance as transmission. 5