Quantum transport (Read Kittel, 8th ed., pp )

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Dwayne Blair
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1 Quntum trnsport (Red Kittel, 8t ed., pp ) Wen we ve structure in wic mny collisions tke plce s crriers trnsport cross it, te quntum mecnicl pse of te electron wvefunctions is essentilly rndomized, nd te semiclssicl pproc tt we ve been following is pproprite. In nnoscle electronic devices, especilly t low tempertures: device dimensions begin to pproc te men-free pt between collisions scttering processes ply n decresingly importnt role Tis is te regime of quntum trnsport. Electron trnsport becomes governed by principles of quntum-mecnicl wve propgtion nd interference, rter tn te clssicl diffusive trnsport. Conductnce quntiztion Consider te limit of one-dimensionl conducting wire wit no collisions. Does it ve resistnce? Connect te wire between to contcts. We model te contcts s lrge, 3-D reservoirs wit voltge difference V between tem. μ 1 μ + qv wire μ left contct rigt contct Te rigt-going sttes of te wire re populted up to te electrocemicl potentil (qusi- Fermi energy) μ 1 nd te left-going sttes will be populted up to te electrocemicl potentil μ were te potentil difference is μ 1 - μ qv. Te net current flowing troug te wire is due to te excess rigt-moving crrier density Δn is limited by te density of sttes in te wire. For te lowest trnsverse sub-bnd (or mode) of te wire:

2 DE ( ) dn de spin ml π _ me 1 -- We obtin te differentil conductnce, di, s te smll cnge in current for smll dv cnge in pplied voltge. Ten we cn write te density of excess rigt-moving crriers s DE ( ) Δn qv. L We divide by to get te density of sttes only for rigt moving crriers, nd we divide by L to get te crrier density per unit lengt. We cn re-write D(E) in terms of te crrier velocity, v: DE ( ) Te net current (for electron flow) is ten: 4L v DE di Δnqv ( )qdv qv -----q vdv L v nd we obtin te fmous result of quntized conductnce: e dv di dv e G. Q Te resistnce quntum, R Q 1/G Q kω. Tis is te conductnce for crriers in te lowest sub-bnd of te wire. As dditionl subbnds re populted, ec dds one dditionl quntum of conductnce, nd step-like rise of conductnce cn be observed s te density is incresed. Tis cn be observed in conductnce mesurements of 1D cnnel formed of D electron gs were te crrier density cn be seprtely controlled by gte voltge. Tis ws elegntly demonstrted in [B. J. Vnwees, L. P. Kouwenoven, E. M. M. Willems, C. Hrmns, J. E. Mooij, H. Vnouten, C. W. J. Beenkker, J. G. Willimson, nd C. T. Foxon, "Quntum Bllistic nd Adibtic Electron-Trnsport Studied wit Quntum Point Contcts," Pysicl Review B, vol. 43, pp , My 1991.] If te cnnel is not perfectly conducting, te conductnce is modified by te trnsmission probbility T(E F ). Wen tere re multiple conducting cnnels (sub-bnds), ec cn ve its own trnsmission probbility. Tus:

3 GE ( F ) e T ( E F ) were i,j lbel te trnsverse sub-bnds. Quntum dots, Coulomb blockde, nd single-electron trnsistors (SETs) Consider 0-D object, te quntum dot. Te electrocemicl potentil for dding te (N+1)t electron to tis dot is were U is te crging energy, α is te gte coupling fctor tt expresses te rte t wic voltge on nerby gte sifts te electrosttic potentil of te dot. As n pproximtion, we cn express te crging energy in terms of te totl cpcitnce of te dot. Wit C g s te gte cpcitnce, we ve U e C nd α C g C. To mke n SET, we couple tis dot wekly vi tunnel brriers to two contcts (cll tem source nd drin), s well s to te gte

4 Electrons will tunnel onto te dot until te electrocemicl potentil for dding one more electron exceeds te electrocemicl potentil of te contct. Tis sets te equilibrium number of electrons on te dot, N. Tis number cn ten be cnged by vrying te gte voltge V g. To dd one dditionl electron, we need gte voltge cnge of: Te crging energy, U, depends on te size of te dot. If te dot is very smll, te cpcitnce cn be smll enoug tt te crging energy cn exceed kt. For dot on te order of 1nm, U cn be on te order of ev. Ten terml fluctutions cnnot vry te mount of crge on te dot. For somewt lrger dot, we cn still operte t low temperture to observe single crging effects. Coulomb blockde occurs wen tis crging energy seprtes te dot energy levels by more tn kt. Wen te Fermi levels of te source nd drin contcts lie between tese levels, trnsport troug te dot is suppressed. Wen te dot cemicl potentil for N+1 electrons lies between te contct Fermi levels, ten n electron cn op onto te dot from te source nd op off te dot to te drin. Tis leds to Coulomb oscilltions in te conductnce s function of V g. Tis device is clled single electron trnsistor (SET) becuse it turns on nd off periodiclly s te occupncy of te dot cnges by one electron crge. It cn be used s igly sensitive detector of electric field cnges

5 Tunneling (reding in supplementry references: Sciff, Sing) Electron tunneling is noter form of trnsport tt is importnt in number of devices. Tunneling occurs over very sort distnces, usully witout te influence of oter scttering processes. It is terefore coerent quntum mecnicl process, similr to bllistic trnsport. Te key fetures cn be obtined using simple quntum mecnicl clcultion. Tunneling troug rectngulr brrier V 0 Te boundry conditions given by te continuity of ψ, nd ψ give te reltions tt deter x 1 3 Consider te cse of tunneling troug te brrier represented ere, wit potentil eigt ev 0, nd widt. We write te wvefunction in ec of regions 1, nd 3 s: ψ 1 A 1 exp( ik 0 x) + B 1 exp( ik 0 x) x < -- ψ A exp( ik 1 x) + B exp( ik 1 x) -- < x < -- Te mplitude of te incident wve is A 1, te mplitude of te reflected wve is B 1, nd te mplitude of te trnsmitted wve is A 3. Te wvevectors inside nd outside te brrier re given by: k 0 m m E k _ ( ev _ 0 E)

6 mine te coefficients A i nd B i. Te trnsmission probbility is: A 3 T e A 1 After solving for te coefficients, te resulting trnsmission probbility is (ssuming k 1» 1 : 1 k T e k 1 k 0 + k exp( k + sin k k 0 k 1 ) 1 1 4k 0 k E ev 0 ev E m exp ( ev _ 0 E) Tunneling troug tringulr brrier (Fowler-Nordeim tunneling) V 0 ΔE 0 x Brriers suc s tis one rise for tunneling troug te gte insultor in MOSFET. Bndbnd tunneling in Zener diode is lso similr to tis. Te trnsmission probbility for wide clss of brriers suc s tis cn be clculted using te WKB pproximtion (see Sciff) s: were pre-fctor of order unity is neglected. Te spe of te brrier comes in vi k(x): kx ( ) m ( ev _ 0 Ex ( ))

7 For te tringulr sped brrier sown: were F is te field strengt in te brrier. Performing te integrl of k: kx ( ) dx Te trnsmission probbility is ten: 0 m _ 3 -- ( ΔE efx) ef 0 T 4 e exp -- 3 m ( ΔE) _ ef For bnd-bnd tunneling, te brrier eigt is just te bnd-gp, E g 4 T eg exp -- 3 m ( E g) _ ef Numericlly: T eg ( [ ΔEV ( )] 3 exp ) ef( V cm) For brrier of 1 Volt, very ig field (> 10 6 V/cm) is needed to get pprecible tunneling currents. Double-brrier resonnt tunneling Te tunneling probbility is proportionl to te density of sttes on te oter side of te brrier. Tis s inspired te following device structure, wit two tunneling brriers nd quntum well sndwiced in between. As te bis cross tis device is vried, te discrete stte formed inside te well sifts reltive to te Fermi level on te emitter side. E c E f E f

8 E c E f E f V b A C V b V b B V b D

9 Te I-V crcteristic for suc resonnt tunneling device is: current (ma) 4 3 B D C 1 A pplied bis (V) It is possible to switc te device from condition B to condition C very rpidly ence devices bsed on tis structure (resonnt tunneling diode or RTD) re receiving considerble mount of ttention

Chapter 2. Numerical Integration also called quadrature. 2.2 Trapezoidal Rule. 2.1 A basic principle Extending the Trapezoidal Rule DRAWINGS

Chapter 2. Numerical Integration also called quadrature. 2.2 Trapezoidal Rule. 2.1 A basic principle Extending the Trapezoidal Rule DRAWINGS S Cpter Numericl Integrtion lso clled qudrture Te gol of numericl integrtion is to pproximte numericlly. f(x)dx Tis is useful for difficult integrls like sin(x) ; sin(x ); x + x 4 Or worse still for multiple-dimensionl