Ballistic electron transport in quantum point contacts

Transcription

1 Ballistic electron transport in quantum point contacts Experimental observation of conductance quantization Wen we discussed te self-consistent calculation of te potential and te modes in an infinite wire (section 8.1), we saw tat te number of occupied modes can be tuned wit te voltage applied between te gate electrodes and te two-dimensional electron gas. Experimentally, sort wires can be realized in split-gate structures (see Fig. 6.11, and te inset of Fig. 11.1) placed on top of a Ga[Al]As eterostructure incorporating a two-dimensional electron gas. If a negative voltage is applied to te gates, te electron gas below te gates can be depleted and a narrow cannel remains connecting te two large two-dimensional electron reservoirs. In 1988 two experiments by van Wees and co-workers, and Waram and co-workers, on te low-temperature conductance of suc quantum point contacts at zero magnetic field sowed remarkable results. Figure 11.1(a) sows te measured resistance as a function of te voltage applied to te split gate. Te measured resistance increases as te voltage is decreased, in agreement wit te intuition tat te widt of te cannel decreases. However, te resistance increase sows pronounced steps once te resistance value exceeds a few kω. Detailed investigations of tis beavior sowed tat te two-dimensional electron gas connecting te quantum point contact to te external omic contacts contributes a gate-voltage independent series resistance of 4 Ω. If tis resistance is subtracted, te resistance plateaus appear at quantized values /2Ne 2, were N is an integer number. Te conductance, determined as te inverse of te resistance is sown in Fig. 11.1(b). It sows pronounced plateau values at G = 2e2 N, (11.1) were N is an integer number. Tis result implies tat te conductance is quantized in units of twice te conductance quantum 11.1 Experimental observation of conductance quantization Current and conductance in an ideal quantum wire Current and transmission: adiabatic approximation Saddle point model for te quantum point contact Conductance in te nonadiabatic case Nonideal quantum point contact conductance Self-consistent interaction effects Diffusive limit: recovering te Drude conductivity 189 Furter reading 192 Exercises 192 G = e2 = Ω 1. (11.2) Since its discovery, te quantization of te conductance as been observed in a large number of experiments on samples of vastly differ-

2 176 Ballistic electron transport in quantum point contacts (a) 15 Resistance (kω) 1 5 Gate W = 25 nm (b) 1 G (2e 2 /) Gate voltage (V) Gate voltage (V) Fig (a) Resistance of a quantum point contact as a function of te gate voltage. Te inset sows a scematic top view of te split-gate structure. (b) Conductance of te same quantum point contact as a function of gate voltage after subtraction of a gate voltage independent series resistance of 4 Ω (van Wees et al., 1988). 4 pg Fig Conductance quantization in a quantum point contact fabricated by AFM litograpy on a p-type GaAs eterostructure (see inset). Te labels pg, S, and D denote te plunger gate, te source, and te drain contacts, respectively. Te 7 mk curve is offset by 2e 2 / for clarity. Conductance (2e 2 /) S D 2 µm T = 5 mk T = 7 mk Plunger gate (V) ent materials. Figure 11.2 sows an example of te effect observed on a quantum point contact fabricated in a two-dimensional ole gas in GaAs. Small kinks on te measured curve are most likely te result of rearrangements of carge in te sample close to te quantum point contact arising as te gate voltage is swept. Te sample sown in te inset was fabricated by AFM litograpy. Te experimental conditions for te observability of te quantization are samples of ig quality in wic te electron (or ole) mean free pat is very large compared to te lengt and widt of te cannel. In order to observe te quantization, te widt of te cannel must be comparable to te Fermi wavelengt of te electrons, and te temperature must be low compared to te caracteristic energy spacing of transverse modes in te cannel.

3 11.2 Current and conductance in an ideal quantum wire 177 (a) L x R (b) R E E 2 E 1 E ev SD k x L Fig (a) One-dimensional cannel connected to left and rigt electron reservoirs (gray). Te electrocemical potentials of te reservoirs are µ L and µ R. Transverse modes are scematically drawn witin te cannel wit arrows indicating teir propagation direction. (b) Dispersion relation in te one-dimensional cannel. For eac mode n, te parabolic dispersion relation as its minimum at energy E n. States wit negative k x propagate from rigt to left and are fed from te rigt reservoir wit electrocemical potential µ R, wile tose wit negative k x travel from left to rigt and are fed from te left reservoir wit electrocemical potential µ L. Te gray-saded energy interval is given by te applied voltage between left and rigt reservoirs (bias window) Current and conductance in an ideal quantum wire In order to understand te experimental finding of conductance quantization, we consider te simple model of a perfect one-dimensional cannel, as produced, for example, in te split-gate device in a Ga[Al]As eterostructure used for te experiments described above. Suc a cannel is scematically depicted in Fig. 11.3(a). In order to simplify te reasoning, we take te cannel to be very long compared to its crosssectional area, suc tat it can be treated in good approximation as being translationally invariant in te x-direction. Our goal is to find te current troug tis ideal wire in response to a voltage applied between two big electron reservoirs connecting to te wire. Te quantum problem for te states in te wire is separable and we can write te wave functions in te wire as ψ nk (r) = χ n (y, z) 1 L e ikxx, (11.3) were L is a normalization lengt (very large compared to te relevant electronic wavelengts), and χ n (y, z) are quantized states normal to te wire direction. Tese states are called te transverse modes of te wire. We assume a parabolic energy dispersion along te wire E n (k x ) = E n + 2 k 2 x 2m, were te E n are contributions to te energies arising due to mode quantization normal to te wire axis. Terefore, te quantum number n labels te modes of te quantum wire. Positive values of k x denote states

4 178 Ballistic electron transport in quantum point contacts propagating from left to rigt, negative k x tose traveling from rigt to left. Tis dispersion relation is scematically sown in Fig. 11.3(b). We now determine an expression for te contribution to te electrical current produced by an electron in a particular state (n, k x ). Te quantum mecanical expression for te current density j nkx (r) is j nkx (r) = 2im ( ψ nkx (r) ψ nkx (r) ψ nkx (r) ψ nk x (r) ). Inserting te wave function (11.3) leads to dj nkx (r) = L χ n(y, z) 2 k x m e x, were e x is a unit vector in te wire direction. Te inverse normalization volume L is related to te k x -interval dk x = 2π/L between two successive k x -values leading to dj nkx (r) = e x 2π χ n(y, z) 2 k x m dk x. (11.4) Te quantity dj nkx is terefore te infinitesimal contribution of a small interval dk x to te current density. Tis equation is equivalent to te well-known expression j = ρv occurring in electrodynamics or fluid mecanics if we identify te carge density to be ρ = dk x χ n (y, z) 2 /2π and te expectation value of te velocity in te x-direction v n (k x ) = e x k x m = e x nk x H 1 E n (k x ) nk x = e x. p x k x If we use te above expression for te velocity, we obtain, for te current density, dj nkx (r) = e x g s χ n(y, z) 2 E n (k x ) k x dk x. Here we ave introduced te spin degeneracy factor g s, wic takes te value g s = 2 in te case of GaAs. We convert te small interval dk x into a small energy interval by using dk x = de k x / E n (k x ), and realize tat te expression k x / E n (k x ) represents te one-dimensional density of states up to a factor 2π, but it is inversely proportional to te velocity E n (k x )/ k x. We terefore obtain dj ne (r) = e x g s χ n(y, z) 2 de, were te energy dependence of te density of states as exactly canceled te energy dependence of te group velocity. Te minus sign is valid for rigt-moving states wit k x >, wereas te positive sign is applicable for k x <. Te exact cancelation of group velocity and one-dimensional density of states leads to te remarkable result tat te contribution of a small energy interval de to te current density is independent of te absolute value of te energy. Small group velocities at low energies are

5 11.2 Current and conductance in an ideal quantum wire 179 exactly compensated by te large density of states at low energies, and large group velocities at ig energies are exactly compensated by te lower density of states at tese iger energies. Tis exact cancelation will turn out to be te key to te quantization of te conductance, as we will see below. Te current contributions di n (E) of states witin a small energy interval are obtained from te previous result by integration over te crosssection of te wire. Using te fact tat te transverse modes χ n (y, z) are normalized, we obtain di n (E) = e x g s de. (11.5) Tis contribution to te current is again independent of te energy around wic states are considered, but also independent of te mode index n. Te fundamental proportionality constant / is te inverse of te magnetic flux quantum φ = /. Taking into account tat de/ as te units of an electric voltage, we can identify e 2 / to be te fundamental quantum unit of electrical conductance G in eq. (11.2). Electrical current in termodynamic equilibrium. Starting from eq. (11.5) we are now able to calculate te total current in te wire by energy integration. It is now important to notice tat rigt-moving states (tose wit positive k x ) are occupied via te left electron reservoir connecting to te wire, wereas left-moving states (negative k x ) are occupied via te rigt electron reservoir [cf., Figs. 11.3(a) and (b)]. If te reservoirs are in termodynamic equilibrium wit eac oter, leftand rigt-moving states will bot be occupied according to te same equilibrium Fermi Dirac distribution function, and te total current is I (eq) tot = g s ( n E n def(e) n E n def(e) ) =. Te current contributions from left to rigt and vice versa cancel exactly, leading to zero total current. Nonequilibrium currents due to an applied voltage. If te left and rigt electron reservoirs are not in termodynamic equilibrium, for example because a voltage is applied between tem via an external voltage source, te distribution functions for left- and rigt-moving electrons will differ and te net current is given by I tot = g s n E n de [f L (E) f R (E)], (11.6) were te subscripts L and R refer to te left and te rigt reservoirs. Te distribution functions are given by te Fermi Dirac distributions f i (E) = 1 exp ( E µi k BT ). + 1

6 18 Ballistic electron transport in quantum point contacts G (2e 2 /) V g Fig Conductance as a function of gate voltage V g according to eq. (11.8), assuming tat te mode energies sift down proportional to te gate voltage. Te spin degeneracy as been assumed to be g s = 2. If te temperature is significantly lower tan te spacing between te mode energies, te conductance is quantized (solid line). If 4k B T is comparable to te mode s energy spacing, te quantization is smooted out (dased curve). Conductance (2e 2 /) K Gate voltage (V) Fig Experimentally observed temperature dependence of te conductance quantization in a quantum point contact (van Wees et al., 1991). Linear response. Assuming tat te applied voltage difference V SD is small, i.e., µ L µ R = V SD k B T, we can expand f L (E) f R (E) = f L(E) µ L (µ L µ R ) = f L(E) E V SD. (11.7) Inserting into eq. (11.6) and performing te energy integration gives e 2 I tot = g s f L (E n )V SD. n As a consequence, we obtain te quantization of te linear conductance of an ideal one-dimensional cannel G = I tot e 2 = g s V SD f L (E n ) (11.8) Te beavior of tis expression is sown in Fig for te case tat te mode energies E n sow a linear dependence on te applied gate voltage. If te energy E n of a particular mode n is well below te Fermi energy, te Fermi Dirac distribution in te sum of eq. (11.8) is essentially one, te mode is occupied, and it contributes an amount g s e 2 / to te conductance. In turn, if te energy E n is well above te Fermi energy, te Fermi Dirac distribution is zero, te mode is not occupied, and it does not contribute to te conductance. Figure 11.4 resembles te most striking features of te experimental results in Fig. 11.1(b). If te separation between te mode energies E n is significantly larger tan k B T, te conductance increases in steps of 2e 2 / wit eac additional occupied mode. Te sarpness of te steps is in tis ideal model given by k B T. If te separation between mode energies E n is comparable to, or smaller tan 4k B T, te quantized conductance is completely smooted out, in agreement wit experimental observations. Figure 11.5 sows te experimental temperature dependence of te conductance of a quantum point contact. Te temperature dependence of te quantization can be used to estimate te energetic separation of te lateral modes in te point contact. In te limit of zero temperature, te conductance becomes n G = I tot V SD = g s e 2 N, were N is te number of occupied modes, in agreement wit eq. (11.1). Tis important result implies tat, in an ideal quantum wire wit N occupied modes at zero temperature, eac mode contributes one conductance quantum e 2 / to te total conductance. Te number of occupied modes can be estimated by comparing te widt of te cannel and te Fermi wavelengt of te electrons to be N 2W/λ F. Resistance and energy dissipation. A remarkable property of te above result is tat te conductance of te ideal cannel is finite even

7 11.2 Current and conductance in an ideal quantum wire 181 toug tere is no scattering inside. From Om s law, owever, we are used to te fact tat any finite resistance comes along wit energy dissipation. We terefore ave to answer te question of were energy is dissipated in our system, and ow tis energy dissipation leads exactly to te conductance quantum e 2 /. We answer tese questions wit te elp of Fig In tis figure we can see tat an electron leaving te left contact from an energy below µ L leaves a nonequilibrium ole beind. In addition, te electron constitutes a nonequilibrium carge carrier in te drain at an energy above µ R, after it as traversed te ideal quantum wire. Te electron in te drain, and te ole in te source contact, will eventually relax to te respective electrocemical potential. Te energy dissipated in te source contact is µ L E and te energy dissipated in te drain contact amounts to E µ R. As a consequence, te total energy dissipated by a single electron aving traversed te wire witout scattering is (µ L E)+(E µ R ) = µ L µ R = V SD. Most importantly, te dissipated energy for one electron is independent of te energy of tis electron. In order to find te total power dissipated in te system due to all electrons traversing te wire at all energies between µ R and µ L we argue as follows: a single electron dissipates a power L Fig Energy dissipated by a single electron tat travels witout scattering from source to drain contact troug te wire. Te wire itself, located between te two reservoirs, is omitted in te drawing for clarity. R dp = V SD, τ were τ is te time tat a single electron needs on average to traverse te wire. We obtain tis time from te electrical current wic counts te number of electrons traversing te wire per unit time. Using eq. (11.5) we ave di tot = g s NdE = τ 1 τ = g snde, giving te contribution of te infinitesimal energy interval de to te dissipated power dp = g snv SD de. We assume zero temperature for simplicity, and integrate tis expression between µ R and µ L. Te resulting total dissipated power is P = g snv SD (µ L µ R ) = g snv SD V SD = V 2 SD /e 2 g s N. Te resistance appearing from tis power dissipation argument is R c = e 2 1 g s N, (11.9) wic corresponds exactly to te quantized conductance tat we ave derived before. We conclude tat te finite quantized conductance of an ideal wire witout scattering arises due to te power dissipation of te carge carriers in te source and drain contacts. Te resistance R c is terefore often called te contact resistance. In linear transport we can define a caracteristic inelastic scattering lengt l i. Closer tan l i to te wire (or te quantum point contact),

8 182 Ballistic electron transport in quantum point contacts tere is no equilibrium distribution of carge carriers in te contacts, wereas at muc larger distances te equilibrium distribution is fully restored.

9 Multiterminal systems Generalization of conductance: conductance matrix Now we consider te more general case of nanostructures wit not just two, but n contacts. Tis situation is scematically sown in Fig In tis generalized situation, te linear expansion of te current in terms of te voltages applied to te reservoirs [i.e., te generalization of Om s law in eq. (1.2)] leads to te matrix equation I 1 I 2 I 3. I n = G 11 G 12 G G 1n G 21 G 22 G G 2n G 31 G 32 G G 3n.... G n1 G n2 G n3... G nn V 1 V 2 V 3. V n. (13.1) 13.1 Generalization of conductance: conductance matrix Conductance and transmission: Landauer Büttiker approac Linear response: conductance and transmission Te transmission matrix 24 Here, te matrix of te conductance coefficients G ij is te generalization of te conductance G. Two very fundamental considerations lead to relations between te conductance coefficients wic ave to be obeyed by all pysically acceptable conductance matrices. Consequence of te conservation of carge. In a transport experiment, electric carge is neiter created nor destroyed: te carge is a conserved quantity. In electrodynamics tis fact is often expressed by writing down te continuity equation ρ t + j = were ρ is te carge density, and j is te electrical current density. In te case of stationary, time-independent problems, suc as conductance measurements at zero frequency, te time derivative of te carge density is zero. As a consequence, te divergence of te current density must be zero. Tis implies tat all currents entering and leaving te structure ave to sum up to zero. Tis is Kircoff s current law tat as to be obeyed for arbitrary applied voltages. Terefore, te conductance coefficients in eac column of te conductance matrix fulfill te sum rule n G ij =. (13.2) i=1 2 1 Φ Fig Semiconductor nanostructure wit four contacts consisting of perfect wires eac connected to a reservoir wit te electrocemical potential µ i (i = 1, 2, 3, 4) (Buttiker, 1986). 4 3 No transport currents witout voltage differences between contacts. Te second relation between conductance coefficients is obtained

10 22 Multiterminal systems from te requirement tat no currents will flow into or out of te structure, if all voltages applied to te contacts are te same. Tis leads to te sum rule n G ij = (13.3) j=1 for te rows of te conductance matrix. Using te two above sum rules we can sow tat te stationary currents in any nanostructure depend only on voltage differences between contacts: I i = j j i G ij (V i V j ) = G ii V i + j j i G ij V j = j G ij V j. (13.4) 13.2 Conductance and transmission: Landauer Büttiker approac In te same way as for structures wit two contacts, also in te general case of many terminals te conductance matrix can be related to te transmission probabilities of quantum states from one contact to te oter. In order to sow tis, we again assume tat te quantum mecanical many-particle problem as been solved for te termodynamic equilibrium situation (i.e., µ 1 = µ 2 = µ 3 = µ 4 =...), for example, using te Hartree approximation. We allow a omogeneous external magnetic field B to penetrate te structure. Corresponding to our previous considerations for te quantum point contact, we write te asymptotic form of te scattering states in te leads wic are assumed to be perfect wires as { ψm(r) α = 1 χ α m (y, z)e ikα m x + n rα nmχ α n(y, z)e ikα n x for x L β,n tβα nmχ β n(y, z)e ikβ n x for x + Here, t βα nm describes te transmission amplitude from mode m in lead α into mode n in lead β. Correspondingly, rnm α is te reflection amplitude from mode m into mode n in te same lead α. Te wave functions χ α m(y, z) are te lateral modes in lead α. Tese scattering states describe a scattering experiment in wic a particle is incident from mode m of lead α and scattered into any oter mode in any oter lead. Te above scattering state represents te current contribution I αα = g s de [N α (E) R α (E)] f α (E) in lead α, were te number of modes at energy E in lead α is N α (E) = 1, m m α

11 13.3 Linear response: conductance and transmission 23 and te reflection back into lead α is R α (E) = n,m n,m α k α n(e) k α m(e) rα nm(e) 2. In addition, tere are currents in lead α tat arise from transmission from oter contacts. An arbitrary contact β α contributes te current I αβ = g s det αβ (E) f β (E), were we ave defined te transmission T αβ from lead β into lead α as T αβ (E) = n m n α m β k α n(e) k β m(e) t αβ nm(e) 2. (13.5) Te total current flowing in lead α is te incoming current minus te reflected current minus all te currents transmitted into lead α from oter contacts. Tis results in I α = g s de { [N α (E) R α (E)] f α (E) T αβ (E)f β (E) β β α }. (13.6) Termodynamic equilibrium. In te case f α (E) = f β (E), i.e., if all contacts are in termodynamic equilibrium, no current will flow in te system and all I α =. It follows tat te transmission and reflection probabilities fulfill te condition N α (E) = R α (E) + β T αβ (E). (13.7) It is te generalization of te condition tat in a system wit two singlemoded leads reflection and transmission probabilities add to one (R + T = 1) Linear response: conductance and transmission For obtaining te linear response we expand te Fermi distribution functions f β (E) in eq. (13.6) for small differences µ β µ α. Te result is I α = g s ( ) fα (E) det αβ (E) (µ β µ α ) µ β = ( µα G αβ µ ) β. β

12 24 Multiterminal systems Denoting te voltage differences between contacts α and β as V α V β, we obtain in agreement wit eq. (13.4) I α = β β α G αβ (V α V β ). (13.8) Here we ave introduced te off-diagonal elements of te conductance matrix in eq. (13.1) e 2 ( ) fα (E) G αβ = g s det αβ (E). (13.9) µ Equations (13.5), (13.8) and (13.9) are te basis of te Landauer Büttiker formalism for te calculation of te electrical conductance from te asymptotic form of te scattering states. Tese equations contain te special case of two-terminal structures suc as te quantum point contact Te transmission matrix For a sample wit n contacts, equation (13.8) can also be written in te matrix form I 1 N 1 R 1 T T 1n T 21 N 2 R 2... T 2n I 2. = e2 V 1 V I n T n1 T n2... N n R n V n (13.1) Here, N α denotes te number of modes in lead α, and te R α are te reflection probabilities. Te transmission probabilities are ere defined as T αβ = g s and according to eq. (13.7) ( ) fα (E) de T αβ (E) µ N α R α = β,β α T αβ. Tis equation ensures tat no current flows wen all contacts are at te same voltage, in complete agreement wit eq. (13.3). If we set te voltage V α to 1 V and all oter voltages to zero, carge conservation [eq. (13.2)] requires N α R α = T αβ. α,α β Wen te Landauer Büttiker formalism is used for calculating currents and voltages in an experiment, current and voltage contacts ave to be distinguised. For te latter, te net current is zero, because

13 13.5 S-matrix and T -matrix 25 te connected voltmeter will ave a very ig internal resistance. Te measured voltage on suc a voltage terminal is ten V α = β( α) T αβ γ( α) T αγ V β, (13.11) te average of te voltages on all oter contacts transmitting into it weigted by te relative transmission strengt. Also relevant for te application of eq. (13.1) to experimental problems is te fact tat not all lines of tis matrix equation are linearly independent. Tis is seen, for example, by adding te first n 1 equations and using te column sum rules. Te resulting equation is equal to te nt equation multiplied by 1. As a consequence, one equation of te system can be omitted. Tis is usually combined wit te freedom of te coice of te voltage zero wic allows us to regard one of te n voltages to be te zero voltage reference. If we cose V α = and omit equation α, te problem reduces to a system of n 1 linear equations wit n 1 unknown quantities.