A Quantum Hydrodynamic Simulation Transport Quantum...

VLSI DESIGN2001, Vol. 13, Nos. 1-4, pp. 237-244Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2001 OPA (Overseas Publishers Association)N.V.Published by license under

the Gordon and Breach Science Publishers imprint,member of the Taylor & Francis Group.

A Simple Model for the QuantumHydrodynamic Simulation of Electron

Transport in Quantum Confined Structuresin the Presence of Vortices

J. R. BARKER*

Nanoelectronics Research Centre, Department of Electronics and Electrical Engineering,University of Glasgow, Glasgow G12 8LT, Scotland, UK

It is demonstrated that the ballistic quantum transport properties of an open quantumdot may be described by an ensemble of spatially correlated virtual classical particlesmoving within self-avoiding strings. The string paths correspond to ray trajectories. Thestrings exhibit the necessary properties of self-avoidance, interference and the non-localcondition fmv dr nh. The formalism suggests that numerical simulation of quantumflows may be constructed ab initio by using the string representation.

Keywords: Quantum transport; Quantum dots; Quantum hydrodynamics; String model

1. INTRODUCTION

In a recent study [1] of steady state quantumtransport through open quantum dots we havedemonstrated that numerical simulations based onthe transfer matrix method may be revealinglymapped into velocity flows where the velocityvector field v(x) is defined as the ratio of thequantum mechanical probability current densityj(x) to the probability density p. The velocitystreamlines do not cross: a property of the single-valuedness of the wave function which leads to thevelocity field being derived from the gradient ofthe phase of the wave function. Consequently,

the qualitative features of the steady flow maybe deduced from the singularity structure in theflow which may be shown to comprise vortexcentres (generally nodes in the wave function)and hyperbolic points (generally at nodes inthe velocity field) (see Fig. 1). This picture isgeneric but we shall focus on a simple two-dimensional Hamiltonian system for simplicity:H pZ/2m+V(x).

If the quantum state of the system is known, forexample, in the polar representation where weknow the amplitude R(x, t) and phase S(x, t) ofthe wave function I,(x, t)= R(x, t)exp[iS(x, t)/h],then we can construct an a posteriori quantum

*Tel.: +44 141 330 5221, Fax: +44 141 330 4907, e-mail: [email protected]

237

238 J.R. BARKER

(a) (b)

FIGURE Velocity fields in the vicinity of the singular points of the quantum flows. (a) Hyperbolic fixed point; (b) Vortex centre.

hydrodynamics in which the probability velocityflow and probability density satisfy a continuityequation (Eq. (1)), an Euler like equation whichcontains the quantum potential (Eqs. (2, 3)).

(1)Ot

rn - + my. Vv -VV(x, t) VVQ(X, t) (2)

2) V2R(x, t)VQ -m R(x, t)

p--R2; my VS(x, t)

(3)

A third non-local condition (Eq. (4)) is alsosatisfied automatically if the wave function isknown; it corresponds to the condition that thewave function is single valued and has deepconsequences if gauge invariance is taken intoaccount:

VS. dr nh (n integer) (4)

The main purpose of a posteriori quantumhydrodynamics is in its interpretative value.The construction of an a priori or ab initio

quantum hydrodynamics or even a quantumMonte Carlo particle ensemble picture has beenan,important target in recent years, particularly asEqs. (1) and (2) seem to provide a simple route tomodelling quantum effects in decanano MOSFETsby using the quantum potential (which is afunction p of only in classical hydrodynamicmodels). The aim of ab initio quantum hydro-dynamics is to use the fluid Eqs. (1) and (2) toconstruct flows knowing only the appropriateinitial conditions and boundary conditions.However, primarily because of condition (4)

which imposes a quantisation of velocity circula-tion (a matter made more complex for mixedstates), it is not possible to reconstruct correctlyeven simple solutions to the Schr6dinger equation[2,3] by ab initio methods. The problem isexacerbated by the difficulty of generating thecorrect singularities in the flow which indeed areonly fixed for stationary flows. The singularitiesare mobile for time-dependent systems such as

QUANTUM HYDRODYNAMIC SIMULATION 239

occur in the transient response of devices or inshort channel devices especially in the presence offluctuations.

It was shown in [1] that the computed carrierflow in open quantum dots may be described interms of meandering open orbits which separateregions of trapped flows around vortices. It wasconjectured in [1] that the computed carrier flow inopen quantum dots may be described in terms ofmeandering open orbits which separate regions oftrapped flows around vortices. It was conjecturedin [1] that the topology of the flow may be derivedab initio from a simple geometrical ansatz in whichthe classical ballistic trajectories are first computedand then all crossings are replaced by hyperbolicflow separation and for which all trapped flowsform quantised closed orbits. In the present paperwe analyse and extend this picture to a computa-tional methodology for developing quantum hy-drodynamic descriptions of flows in the presenceof quantised vortices. The outcome is stronglyreminiscent of a primitive string theory.

2. SINGULARITY STRUCTURE

The steady flow corresponding to (2) and (3) isdescribed by two autonomous coupled equationsfor the particle position and velocity, which for 2Dcorrespond to:

dx_

OS dy_

OS(5)

dt Vx rnOx d-= Vx rn

Oy

The autonomous system (5) describes velocitystreamlines or trajectories, which cannot crossbecause the wave function is single-valued. Thetopology of the velocity flow may be established byusing stability analysis to characterise the flow inthe vicinity of the fixed points. There are twoclasses of singularity in the flow. Linear stabilityanalysis around the velocity nodes, points x0 forwhich v 0, reveals that for zero magnetic field thelocal flow pattern is hyperbolic; for non-vanishingmagnetic field there occur in addition closed

elliptical orbits due to the Lorentz force. Thehyperbolic fixed points (saddle points) act to

seperate the flow, whereas the closed orbitssurrounding fixed points which are centres corre-spond to Lorentz vortices. A second class of vortexflow occurs even in the absence of a magnetic fieldwhen V x v--0 everywhere except at the nodes inthe amplitude of the wave function [1, 5]. At theamplitude nodes the phase and hence the velocityis not defined. This is a purely quantum effect. Theclosed orbital motion around the centre of thequantum vortex is due to the quantum potentialVQ (see Eq. (3)), which in the vicinity of pointswhere x/- 0, may exhibit a Ix-x01 depend-ence leading to closed orbits. In the case of the2D open quantum dot structures of Figure wetherefore expect three classes of flow: (i) openorbits which connect the entry and exit leads of thedot; (ii) hyperbolic orbits which link to the saddlepoints or terminate at hard wall boundaries; (iii)Lorentz and quantum vortices. In the sequel weshall ignore the Lorentz vortices by setting B 0.The wave function may be reconstructed in thevicinity of the singular points from small signalanalysis of the flow and the polar form

I’(x, t)-- R(x, t)exp[iS(x, t)/h].

2.1. Ab Initio Quantum Hydrodynamics

Numerical studies of the rectangular open quan-tum dot [1], essentially a double aperture device,reveal that, standing wave patterns are formedupstream of the first aperture and within thesquare dot itself. At the first aperture a strongpropagating wave is injected which is approxi-mately confined to a ray path of width determinedby the aperture and which involves multiplereflections of the boundary walls before reachingthe exit aperture. The reflecting ray paths interferewith the standing wave in the dot to form micro-vortices as with the upstream case in the singleaperture. In addition, new and larger vortices formwhere the ray path loops back on itself due toreflections at the walls. At high wave vectors this isthe dominant feature. The origin of the large

240 J.R. BARKER

(a)

FIGURE 2 The computed flow of a long wave packet in an open rectangular dot. (a) The probability density; (b) the velocity field.

vortices may be seen in Figure 2 where we show asimple reconstruction based on evolving an initialinjected long wave packet with a plane wavemotion along the direction of the packet and astanding wave in the perpendicular direction.Neglecting dispersion, the evolution of the wavepacket involves multiple reflection off the wallsand when the phase condition (4) is satisfied aclear vortex develops in the derived velocityflow (Fig. 2).To construct the vortices we imagine a classical

path formed by a free particle bouncing betweenthe walls of the dot where the injection directionis determined from the incident wave-vector com-ponents (Fig. 3a). Because quantum trajectoriescannot cross, the quantum case is derivablequalitatively from the classical cases by lettingthe classical streamlines repel each other nearcrossing points: bifurcation of the flow occurs. Theresulting flow topology contains hyperbolic andvortex flows as shown in Figure 3b. This result isconfirmed by analytical and numerical calcula-tions. The net effect is that the complex classicalray trajectories associated with the quantum raypropagation separates the flow into localised flow

FIGURE 3 Classical ray paths corresponding to Figure 2.(a) Purely classical: path crossing displayed. (b) Quantumcase: paths cannot cross: bifurcation occurs.

around vortex centres and meandering open orbitswhich thread between them. Typically there areseveral open orbit paths some with minimumtraversal times, others with very long paths andhence long traversal times. However, this proce-dure is qualitative and we turn now to aquantitative algorithm. There are two possibleroutes: (a) develop a field theory of self-avoiding


streamlines; (b) develop a particle theory ofinteracting trajectories. We follow the latter here.

3. MATERIAL FLUID MODEL

A classical trajectory is defined as the pathfollowed by a real particle moving through space;the particle is at the head of the trajectory. Asimple representation of the quantum flow (quan-tum trajectories) is obtained by representing eachstream line as the path of a stream of non-

interacting virtual particles moving classically with-in the system and distributed according to theprobability density p. If a real particle is locatedsomewhere in space, it is then possible to re-construct its history by following the streamline

back in time. This picture carries precise predictivepower but depends upon being able to computethe streamline in the first place from a wavefunction. To see if this approach will work withoutrecourse to the wave function we might calculatethe evolution of a finite bundle of classical particlepaths corresponding to an initial set of positionsand momenta consistent with the initial wavefunction (for example for injection into the dot).This is shown in Figure 3 but no interference or

vortex motion occurs and indeed the trajectoriescross. To extend the model we now let the virtualparticles interact through a repulsive hard sphereinteraction, which prevents trajectory crossingwhere two virtual particles collide. If a finitenumber of particles are used we must maintaincommensurability of position to ensure that all

-12

-14

(a)

10

(b)

(d)

FIGURE 4 Spatially correlated virtual particle model corresponding to Figure 2 under vortex forming conditions (commen-surability of virtual particle spacing with closed path length). (a) Positions of virtual particles shown a short time after the flow hasdeveloped. (b) Enlargement of interaction of three streams of virtual particles at the wall reflection. (c) Enlargement of interaction ofthree streams of virtual particles at a cross over region. (d) Flow separation in limit of large number of virtual particle streams.

242 J.R. BARKER

crossing trajectories are used we must maintaincommensurability of position to ensure that allcrossing trajectories are deflected. In the presentcase the virtual particles are initially out onewavelength A h/mv apart along the path and thestreams are separated initially by the correspond-ing lateral wavelength. Figure 4 illustrates theeffect of three such virtual particle streamsreflecting off a hard wall (Fig. 4a) calculated bythe algorithm. An enlargement (Fig. 4b) showsthat the reflection off the wall involves the upperleading particle bouncing off the wall but collidinglater with the leading particle in the second streamresulting a return to the wall and a furtherreflection sequence. As we increase the numberof particle streams the net effect is for the loweredge stream of particles to reflect well before thewall, whilst the other trajectories follow a pathrunning parallel to the wall before reflecting. Thisagrees well with the a posteriori results. Similarly,collisions between particles where trajectoriescross show clear flow bifurcation and the appear-ance of a vortex provided the closed path of eachvortex line is commensurable with the particlespacing. This condition is-- n => k. dr 27rn => hk. dr- nh

mv. dr nh

(6)

Conditions (6) correspond to the phase condition(4). If (6) is not satisfied, the long-time evolution ofthe virtual particles leads ultimately to a stationarydistribution in the lower part of the dot and anopen set of orbits entering and leaving the dot inthe upper half.

4. STRING FLUID MODEL

The model described in Section 3 is very unstableand of poor practical application. It is also crucialto build in the coherence of the wave function viathe spatially correlated initial virtual particledistribution. However, the algorithm may be

stabilised by a string model (Fig. 5a) in whicheach virtual particle is bonded into a string ofvirtual particles, each of which, is bound by anearest neighbour attractive potential at largeparticle separation but has a repulsive core atsmall separation (Fig. 5d). The model interactionpotential for neighbouring virtual particles at xjand xj+l is chosen as:

)V--Z og(xk)g(Xk+l)g(xk) g(X)k+l

(7)k

where the functions g(x) are generalised densityfunctions with a broadened 5-function character-istic describing the smearing out of the virtualparticles along the string. String interactions maybe modelled by two processes: string repulsion(Fig. 5b) along the bodies of two strings; bondbreaking leading to bifurcation of the flow when a

string head encounters a string body (Fig. 5c). Thismodel has real computational power and containsseveral new features. We note first that the virtualparticle interactions lead to waves along thestring. The parameter a may be chosen to give thefamiliar quantum relation between wave vector kfor these waves and the particle velocity. Thecommensurability needed to obtain the quantisa-tion of circulation (a closed string) has its originshere. In the continuum limit the inter-particlepotential may be related to the quantum potential.Denoting the average density of virtual particlesby p(x) in the continuum limit we obtain:

Xk --- X Xk+l - X + 6X, c16Xl-- /3

Vfd3xp/(p2--p-) (8)

By choosing the parameter /3 appropriately (7)becomes

h-m D-l (Vlg)2d3x (9)

The effective Lagrangian for this model maybe represented as: L=fd3xL where theLagrangian density is written in terms of the


1.75

1.25

(a) (b)

(c)

FIGURE 5 The string model. (a) String bundle model. (b) String repulsion. (c) String collision: bond breaking. (d) Effective inter-particle interaction potential.

classical action S as

L--R2(xt)l(VS(x’t))2’2m +-mh2(VR(x’t))2---tOS(x, t) }+ V(x, ) + 0---T- (]0)

and we have written p-R2. The Lagrangianequations of motion are easily obtained for thetwo fields R and S and correspond to Eqs. (1)and (2) consistent with the Schr6dinger equation.For a closed string it is essential in this modelthat Eq. (6) holds. Thus the string model reducesto the conventional Schr6dinger picture in the

244 J.R. BARKER

continuum limit. It is also relatively easy to velocity circulation quantisation. It is clear thatcalculate the adiabatic evolution of bound states practical difficulties remain to be resolved butwithin the string model for example by moving the full string model in the non-continuum limitthe hard wall boundaries of a stationary confined holds considerable promise. Furthermore dissi-system, pative processes are easily incorporated followin

the prescription outlined in [4]. The formalismdescribed here has described coherent quantum

5. DISCUSSION AND CONCLUSIONS transport but provides a basis for the onset ofphase breaking processes if a non-Hermitean

We have outlined a route towards an effective quantum Hamiltonian is used to derive theparticle representation of simple quantum sys- string equations of motion.tems which resemble a self avoiding walkproblem or a string theory. The virtue of the

Refe’encesmodel is that for the first time it possible in anab initio calculation to predict vortex formation [11 Barker, J. R., Akis, R. and Ferry, D. K. (2000). Super-and the occurrence of all the topological features lattices and Microstructures, 27, 319-325.

[2] Barker, J. R. and Ferry, D. K. (1998). Semiconductorof quantum flows. The inter-particle potential Scienee andTechnology, 13A, 135-139.plays a similar role to the quantum potential VQ [3] Barker, J. R. (1998). Extended Abstracts, 1998 Sixth

International Workshop on Computational Electronicsin a posteriori quantum hydrodynamics but the (IWCE-6) IEEE Catalog No. 98EXl16, pp. 1-4.incorporation of coherence via the correlated [4] Barker, J. R. and Watling, J. (2000). Superlattiees and

particles in the strings provides the basis Microstructures, 27, 347- 351.[5] Hua Wu and Sprung, D. W. L. (1994). Phys. Letters A,

for describing interference phenomena and the 196, 229.

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