theory transport devices using the pauli...

VLSI DESIGN1998, Vol. 8, Nos. (1-4), pp. 173-178Reprints available directly from the publisherPhotocopying permitted by license only

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Printed in India.

Theory of Electron Transport in Small SemiconductorDevices Using the Pauli Master Equation

M. V. FISCHETTI*

IBM Research Division, Thomas J. Watson Research Center, P. O. Box 218, Yorktown Heights, NY 10598, USA

It is argued that the Pauli master equation can be used to simulate electron transport invery small electronic devices under steady-state conditions. Written in a basis of suitablewavefunctions and with the appropriate open boundary conditions, this equationremoves some of the approximations which render the Boltzmann equationunsatisfactory at small length-scales. The main problems consist in describing theinteraction of the system with the reservoirs and in assessing the range of validity of theequation: Only devices smaller than the size of the electron wavepackets injected fromthe contacts can be handled. Two one-dimensional examples solved by a simple MonteCarlo technique are presented.

Keywords: Master equation, electron transport, small devices

1. INTRODUCTION

As noted by Frensley discussing quantum trans-port in open systems [1], the Pauli master equation[2], (PME) could constitute an intuitive descriptionof electron transport in semiconductor devices ofsize comparable to the electron wavelength: Onecould capture the wavelike nature of transport,lost in the Boltzmann-transport-equation (BTE)picture, so bypassing the weak-field, long deviceslimitations. In retaining the weak-scattering andcompleted-collision limits one would not reach afull-fledged quantum description, as required tostudy very fast time transients [3, 4]. Yet, asdevices approach the sub-50 nm length-scale, the

advantage of avoiding the concept of point-likeelectrons appears overwhelming.Although arguments have been raised against

the correctness of the PME [1], it is applicable tovery small devices and to steady-state phenomena:Small devices, because the PME rests on theabsence of off-diagonal elements of the densitymatrix p. In turn, their absence implies that thecontacts inject ’plane waves’, which correspondsphysically to the injection of spatially highlydelocalized wavepackets. Hence, the device mustbe smaller than the ’dephasing length’ in thecontacts [5]. For heavily doped Si, this is of theorder of 50 nm, which is of technological interest.Steady state, since any nontrivial time transient

* Corresponding author.

173

174 M.V. FISCHETTI

(e.g., relaxation of photo-excited carriers, fastswitching of a device) would excite those off-diagonal terms of p which the PME cannot handle.Indeed, as also note by Frensley [1], the PME isinconsistent with current continuity in this case.

2. MASTER EQUATION AND CONTACTS

Let’s consider a semiconductor region unboundedin the (x, y)-plane, homogeneous in this plane, incontact with two reservoirs at z 0 and z L. Let’swrite the single-particle Hamiltonian as:

e V(z) + Hint -[- Hres H0 t_ Hint -[- gres,

(1)

within the envelope/effective-mass approximation,m* being the effective mass. Hint describes theelectron-phonon interaction or interactions withimpurities, etc. nre represents the interactionbetween the device and the external reservoirs.The time evolution of the system is given by the’transport’ equation for the ’reduced’ (electronic)density matrix p:

0-- [p,H] + res" (2)

The last term on the right-hand-side describes theeffect of the reservoirs. Assuming Hint a H’, wecan derive a transport equation from Eq. (2), byconsidering only terms to the leading order in a,and following the ’standard’ derivation of Kohnand Luttinger [7], but now using as basis functionsnot plane waves, but the eigenstates IK) of H0,such that HoI#K)= [E, + h2K2/(2m*)]l#K)=E,(K)I#K). This simplifyies Eq. (2), since thecommutator now contains only the interactionterm Hint. The wave-functions are simply of theform u(z) exp (i K. R)/(27r), where Kand R are thewavevector and position on the (x, y)-plane, andthe functions (u(z) obey the one-dimensional

Schr/Sdinger equation for the potential V(z),subject to the boundary conditions V(0)= VL andV (L) VR. Taking -eVr > -eVn, for any givenenergy Eu>-eVr, two independent solutionsrepresent waves incident from the left ( + ) or right(-), and partially reflected and transmitted, withreflection and transmission amplitudes r+ and +.For -eV > Eu > -eV, the nondegenerate solu-tions represent non-current-carrying states. In thefollowing the index a 4- will label left- andright-traveling waves in the energy range Eu>e VL, and (zl#a) (z).To the leading order in a, the diagonal elements,

pux, (symbol used in place of puxux,) of thedensity matrix obey the Eqs. 7- 9]:

uK’a’##Ka

( Op,x+ \ Ot Je"

here Wx,,;ux represents the probability per unittime of a transition from the state [#Ka) to thestate [u K’a) using Fermi golden rule. While thederivations provided in [7-9] can be followed veryeasily, now one must also assume that thereservoirs do not alter the ratio a between off-diagonal and diagonal matrix elements. Van Hove[8] has shown the validity of Eq. (3) in describingapproach to equilibrium, in the case of closedsystems (for which the term (Op/Ot)res is absent),provided the initial state is chosen as ’quasi-diagonal’: At t=0 we must have puxvX,o=O forEu(K) Ev(K’) > 6, where 6/h is of the order ofthe relaxation rate, 1/r. Similarly, Eq. (3) is validas long as the contacts do not ’inject’ off-diagonalelements, that is (Opuxx,,/Ot)res 0 for Eu(K)E(K) > 6. This is the case for very delocalized

wavepackets entering the contact, plane wavesbeing the ideal limiting case. In this limit, the effectof the contacts can be modeled phenomenologi-cally by assuming that the left reservoir attemptsto restore charge neutrality by injecting a flux of

ELECTRON TRANSPORT AND MASTER EQUATION 175

particles of any K into the right-bound componentof the states associated to waves incident from theleft, +, fixed by the Fermi level at the cathode, E.Accounting also for the flux out of the device viaeither contact, one can write"

Ot res(Eu) pu+ T-pu_] m----2

(4)

where[2m*(E. +eVR/L)] 1/2, and the A’s are normali-zation constants. Similar expressions also hold forstates traveling from the right. The occupationof the states at thermal equilibrium, as determinedby the ’left’ reservoir, has been expressed in termsof the equilibrium Fermi function (integrated overthe ’transverse’ wavevectors K), f(th,L) (Eu) pos-sibly shifted in k-space, in order to account for theopen nature of the contacts [1, 6]. By analogy, asimilar expression (with the terms p,+- Tp,_replaced by Pu) is also used to express the linearrelaxation of the of states with energy -eVn < Eu< -e VL. Finally, the net current flowing throughthe device is given by:

//kS + + 2J -e p.+,m--7 It. A,

where obviously only the doubly-degenerate stateswith E, >-eVr contribute to the sum.

Criticisms have been raised against Eq. (3) basedon the violation of continuity [1]. Yet, it should benoted that at steady state the reservoirs and thereverse scattering-induced transitions provide the’missing’ current and current continuity is triviallysatisfied. However, continuity is indeed violatedwhen using the PME to describe ’fast’ time-dep-endent phenomena: For example, any time depen-dent perturbation V (t) will cause an electron inI#Kr) at 0 to have nonzero amplitude also in astate IK’r’). At long times the requirement thatthis amplitude be negligible translates into thecondition leO6V/Otl << h/7", expressing the fact

that the applied bias cannot change appreciablyover a relaxation time, so that the PME is unableto handle nontrivial time-dependent phenomena.

3. MONTE CARLO SOLUTIONOF ONE DIMENSIONAL EXAMPLES

In order to implement Eq. (3) in one-dimensionalsituations, starting from a given potential V(z),such as a Thomas-Fermi solution, the Schr6dingerequation is solved with vanishing Neumannboundary conditions. The eigenvalues {E} willbe spaced densely enough (to accurately reproducethe bulk density of states in the reservoirs) if thelength L of the device is selected appropriately.The extended states can be finally decomposedinto left- and right-traveling waves following the’quantum transmitting boundary method’ by Lentand Kirkner [10]. The levels I#) are populatedfollowing P6tz [6] and the Poisson equation can besolved again, using well-known potential-dampingschemes to improve convergence. Iterating thisPoisson/Schr6dinger scheme yields a solution inthe ballistic case. Scattering is introduced bycalculating all transition rates Wx,,;ux betweenthe levels, and adding the PME as a third equationover which one must iterate. A Monte Carloalgorithm is used here. Scattering and injection/exit from contacts are treated stochastically inorder to determine the new populations Pux.Typically, 300,000 ’electrons’ are employed, usinga time step t 10-16 S in the Monte Carloalgorithm. After about 100-500 Monte Carlosteps, the Poisson and the Schr6dinger equationsare solved again until convergence is obtained.Finally, the current flowing through the device isobtained from Eq. (5), while information aboutcarrier density, kinetic energy, and velocity as afunction of position inside the device can beobtained from the expectation values of thecorresponding operators as traces over theirproducts with the density matrix.As a first example, let’s consider a simple nin

resistor at 300K consisting of a 300 nm-long

176 M.V. FISCHETTI

region between z 0 and z L with n-type regions100 nm-long n-type doped to 1017 cm-3, and anintrinsic region also 100 nm long. A uniformeffective mass of m* 0.32 m0 (where m0 is thefree electron mass) and a dielectric constant 11.7

eva (where Cvao is the permittivity of vacuum) areused. Nonpolar scattering with optical phonons isincluded using a photon energy 60 meV, an opticaldeformation potential of 5 108 eV/cm, a crystaldensity Px 2.33 g/cm3. These parameters imply a’dephasing length’ , in reservoirs of the samematerial of the order of 785 nm, larger than theactive region of the device, about 100 nm long.

Figure shows the potential, carrier density,kinetic energy, and velocity at the end of theiteration. The dashed lines are the corresponding

0.2 12.0master EBoltzmann |

1.0 z

0.

0.00 50 00 50 200 250 a00

2

0 0.00 50 100 150 200 250 300

POSITION (nm)

FIGURE Potential and charge density (top frame), kineticenergy and drift-velocity (bottom frame) as a function ofposition z inside an nin device calculated using the master (solidlines) and Boltzmann (dashed lines) equation. The chargeaccumulation at the device boundaries is an artifact caused byboundary conditions employed to solve the Schr6dingerequation.

quantities obtained from a conventional MonteCarlo solution of the BTE using identical para-meters. The current density is about 6.78 x 104 A/cm2, about 10% smaller than the value obtainedusing the BTE, 7.58 x 104 A/cm2. This differenceoriginates mainly from a slightly higher built-inbarrier at the cathode/intrinsic-region junctionseen in the ’quantum’ results, caused by thepenetration of wavefunctions into the barrieritself. The penetration of charge into classicallyforbidden region is also the origin of the redis-tribution of charge seen in the top frame ofFigure 1" The charge density obtained using thePME is larger in the intrinsic region, as wavefunc-tions penetrate into the gap beyond the classicalturning point. This charge redistribution is themajor factor responsible for the differences seen inaverage kinetic energy and drift velocity.The second example is a simple GaAs/A10.3_

Ga0.TAs resonant tunneling structure. Two 150 nmthick layers of GaAs-like semiconductor with n-type doping (2 x 1017 cm-3) are separated by twoA1GaAs tunnel barriers, each 2.8 nm thick,enclosing a 4.5 nm wide semiconductor well.Two 3 nm-thick undoped ’spacer’ layers, in turn,enclose the double-barrier region. ’Conventional’parameters for GaAs and A10.3Ga0.7As are used[12], also including nonpolar scattering withacoustic phonons and polar scattering with long-itudinal optical phonons. A lattice temperature of300 K is used. The contact mobility used todisplace the distributions injected by the reservoirsis taken to be el 5000 cm2/Vs, corresponding toa dephasing length in the reservoirs, 130 nm,larger than the active region of the device. Sinceeven at room temperature scattering is notsufficiently strong to fully populate the states inthe accumulation layer, as discussed by Frensley[1], the curves in Figure 2 show that a largefraction of the applied bias drops over the cathoderegion to the left of the double-barrier region.Although similar results have been obtained byFrensley [1] and Kluksdahl et al. [11], they appearpoorly credible, since the potential profile clearlydepends on the size of the simulated region.

ELECTRON TRANSPORT AND MASTER EQUATION 177

0.30

0.20

0.10

o -0.00

z -0.1.0

-J --0.20

z -0.30

-0.40

-0.50

-0.60-150-100 -50 0 50 100 150

POSITION nrn

FIGURE 2 Potential energy vs. position in a resonant tunneldiode calculated including electron-phonon scattering. Thecurves are parametrized by the applied bias from 0.0 to 0.6 Vin steps of 0.05 V. Note the ’gaps’ at the left of the barriersaround 0.2 and 0.45 V. Convergence could not be obtainedclose to these biases, because of an instability caused byscattering, as discussed in the text. The dotted solutions forbiases of 0.2 and 0.45 V are only representative of the averagepotential during the iteration.

Figure 3 shows the current-voltage character-istics: As the resonance is approached (at about0.2 V), the associated build-up of charge in thewell screens the cathode field, increasing the fieldacross the collector barrier. A critical point is soon

o.o’o v

z i, )’o s o s I /

0.00 0.10 0.20 0.30 0.40 0.50 0.60APPLIED BIAS (V)

FIGURE 3 Current density in the GaAs/A1GaAs resonanttunneling structure with the inclusion of electron-phononscattering. Note the ’snap back’ of the current at biases of 0.2and 0.45 V, corresponding to the instability regions of Figure 2.The error bars are used to illustrate the range of the oscillationof the current. The inset shows the Calculated current densityvs. simulation time at two bias points, slightly away from(0.175 V) and close to (0.200 V) resonance. Note the instabilityof the current at the bias point near resonance.

reached at which the rate at which scattering feedscharge into the states localized in the well becomessmaller than the rate at which the well-charge leaksto the right. Charge is removed from the well, andthe potential profile now ’snaps’, a larger voltagenow dropping at the left of the barriers. Thepopulation of the current-carrying scatteringstates, and so the current itself, now drops. Asthe applied bias is increased further, another’snapping point’ is reached, at about 0.45 V. Thesecritical bias points can be seen as ’gaps’ in thecathode potential-profile in Figure 2.As loosely indicated by the error bars in Fig-

ure 3, for biases close to and slightly beyondresonance the situation is unstable: As ’resonantwavefunctions’ leak out to the anode, scattering tonon-current-carrying states in the collector regionis enhanced, thanks to a larger overlap (form)factor. The resulting redistribution of charge altersthe potential profile, driving the system away fromresonance. As this happens, the charge in theanode side of the barriers drops once more, andthe system is now driven back towards a resonantcondition, and the cycle repeats once more. Theinset of Figure 3 illustrates some features of thisinstability. The good convergence properties of thecomputed current density for a bias slightly awayfrom resonance (0.175 V) are compared to theunstable behavior for a bias close to the firstresonance (0.200 V). In the latter case, with someimagination one may detect an oscillatory behav-ior between two states [11, 13].

4. SUMMARY AND CONCLUSIONS

The main conclusion of this paper is that there isan interesting class of problems for which the Paulimaster provides a solution of the electron trans-port problem overcoming some of the strongestlimitations of the semiclassical Boltzmann equa-tion: For devices with active regions smaller than afew hundreds of nm, the master equation makes itpossible to account for the wave nature of theelectrons, although only in the weak scattering

178 M.V. FISCHETTI

limit and in steady state or slow (adiabatic) timetransients. The major price one pays is the inabilityto handle fast time-transients and strong scatteringsituations, collisional broadening and coherenttransport in the femtosecond time scale being themost notable examples. The advantages consist inthe ability to treat inelastic scattering processes,arbitrarily strong electric fields (including intra-collisional field effects) tunneling phenomena,elastic and not, and, in general, all those effectswhich depend on the wavelike nature of theelectrons.

Acknowledgements

The author is deeply indebted to W. R. Fresnley,A. Kumar, R. Landauer, S. E. Laux, P. J. Price,N. Sano and F. Stern for many discussions,suggestions, and criticisms, and to L. Reggianifor having drawn attention to the important workby van Hove.

semiconductors: Coherent and incoherent dynamics",Phys. Rev. B, 46, 7496-7514.

[5] Thanks are due to R. Landauer for making this sug-gestion, with the strong warning, however, that no formalproof has ever been given that this is indeed true.

[6] P6tz, W. (1989). "Self-consistent model of transport inquantum well tunneling structures", J..Appl. Phys., 66,2458-2466.

[7] Kohn, W. and Luttinger, J. M. (1957). "Quantum Theoryof Electrical Transport Phenomena", Phys. Rev., 108,590-611.

[8] L6on, Van Hove (1955). "Quantum-mechanical perturba-tions giving rise to a statistical transport equation",Physica, XXI, 517- 540.

[9] Jones, W. and March, N. H. (1973). "Theoretical SolidState Physics, Volume 2: Non-equilibrium and Disorder"(Wiley-Interscience, Bristol), p. 736 ft. and AppendixA6.1.

[10] Lent, G. S. and Kirkner, D. J. (1990). "The quantumtransmitting boundary method", J. Appl. Phys., 67, 6353-6359.

[11] Kluksdahl, N. C., Kriman, A. M., Ferry, D. K. andRinghofer, C. (1989). "Self-consistent study of theresonant-tunneling diode", Phys. Rev. B, 39, 7720-7735.

[12] Fischetti, M. V. (1991). "Monte Carlo Simulation ofTransport in Technologically Significant Semiconductorsof the Diamond and Zinc-Blende Structure", IEEE Trans.Electron Devices, 38, 634-649.

[13] Buot, F. A. and Rajagopal, A. K. (1995). "Theory ofnovel nonlinear quantum transport effects in resonanttunneling structures", Mat. Sci. Eng. B, 35, 303-317.

References

[1] Frensley, W. R. (1990). "Boundary conditions for openquantum systems driven far from equilibrium", Rev. Mod.Phys., 62, 745-791.

[2] Pauli, W. (1928). In "Festschrift zum 60. Geburtstage A.Sommerfeld" (Hirzel, Leipzig), p. 30.

[3] Jacoboni, C. (1992). "Comparison between quantum andclassical results in hot-electron transport", Semicond. Sci.Technol., 7, B6-- B11.

[4] See, for instance, Khun, T. and Rossi, F. (1992). "MonteCarlo simulation of ultrafast processes in photoecited

Author Biography

Massimo V. Fischetti is a Research Staff Memberat the IBM T. J. Watson Research Center inYorktown Heights, New York. His researchinterests include electronic transport in semicon-ductors and insulators and the simulation ofsemiconductor devices. He is a Fellow of theAmerican Physical Society.

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