Resonant Tunneling in the Quantum Hydrodynamic Model

The phenomenon of resonant tunneling is simulated and analyzed in the quantum hydrodynamic (QHD) model for semiconductor devices. Simulations of a parabolic well resonant tunneling diode at 77 K are presented which show multiple regions of negative differential resistance (NDR) in the current-voltage curve. These are the first simulations of the QHD equations to show multiple regions of NDR. Resonant tunneling (and NDR) depend on the quantum interference of electron wavefunctions and therefore on the phases of the wavefunctions. An analysis of the QHD equations using a moment expansion of the Wigner-Boltzmann equation indicates how phase information is retained in the hydrodynamic equations.

INTRODUCTION sonant tunneling of electrons in quantum emiconductor devices can be modeled by adding quantum corrections to the classical hydrodynamic equations.The leading O(h2) quantum corrections include the effects of particle tunneling through potential barriers and particle buildup in potential wells.
ers.Sequential tunneling occurs if electrons tunnel through the first barrier, undergo scattering, and then tunnel through the second barrier.Coherent tunneling may be interpreted as the constructive interference of multiply reflected electron waves within the well.Resonant tunneling in actual devices usually involves a mixture of coherent and sequential tunneling.Scattering broadens the resonances and reduces the peak to valley ratios in the current- voltage curve.In this investigation, I will present two sets of simulations of the resonant tunneling diode: in the first set, scattering takes place throughout the device, while in the second set the collision terms in the QHD model are set to zero in the barriers and the well, to insure that the electron tunneling is coherent.
Coherent resonant tunneling and negative differ- ential resistance depend on the quantum interfer- ence of electron wavefunctions, and therefore on the phases of the wavefunctions.I will indicate how phase information is retained in the hydrodynamic equations through an analysis of the OHD equa- tions using a moment expansion of the Wigner- Boltzmann equation.202 CARL L. GARDNER

THE QUANTUM HYDRODYNAMIC MODEL
The QHD equations have the same structure [1] as the classical hydrodynamic equations: the stress tensor by h2n 0 2 eij -nTtij -t-12-OxiOx log(n) + O(h4) (6)   On where n is the electron density, rn is the effective electron mass, II is the momentum density, u is the velocity, Pij is the stress tensor, V= -eb is the potential energy, b is the electric potential, e > 0 is the electronic charge, W is the energy density, q is the heat flux, T O is the temperature of the semicon- ductor lattice in energy units (k B is set equal to 1), s is the dielectric constant, N o is the density of donors, and NA is the density of acceptors.Spatial indices i, j equal 1, 2, 3, and repeated indices are summed over.Eq. (1) expresses conservation of electron number, Eq. ( 2)expresses conservation of momen- tum, Eq. (3) expresses conservation of energy, and Eq. ( 4) is Poisson's equation.The classical collision terms in Eqs. (2)and (3) are modeled by the relax- ation time approximation, with momentum and en- ergy relaxation times Tp and "r w.The heat flux is specified here by Fourier's law q -KVT, where T is the electron temperature.
The quantum corrections to the classical hydrodynamic equations appear in the stress tensor and the energy density.As h --* 0, the quantum corrections can be developed in a power series in h2.The actual expansion parameter is h2/8mTl 2, where is a characteristic length scale of the problem [2, 3].For the resonant tunneling diode in section 3 with T --T O 77 K and 100 A, the expansion parameter --0.23.
In Ref. [1], I showed that for the O(h2) momen- tum-shifted thermal equilibrium Wigner distribution function, the momentum density is given by 1-I mnu (5)   and the energy density by Wigner [4] derived the quantum correction to the energy density (in a different form).Ancona and Tiersten [5] proposed the quantum correction to the stress tensor on general thermodynamical grounds, and Ancona and Iafrate [2] later derived the correc- tion in the Wigner formalism.A one-dimensional version of the QHD equations was derived by Gru- bin and Kreskovsky in Ref. [6].

SIMULATIONS OF MULTIPLE RESONANCES
I will simulate a GaAs resonant tunneling diode with double Alo.8Gao.2Asbarriers and a parabolic AlxGaa_ x As quantum well.The diode consists of an n + source (at the left), an n channel, and an n / drain (at the right).The doping density No 10 TM cm -3 in the n + source and drain, and No 5 1015 cm -3 in the n channel (see Figure 1).The channel is 500/ long, the barriers are 30 , wide, and the well between the barriers is 300 , wide.
The device has 70 .spacers between the barriers and the contacts (source and drain) to enhance NDR.The barrier height ' is set equal to 0.7 eV.
In the well, the AI mole fraction X varies between 0.8 and 0, so that the potential barrier height in the well is (see Figure 2) [ ] 2 x + x.) "well 0.7 eV 1/2( XR (8)   where x/ and x R are the coordinates of the left and right edges of the well.The left edge of the first barrier and the right edge of the second barrier are modeled as step functions.(Computationally the step function goes from 0 to 1 over one Ax.)For the classical momentum and energy relax- ation times, I use modified Baccarani-Wordemann models: r0 'p "rpo T (9)   (3) Zp T 'w---1 + 2 (10)   -mv where the low-energy momentum relaxation time "rpO is set equal to 0.9 picoseconds from the low-field electron mobility in GaAs at 77 K.For lower valley electrons in GaAs at 77 K, the effective electron mass rn 0.063 me, where m e is the electron mass, and the saturation velocity v = 2 10 7 cm/s.I set K0 0.4 in the Wiedemann-Franz formula for thermal conductivity q KVT, r, XOZponTo/m.( The dielectric constant e 12.9 for GaAs.I discretize the 1D steady-state QHD equations using the second upwind method and compute the solution using a damped Newton method (see Ref. [1] for details).The barrier height ' is incorporated into the QHD transport equations ( 1)-( 3) by replacing I/ I/+ '.Poisson's equation is not changed.
The current voltage curve for the resonant tunnel- ing diode at 77 K is plotted in Figure 3, using a grid with 300 Ax.For these simulations, scattering takes place throughout the device.There are seven reso- nances and seven regions of negative differential resistance in the current-voltage curve between 0 and 0.5 volts.Figure 4 presents the current-voltage curve for the same device with the classical collision terms in the momentum and energy equations ( 2) and ( 3) turned off in the barriers and the well.Electron tunneling in this case must be purely coherent rather than sequential plus coherent, since there is no scattering in the well or barriers.There are twelve resonances and seven regions of negative differen- tial resistance in the current-voltage curve between 0 and 1 volt.
Note that there are five resonant "shoulders" in the current-voltage curve between 0 and 0.33 volts, and that the sixth through twelfth resonances be- come more pronounced as V increases.Ref. [7]  explains this effect, which was observed in an experimental device: Below 0.33 volts, electrons tunnel out of the well through the thick portion of the parabolic well in Figure 2, and the transmission resonance widths are small.As the voltage bias increases above 0.33 volts, the transmission reso- nance widths rapidly increase because (i) the right barrier height is progressively reduced and (ii) elec- trons tunnel out of the well through the thin portion of the parabolic well.
The resonant peaks of the current-voltage curve occur as the electrons tunneling through the first barrier come into resonance with the energy levels of the quantum well.The locations of the reso- nances can be qualitatively understood from the 0 0.i 0.2 0.3 0.4 0 Volts FIGURE 4 Current density in milliamps/cm vs. voltage for the resonant tunneling diode at 77 K with scattering turned off in the barriers and the well.The dots represent computed solution points.FIGURE 5 Time (solid line) in 10 -4 seconds spent by electrons in the well vs. voltage with scattering throughout the device.For reference, a scaled version of the current density (dots) is also shown.energy levels of an infinite parabolic well.For an infinite well, E,= A+ h r n x ----= A+-0.087eV (12) where x 0 300 is the width of the finite well and A is a non-negative integer.The energy levels of the infinite parabolic well are linearly spaced in A with a spacing of 87 meV, compared with the linear spacings of approximately 70 mV (with scattering throughout the device) and 100 mV (with scattering turned off in the barriers and the well)for the computed peaks in the current-voltage curves.The first peak with scattering throughout the device oc- curs at 0.030 volts, compared with E 0 = 0.044 eV.
The overall shape of the current-voltage curve in Figure 3--the rise to a peak at 0.11 volts and the fall to the valley at 0.58mis determined by the negative differential resistance of a 0.7 eV high double barrier square well with an effective width of approximately 75 A. The energy levels of the parabolic well modulate this basic shape.
The time spent by electronics in the well is shown in Figure 5.The "dwell" time has relative maxima at voltages corresponding to valleys (relative minima) of the current-voltage curve.The absolute maximum of the dwell time occurs at the global valley of the current-voltage curve at 0.58 volts; the dwell time subsequently decreases rapidly toward zero (Cf.Ref. [1], Figure 5).The dwell time mimics 1/Ijl, where j =-enu is the current density, since the peak electron density in the well increases monotonically with applied voltage.Microscopic quantum calcula- tions predict that the electron dwell time is maxi- mum at resonance.The QHD model presents a different "macroscopic" interpretation of the dwell time (see the discussion in Ref. [1]).

PHASE INFORMATION IN THE QHD MODEL
The computations of multiple resonances and re- gions of negative differential resistance naturally raise the question: how do the hydrodynamic equations, expressed in terms of macroscopic variables n, u, and T, "know" about quantum interference, which depends on the microscopic phases of the electron wavefunctions?By analyzing the first three moments of the Wigner-Boltzmann equation, I will show how the phases of the electron wavefunctions are present in the hydrodynamic equations.
For a mixed quantum mechanical state described by wavefunctions (x, t)(A 1, 2,... ) with occupa- tion numbers ax, the Wigner distribution function is defined as fw(X, p, t) (Trh) -3 Ea f day The sum of the occupation numbers Y'.xa M, where M is the total number of electrons in the system.In general the ax's may vary slowly with x and t.For the purposes of the derivation below, I will assume the aa's are constant. (For a mixed state in thermal equilibrium with a heat bath at tempera- ture T--1/fl, aa t e-te.)I will normalize the wavefunctions so that fd3x 1 1.
I will assume that the electron flow can be ap- proximated by a single-particle effective mass Schr6dinger equation with a self-consistent manybody field: o h 2 ih Exqa V2 + Vx (14) at where the wavefunctions have energies Ex and V(x) is the self-consistent potential ener.In the effec- tive mass approximation, Schr6dinger's equation with the free electron mass and the total potential ener (the periodic potential ener of the semiconductor lattice plus V) is rewritten with the total potential ener replaced by V and the electron mass re- placed by the effective electron mass.
The local average value of an obseable X is defined in the Wigner formalism as < x(x,t)) fd3px(x,p,t)fw(x,p,t). (15)e first three moments of the Wigner-Boltzmann equation are obtained [8,1]   define lIi, eij, W, and q.The quantum conservation laws have the same form as their classical counter- parts.Explicit factors of h enter only at the fourth and higher moments.
To calculate the average values in the first three moment equations, in Ref. [1] I used the momen- tum-shifted version of the O(h2) thermal equilib- rium solution [4] to the Wigner-Boltzmann equation.
The momentum-shifted O(hz) fw need only approx- imate the actual Wigner distribution function closely enough for the average values in Eqs. ( 16)-( 18) to be close to the actual values.Using this approach, I derived the three-dimensional QHD equations ( 1)- (3) with Hi, Pij, and W given by Eqs. ( 5)- (7).
Since here I am concerned with how phases ap- pear in the QHD equations, I will instead evaluate average values by writing the wavefunction q,x in terms of its magnitude A a and phase 0" qa(x, t) Ax(x, t)e i(x't). (23 Using this decomposition of the wavefunction, the density is given by n fd3pfw(X, p, t) E a f x (rh) 3d3yA:t(x + y)Aa(x y) Xe-iOa(x+y) +iOa(x-y)e2ip.y/ h EaaA2(x,t)= En (24 where the time-dependence in A a and 0 h is left implicit except when needed for the sake of clarity. n is the electron density for state A, and the total electron density is obtained by summing over all states.The momentum density equals < Pi) Hi fd3pPifw( x, P, t) ) A;(x + y)A(x-y)e -iOa(x+y)+iO"(x-y) o OYi {Ax(x + y)A,(x y)e -i^(x+y)+ia(x-y)} h EaxA2(x, t)V0(x, t) .,mnxuxiwhere I have integrated by parts and where the velocity u is defined by h u a --VO a.
(26) m This definition agrees with the standard expression for the semiclassical momentum pa mu hVO, of an electron in state A. Next I calculate From Eq. ( 27), the energy density equals uiW ujPij + qi dap 2m 2 fw(x, P, t) OYiOY {A,(x + y)Aa(x y) e 0x(x + y) + 0^(x y)  ., u ;t mn u A The results in Eqs. ( 24)-( 29) are exact, and do not involve an expansion in h.However the moment equations ( 16)-( 18) with expressions (24)-( 29) do not form a closed set of equations in terms of hydrodynamic state variables (say n, u and T or n, II, and W).Note that although the only explicit occurrence of h is h2, the sums over A will in general involve a functional dependence on h.Spatial derivatives of the wavefunction phases 0x(x, t) appear in the expressions for H i, P/y, W, and q through ux.Only partial phase information is contained in the first three moment equations, since the fluid dynamical quantities (25) and ( 27)-(29) involve weighted sums of spatial derivatives of 0 h.To make further progress, define u, u + f (30 where u is the macroscopic fluid velocity and fi is the velocity with respect to the macroscopic fluid flow.I will assume that the fix are random, i.e., that nxa 0 (31) A and that the temperature is defined to leading order by the average of the microscopic kinetic energies of motion relative to the macroscopic fluid flow: _.,na-mfi -nT + O( hZ). (32) Then I obtain Hi Emn(ui + Ai) --mui (33) Pi uilI _.,(ui + txi)mn,(uj + taj) x 8m h2n --/VT V211 -1--O(h2). (36) Eq. ( 35) follows if the Wigner distribution function in the classical limit h 0 is a momentum-shifted Maxwell-Boltzmann distribution (see e.g.Ref. [9], pp.99-100).In writing the last equation, I have assumed that E xna 72 log(nA) 0, EfiajnyV.log(ha) 0, EnxV2fia 0. (37) A a The classical heat flux vanishes if the distribution function is a momentum-shifted Maxwell-Boltzmann distribution.The Fourier term -KVT in Eq. ( 36) is obtained if the Wigner distribution function in the classical limit is the momentum-shifted Maxwell-Boltzmann distribution plus the first-order Chapman-Enskog correction (see Ref. [9], pp. 99-100and 103-106).Note that I have not used the quantum correction to the heat flux in the simulations presented here.
Even with these simplifications and with q rVT, the moment equations ( 16)-( 18) with ex- pressions ( 24) and ( 33)-( 35) do not form a closed set of equations in terms of hydrodynamic state vari- ables, due to the quantum terms in Pij and W.
The quantum terms in the .stresstensor and the energy density were evaluated in Ref. [1] to leading order in h 2 by Eqs. (6)and (7).Then the moment equations ( 16)-( 18) do form a closed set in terms of hydrodynamic state variables (say n, u, and T).
The moment equations simplify to a great extent for a pure quantum mechanical state, in which all the ax's except one (say a 1) vanish, and al 1.For the pure state, The expression (42) for q is written down in Ref.
In this case, the moment equations ( 16)-(18) with expressions (25) and ( 40 Note that T 0 for the pure state and that the conservation of energy equation (45) follows from the equations for conservation of particles (43) and momentum (44).
For the pure state, then, we have the "hydrodynamic" formulation (see e.g.[11])of the quantum mechanics of a pure state: and reproduces the bound-state solutions for q,x(x, t) for 0 < E < U, where E(A 0, 1, 2,...) is the energy of the particle (see e.g.Ref. [12], pp. 152-155).The solution in the square well may be thought of as resulting from the interference of a right moving and a left moving wave.In an analo- gous but more complicated way, the full quantum hydrodynamic equations "sense" the width of the quantum well in the resonant tunneling diode.

CONCLUSION
The macroscopic quantum hydrodynamic equations are capable of modeling effects in quantum semi- conductor devices that depend on quantum interfer- ence.The interference effects appear in the hydro- dynamic expressions for the velocity, stress tensor, energy density, and heat flux through sums involving the semiclassical velocities u hVO/m.
The QHD simulations of multiple resonances and multiple regions of NDR in the parabolic well reso- nant tunneling diode clearly demonstrate the effects of quantum interference and present an intuitive macroscopic picture of electron dwell times in the quantum well.

FIGURE 2 FIGURE 3
FIGURE 2Barrier and well heights in eV.

1
)-(42) form a closed set in terms of hydrodynamic state variables n and u.Eqs.

V
46) and (47) are exact, and follow directly from Schr6dinger's equation.to the "hydrodynamic" equations (46) and (47) for the 1D finite square potential well U x< -a