NEW SOLUTION METHOD OF TIME-DEPENDENT SCHRODINGER EQUATION USING IMPROVED BOUNDARY CONDITIONS

Numerical solution of the time-dependent Schrodinger equation for open system structures has been impeded by the difficulty in handing open-system boundary conditions. This paper presents a new numerical method for time-dependent Schrodinger equation and a boundary condition method to simulate the interaction with ideal particle reservoirs at the structure boundaries.


INTRODUCTION
The time-dependent Schr6dinger equation is readily solved numeri- cally for the case where ,I, may be set to zero at the boundaries of the simulation [1], the open nature of the semiconductor device problem requires the formulation of nonzero boundary conditions to model the interaction of the device with particle reservoirs at the contacts, for both the time-dependent and the time-independent case [2].The implementation of this type of boundary condition for the time- dependent problem proves to be a formidable task.*Address for correspondence: P.O.Box 345, Tainan, Taiwan 704, Republic of China.Fax: 886-6-2022377.
In this paper, boundary conditions are imposed which model to second-order plane waves of constant amplitude incident at the contacts, and waves with modulated amplitude and phase exiting the contacts without reflection.This boundary condition scheme intro- duces irreversibility in the problem and thereby allows stable, stead- state solutions to be obtained, as well as transient solutions.

FINITE-DIFFERENCE NUMERICAL METHOD
FOR THE TIME-DEPENDENT SCHRODINGER EQUATION The evolution of a one-dimensional wave function (x, t) is deter- mined by the time-dependent Schrtdinger equation o t) t) the wave function ,(x, t)can be given by where At, n, and j are the time interval, time index, and space index, respectively.Thus Eq. ( 2) can be approximated by with BenDaniel and Duke's effective Hamiltonian [3] if/= 2--7 0-m*(x) gx + V(x) (4) so as to preserve the continuity of the wave function.
The additional potential leiFx due to the external electric field is added directly to the potential profile of the structure.Equation (3) can be discredited with respect to time and space At _h 2 tll.+ tll.#n_2 2Ax 2 Ax2 ma(j )mb (j .[ where 1 ma(j) + m; Uj [ma(j) -I-mb(j)] -t" Vj.
Equation (5) is the numerical solution of the time-dependent Schrtdinger equation.

BOUNDARY CONDITIONS FOR THE TIME-DEPENDENT ANALYSIS
In solving the Eq. ( 5), boundary conditions must be specified at the left and fight simulation boundaries.It is desirable to set boundary conditions at the inflow point which correspond to incident plane waves ofconstant amplitude, to model the incident flux ofcarriers from particle reservoirs.However, at the inflow point there also exists a reflected wave whose characteristics are not known until the simulation develops.Therefore, the boundary condition should also minimize reflection of this wave at the boundary, to model particles leaving the simulation region.In this manner irreversibility is introduced, since the system loses memory of particles that have exited from the boundary, while the incident distribution is unaffected by these particles.
The approach taken here is to assume that at t 0 (time is redefined to reference zero at the beginning of each time step), the wave function at the boundary is given by (for the case of particles incident from the left) where here represents only a single incident state, &i, in the ensemble.It is assumed that A is a constant to model the interaction with the reservoir, however, the envelop function of the reflected wave, B(x), is allowed to vary with x, since, in general, the amplitude of the wave reflected from structure will vary with space and time.If the further assumption is made that to a good approximation the variation of B(x) near the boundary may be regarded as a second- order term, then the time-dependent Schrfdinger equation at the boundary becomes where the potential energy at the boundary does not appear since V is zero at the inflow boundary.The solution for at the future time may be approximately written as In this equation, the first term on the right gives the time advancement expected for an energy eigenstate.The second term gives the modi- fication of due to the local variation of the reflected wave.
At the outflow point, only a transmitted wave exiting the boundary is expected, however the amplitude of this wave will, in general, not be constant.Therefore at the outflow, the wave function is assumed to be C(x)e -'kx (10) where the wave vector ko differs from k appearing in Eq. (7) as follows (again for the case of an incident wave from the left): ko v/2m*(e-Vmax)/h (11) Again assuming that to a good approximation the variation of C(x) near the boundary may be regarded as a second-order term, then the time-dependent Schr6dinger equation at the boundary becomes where t(x) li2k 0 C(x) "4- The time-dependent Schr6dinger equation may be approximately solved to give the update: (t At) (t O)e (-/)At-T(x)e-aXAt The boundary updates given by Eqs. ( 9) and ( 14) are used to set boundary conditions of Eq. (5).It is seen that in order to use these expressions, R(x) and T(x) must be determined.

RESULTS
A psudomorphic GaAs/AIGaAs double-barrier quantum well structure, the well width is 50A and the effective mass used is m*= 0.067 mo, the barrier widths are 30 A and the effective mass used is m* =0.097mo, the conduction-band discontinuity is assumed 0.212eV.Figure l(a) shows the potential profile of the structure.zero bias.When the transmission coefficient has maximum, that the energy state is a quasi-bound state.Figures 2 and 3 show the [1 , real() and image() varies of the quasi-bound state(E=O.O66eV) FIGURE 4 (a) potential profile (b) the transmission coefficient-electron energy curve at -0.04V bias.
and unquasi-bound state(E=0.0455eV),respectively.The 112s of Figures 2 and 3 do not change in any time (steady solution) but the real()s and image()s change with cycles, when E 0.066 eV the cycle approximate to 60fsec(numerical solution) and then E=0.0455eV the cycle approximate to 90 fsec(numerical solution).The analytical solutions are 62.75fsec and 91.02fsec for E=0.066eV and E= 0.0455 eV, respectively.Figure 4(a)shows the potential profile of the structure.Figure 4(b) show the transmission coefficient-electron energy curve at -0.04 V bias, and then E=0.0455eV become quasi-bound state and E 0.066eV become unquasi-bound state.Figures 5 and 6 show the [(I)12 of E 0.066 eV and E 0.0455 eV from zero bias to -0.04 V bias, respectively.Figure 5 shows the wave function of E =0.066eV from quasi-bound state become unquasi-bound state and Figure 6 shows  FIGURE 6 The I12 of E=0.0455eV varies of time at -0.04V bias.
the wave function of E= 0.0455 eV from unquasi-bound state become quasi-bound state.The changes of Figures 5 and 6 accord with the physical requirements.

CONCLUSIONS
A method of solving the time-dependent Schr6dinger equation with boundary conditions modeling the interaction of the structure with FIGURE (a) potential profile (b) the transmission coefficient-electron energy curve at zero bias.

EFIGURE 2
FIGURE 2 The I)1 , real() and image() of E=0.066eV varies of time at zero bias.

FIGURE 5
FIGURE 5The I[ 2 of E=0.066eV varies of time at -0.04V bias.