Finite Difference Scheme for Semiconductor Boltzmann Equation with Nonlinear Collision Operator

Transcription

1 Applied Mathematical Sciences, Vol. 7, 2013, no. 15, Finite Difference Scheme for Semiconductor Boltzmann Equation with Nonlinear Collision Operator C. L. R. Milazzo Kore University of Enna Faculty of Engineering and Architecture Cittadella Universitaria Enna, Italy cristina.milazzo@unikore.it Abstract In this paper the Boltzmann equation describing electron flow in a semiconductor device is considered. The doping profile and the self consistent electric field are related by the Poisson equation. Time dependent solutions to the Boltzmann-Poisson system are obtained by using a finite difference numerical scheme. The collision operator of the Boltzmann equation models the scattering processes between electrons and phonons which are assumed in thermal equilibrium, and takes into account the Pauli s exclusion principle. Numerical results for one dimensional n + n n + silicon device are shown. Keywords: Semiconductor device, Boltzmann-Poisson system, Finite Difference Scheme 1 Introduction The invention of the semiconductor devices has drastically transformed both conditions of living and society structures in the past and present decades. Since then, a large variety of semiconductor devices have been designed and produced. A universal trend in the evolution of this industry has been the search for smaller, quicker devices and higher integration of these on a single semiconductor chip. Hot electron transport has been a fruitful subject in the field of solid state physics both for theory and experiments [9], [10], [8] and [7]. The description of carrier transport in semiconductor is often too costly for industrial purpose. Many numerical codes have been developed to solve it,

2 736 C. L. R. Milazzo mostly based on Monte-Carlo or particle methods. However, these codes are very time consuming and cannot be routinely used to design components. A semi-classical kinetic description of the electron and hole population in semiconductor devices can be done using the Boltzmann Transport Equation (BTE) coupled to the Poisson equation. To solve this system numerically is not an easy task. Indeed, it involves the resolution of a problem posed on 6-dimensional space (3 dimensions in position x and 3 dimensions in wave-vector k) plus time t. The collision operator of the BTE describes the scattering processes inside the crystal, and its expression is obtained by Fermi s golden rule [2] [3]. The main scattering mechanisms, to which electron are subjected in semiconductors, are the ones with phonons, impurities, other electrons and stationary imperfections of the crystal such as vacancies and external and internal crystal boundaries. In the present paper we neglect the scattering with impurities and stationary imperfections. The electron-electron interaction is taken into account only in average through the Poisson s equation. The electron in motion must find free states in the neighborhood of the one they occupy initially, to move into. In particular, they deal with continuous x and k variables. In reality the states k are quantized, which means that electron motion is a succession of jumps from one state to a neighboring one. However, Pauli s exclusion principle forbids such a jump if the neighboring state the electron is aiming at, is already filled by another electron. The deformation of the lattice, due to vibrations of the ions about their equilibrium positions, is described by deformation potentials. Either the acoustic or the optical modes can be excited. The scattering arising from the excitation of the acoustic mode is typically called acoustic scattering, while the one arising from the excitation of the optical mode is called non-polar optical scattering. In [5] and [4] a new difference scheme based on the box method and the upwind scheme was proposed and validated. In those papers the scattering terms are tackled rigorously but the collision operator is linearized. The task of this paper is to solve numerically the Boltzmann-Poisson system with a non linear collision operator and compare this results with those obtained by using a linear collision operator. The plan of the paper is the following. In section 2, we set up the basic formalism, we introduce the BTE, and we specify the scattering mechanism. In Section 3 we describe the numerical scheme and in Section 4 we compare the simulation results for the one dimensional n + n n + silicon device obtained by using the two collision operator (linear and non linear), in section 5 some comments and conclusions are drawn.

3 Finite difference scheme for semiconductor Boltzmann equation Basic Equation The BTE for electrons and one conduction band writes [3], [6]: f t + v(k) xf q h E kf = Q(f). (1) where the unknown function f(t, x, k) represents the probability density of finding an electron at time t in the position x, with the wave vector k. The domain of k can be the full dimensional space or the first Brillouin zone. We denote by Ω the kdomain. The electron group velocity v is given by: v(k) = 1 h kε, where ε(k) is the energy of the considered crystal conduction band structure measured from the band minimum, and h is the Planck constant divided by 2π. In the neighborhood of the band minimum a good dispersion relation is given by the quasi parabolic approximation: ε(k)[1+α ε(k)] = h2 k 2, k Ω, (2) 2m where α is the non parabolicity parameter (for silicon α = 0.5 ev 1 ), and the m is the effective electron mass. The constant q is the absolute value of the electron charge (q = e 19 C), The electric field E(t, x) is related to the electronic distribution f by the Poisson s equation: [ ] ɛδφ(t, x) =q n(t, x) N(x), (3) E = x φ, where φ is the potential, N is the donor density (which is a positive function), ɛ is the dielectric constant and n is the particle density, given by n(t, x) = f(t, x, k) dk. Ω The collision operator can be schematically written as { Q(f) = S(k, k)f(k )[1 4π 3 f(k)] S(k, k )f(k)[1 4π 3 f(k )] } dk, (4) Ω where S(k, k ) is the transition probability, per unit time, from the state k to the state k. If the states k are already occupied, Pauli s exclusion principle prevents new electrons to reach them, and the transition from k to k is not

4 738 C. L. R. Milazzo possible. Therefore, the term [1 4π 3 f(k )] represents the proportion of occupied states k to the total number of states. The linearized collision operator is given by: [S(k, k)f(k ) S(k, k )f(k)] dk. (5) Ω The first term in (4) and (5) represents the gain and the second one the loss. Silicon electron-phonon scatterings can be modeled as in [3], i.e. S(k, k ) = K 0 (k, k )δ(ε(k ) ε(k)) K i (k, k )[δ(ε(k ) ε(k)+ hω i )(n qi + 1)+ i=1 +δ(ε(k ) ε(k) hω i )n qi ]. where hω i is a longitudinal optical phonons energy and n qi the phonons equilibrium distribution which, according to the Bose-Einstein statistics, is given by 1 n qi = exp( hω i /k B T L ) 1 here k B is the Boltzmann constant and T L is the lattice temperature, which will be considered constant in the following. The function K 0 represents the elastic scattering transition probability and it writes K 0 (k, k )= k BT L Ξ 2 d 4π 2 hρ, vs 2 where Ξ d is the acoustic-phonon deformation potential, ρ is the silicon mass density, and v s is the sound velocity of the longitudinal acoustic mode. The inelastic scattering probability is K i (k, k )= Z f(d t K i ) 2 8π 2 ρω i, where D t K i is the deformation potential for the i-th optical phonon and Z f is the number of final equivalent valleys (the lowest energy local minima) for the considered inter-valley scattering. In the following we will consider one-dimensional device, where the electric field is in the z axis direction. Then, the probability density function is invariant under rotation around the z axis and it depends only on (t, z, ɛ, μ) where μ is the cosine of the angle between the wave-vector k and the z axis. So Eq.(1) is transformed by using (ɛ, μ) variables instead of k B. After performing a discretization of (t, z, ɛ, μ) variables, Eq.(1) and Eq.(3) are approximated by using finite difference formulas. Details of the numerical scheme are shown in the next section.

5 Finite difference scheme for semiconductor Boltzmann equation The Boltzmann-Poisson solver For sake of simplicity it is useful to use the following coordinate transformation: k = m k B T L ( ) 2 w 1+αk w 1 μ h 2 cos ϕ, 1 μ 2 sin ϕ, μ (6) where α k = k B T L α, w [0; + [ represents the dimensionless energy, μ [ 1; +1] is the cosines of the angle between the z axis and the wave vector k, ϕ [0; 2 π] is the angle between the x axis and the projection of the wave vector k over the plane xy. The Jacobian of the transformation k (w, ϕ, μ) is: ( 1 2 m ) 3/2 k B T L 2 h 2 w(1 + α k w)(1 + 2 α k w). By using the above change of variable, it is possible to deduce that ε = k B T L w thus thanks the distribution properties it is possible to calculate all the integrals respect to the variable w. If K is a constant of the same order of the kernel K(k, k ), it is useful to introduce the following dimensionless variables: α = hω, a = n q +1 = e α, ξ =(w, ϕ, μ), k B T L n q K(k, k )=K K(ξ,ξ ), K 0 (k, k )=K n q K0 (ξ,ξ ), [ t = 4 π m ] 2 m 1 k B T L kb T L h 3 n q K, l = t m, t = t t, x = ( ) 3 2 m k B T L 2 l x, N D (x) = Ñ D (x), h c p = 2 2 ɛ e 2 l 2 m m h 3 E(t, x) = 2 k BT L e l k B T L, V(t, x) =2 k BT L Ψ(t, x), e ). ( E 1 (t, x), E 2 (t, x), E 3 (t, x) Where x are the new spatial coordinates, t is the dimensionless time and F the unknown function in terms of the new variables. Since we are taking into account only one spatial coordinate, the unknown function will depend only on (t, z, w, ϕ, μ). Also, due to simmetrie of the problem, the dependence on the variable ϕ will disappear. We are taking into account a silicon device so the kernels K and K 0 can be

6 740 C. L. R. Milazzo considered constant thus it is possible to chose K such that the β 5.986, thus t 3.6 ps and l 0.43 μm. After some algebra, it is possible to obtain k 2 =2 m k B T L h 2 w(1 + α k w). Due to the fact that we are taking into account just one spatial dimension, it is possible to suppose k =(0, 0,k 3 ). This leads to: k 2 k 3 =2 m k B T L h 2 (1 + 2 α k w) w k 3, and Also and 1= w w(1 + α k w) =2μ k 3 1+2α k w k k 3 = k 3 [ μ 2 m k B T L h h 2 m k B T L. ] w(1 + α k w), 2 m k B T L w(1 + α k w) μ + (1 + 2 α kw) h k 3 2 w(1 + α k w) μ w, k 3 So it is possible to deduce that μ k 3 = h 2 m k B T L 1 μ 2 w(1 + α k w). In terms of the new variables, the Boltzmann equation (1) is F t + μ w(1 + α k w) F (1 + 2 α k w) z E 2 μ w(1 + α k w) F (1 + 2 α k w) w (1 μ 2 ) F + = Q(F ). w(1 + α k w) μ (7) The function is the new unknown, where Φ(t, z, w, μ) =s(w)f (t, z, w, μ). (8) s(w) = w(1 + α k w)(1 + 2 α k w). (9)

7 Finite difference scheme for semiconductor Boltzmann equation 741 The collisional operator can be written as follows: Q(Φ) = 1 t { The Poisson equation becomes: 2 Ψ z 2 [ ] β Φ(w)+aΦ(w + α)+φ(w α) 1 [ ] βs(w)+as(w α)+s(w + α) s(w) = c p [ + ND (z) π 0 +1 dw 1 dμ (10) } Φ(t, z, w, μ) ] dμ Φ(t, z, w, μ). (11) In order to solve numerically the system (7)-(11), we follow [5] by discreti zing the variables w e μ. Let Δw and Δμ the constant step sizes, we choose Δw such that α/δw is integer, because in such a way the terms Φ(t, z, w ± α, μ) in (10) can be treated correctly. The grid points in the w μ space are: w i = i Δw, μ j = 1+j Δμ i,j =0, 1, 2,... For each grid point (w i,μ j ), we consider the rectangle R ij = [w i 1 ; w i+1 ] [μ j 1 ; μ j+1 ], now by multiplying both side of the (7) for s(w) and integrating over R ij respect to the variables w e μ, it is possible to obtain: w(1 + α k w) Φ(t, z, w, μ) dw dμ + t R ij R ij (1 + 2 α k w) μ Φ dw dμ z w i+1 μj+1 w(1 + α k w) E(t, z) dμ 2 μ Φ(t, z, w, μ) μ j 1 (1 + 2 α k w) w i 1 wi+1 1 μ + dw 2 μ j+1 Φ(t, z, w, μ) (12) w i 1 w(1 + α k w) wi+1 =Δμ w i 1 μ j 1 +1 { dw dμ K0 Φ(t, z, w, μ ) 1 + K i [a i (1 β i )Φ(t, z, w + l i Δw, μ ) i. 1 2 R ij +1 dw dμ 1 +a i β i Φ(t, z, w +(l i + 1)Δw, μ ) +β i Φ(t, z, w (l i + 1)Δw, μ ) ]} +(1 β i )Φ(t, z, w l i Δw, μ ) s(w) dμ { K0 Φ(t, z, w, μ )

8 742 C. L. R. Milazzo + i K i [a i (1 β i )Φ(t, z, w + l i Δw, μ ) +a i β i Φ(t, z, w +(l i + 1)Δw, μ ) +β i Φ(t, z, w (l i + 1)Δw, μ ) ]} +(1 β i )Φ(t, z, w l i Δw, μ ) Φ(t, z, w, μ) { dw dμ K0 s(w)+ K i [a i β i s(w (l i + 1)Δw) R ij R ij Φ(t, z, w, μ) dw dμ i i +a i (1 β i )s(w l i Δw) +(1 β i )s(w + l i Δw) ]} +β i s(w +(l i + 1)Δw) Φ(t, z, w, μ) dμ { K0 Φ(t, z, w, μ ) K i [a i (1 β i )Φ(t, z, w l i Δw, μ ) +a i β i Φ(t, z, w (l i + 1)Δw, μ ) +β i Φ(t, z, w +(l i + 1)Δw, μ ) ]} +(1 β i )Φ(t, z, w + l i Δw, μ ). This approach is commonly called box scheme. For the first term of (12) we use the mid point rule: R ij Φ(t, z, w, μ) dw dμ 4Δw Δμ Φ(t, z, w i,μ j ), the other integrals are approximated by using the Simpson rule, except for the integral: wi+1 1 μ 2 μ j+1 Φ(t, z, w, μ) dw, (13) w i 1 w(1 + α k w) μ j 1 where the function Φ(t, z, w, μ)/ w(1 + α k w) presents a singularity in w = 0. To overcome this problem, the integral (13) is approximated by using a parabolic interpolation [1]. The Poisson equation can be formally solved by using the Trapezoid rule. The partial derivatives of the equation (12), are approximated by using the following formulas: Φ(t, z, w, μ) z 1 Δz [ 1 6 Φ(t, z 2Δz, w, μ) Φ(t, z Δz,w, μ) Φ(t, z, w, μ)+1 3 Φ(t, z +Δz,w, μ) ] for μ>0,

9 Finite difference scheme for semiconductor Boltzmann equation 743 Φ(t, z, w, μ) z 1 [ 1 Δz 3 Φ(t, z Δz,w, μ) 1 Φ(t, z, w, μ) 2 +Φ(t, z +Δz,w, μ) 1 ] 6 Φ(t, z +2Δz, w, μ) for μ<0, After performing the numerical approximation and some algebras the (12) becomes an ordinary differential equation in time. In order to find the solution of this equation, we used the explicit multi step Adams-Bashforth algorithm: Φ(t, z, w, μ) = Φ(t 0 +(l 1)Δt, z, w, μ)+ Δt 24 [55A(t 0 +(l 1)Δt, z, w, μ) 59A(t 0 +(l 2)Δt, z, w, μ)+37a(t 0 +(l 3)Δt, z, w, μ) 9A(t 0, (l 4)Δt, z, w, μ)], t = t 0 + lδt, here the time step Δt is constant. The values of Φ on the boundary of the w μ domain are determined as follows. Φ(t, z, 0,μ)=0, Φ(t, z, w max,μ) = 0 for every (t, z, μ) (14) The values of Φ for μ = ±1 are easily obtained by using the following relations: Φ(t, z, w, 1) = Φ(t, z, w, 1 Δμ) Φ(t, z, w, 1) = Φ(t, z, w, 1+Δμ) for every (t, z, w). 4 Numerical Results We choose the n + n n + silicon diode as test problem. This is the simplest one dimensional inhomogeneous structure. This device has often been simulated when new device modeling techniques are discussed, because its internal potential bears resemble to the potential in a MOSFET channel or bipolar junction transistor active region. This diode consists of two highly doped regions (n + called cathode and anode) connected by a less doped region n called channel. In our simulations, the n + regions are 0.1 μm long, while the channel length is 0, 4 μm long. The device is at room temperature T L = K and the applied bias voltage is V bias = 2 Volt. The doping densities of our simulations are: N + D = cm 3 N + D = cm 3 N D = cm 3 N D = cm 3

10 744 C. L. R. Milazzo The doping profile is regularized according to the function ( N(z) =N D d 0 tanh z z 1 tanh z z ) 2 s s where d 0 = [N + D N D]/2, z 1 = 0.1 μm, z 2 = 0.5 μm the parameter s is considered equal to μm. The values coupling constants and the other parameter for silicon are given in the following tables: Table I. Phonon energies deformation potentials and number of equivalent valley used for silicon hω(mev ) (D t K)(10 8 ev/cm) Z f Table II. Silicon constants m e electron rest mass g m effective mass 0.32 m e ρ density 2.33 g/cm 3 v s longitudinal sound speed cm/sec Ξ d acoustic-phonon deformation potential 9eV At first we simulated the system (1)-(3) using the linear collision operator (5), after we simulated the same system using the non linear collision operator (4). This results are compared in the following pictures. Figure 1-5 show simulation results for the first test case (low doping density), figure 7-11 show the simulation results for the second test case (high doping density). In figure 6 and 12 are represented the error between the density results of the two simulations. The error is calculated by using the following formula: E(x) = 2 n 1(t, x) n 2 (t, x) n 1 (t, x)+n 2 (t, x) where n 1 (t, x) is the density results obtained by using the non linear collision operator, and n 2 (t, x) is the density results obtained by using the linear collision operator.

11 Finite difference scheme for semiconductor Boltzmann equation 745 Figure 1: Electron velocity (10 6 cm/s). Figure 2: Energy (ev). Figure 3: Electric field (Volt/cm).

12 746 C. L. R. Milazzo Figure 4: Electric Potential (kv). Figure 5: Electron Density (10 6 cm 3 ).

13 Finite difference scheme for semiconductor Boltzmann equation 747 Figure 6: Error velocita (10 6 cm/s) non lineare -0.2 lineare z (μm) Figure 7: Electron velocity (10 6 cm/s).

14 748 C. L. R. Milazzo non lineare lineare 0.25 energia (ev) z (μm) Figure 8: Energy (ev) campo elettrico (Volt/cm) non lineare -100 lineare z (μm) Figure 9: Electric field (Volt/cm). 2 potenziale elettrico (kv) z (μm) non lineare lineare Figure 10: Electric Potential (kv).

15 Finite difference scheme for semiconductor Boltzmann equation non lineare lineare densita (10 16 cm -3 ) z (μm) Figure 11: Electron Density (10 6 cm 3 ) Error z (μm) Figure 12: Error. 5 Conclusions In all of these picture is possible to see that the influence of the nonlinear terms is not significant both for low and high doping density. This is reasonable if we consider the fact that the two collision operator differ only through the term [1 4π 3 f(k )] and f(k ) is very small in comparison with 1. Finally we can conclude that our results indicate that the linear BTE is a good model form simulating sub-micron silicon diodes and the treatment of the non linear term seems to be not necessary

16 750 C. L. R. Milazzo References [1] Abramowitz and I. E. Stegun, Handbook of Mathematica Functions with Formulas, Graphs and Mathematical Tables, Washington, U. S. Department of Commerce, National Bureau of Standards, (1972). [2] C. Jacoboni and P. Lugli, The Monte Carlo Methods for Semiconductor Devices Simulations, Springer-Verlag, (1989). [3] C. Jacoboni and L. Reggiani, The Monte Carlo Method for Solution of the Charge Transport in Semiconductor with Application to Covalent Materials, Review Modern Phys, 55 (1983), [4] A. Majorana, O. Muscato and C.L.R Milazzo, Charge transport in 1D silicon devices via Monte Carlo simulation and Boltzmann-Poisson solver, Compel, 23 (2004) No.2, [5] A. Majorana and R.M. Pidatella, A Finite Difference Scheme Solving the Boltzmann Poisson System for Semiconductor Devices, J. Comp. Phys., 174 (2001) No.2, [6] P.A. Markowich, C. Righonfer and C. Schmeiser, Semiconductor equations, Springer-Verlag, (1991). [7] O. Muscato, C.L.R. Milazzo and V. Di Stefano, Extended hydrodynamic model describing heat generation in thin body silicon semiconductor devices, J. Comput. Electron., 7 (2008), [8] S.V.J. Narumanchi, J.Y. Murthy, C.H. Amon, Comparison of different phonon transport models for predicting heat conduction in silicon-oninsulator transistors, J. Heat Transf., 127, 713, (2005). [9] P.G. Sverdrup, S. Sinha, M. Asheghi, S. Uma and K.E. Goodson, Measurement of ballistic phonon conduction near hotspots in silicon, Appl. Phys. Lett., 78, 3331, (2001). [10] P.G. Sverdrup, S. Ju and K.E. Goodson, Sub-continum simulations of heat conduction in Silicon-on-Insulator transistors, J. Heat Transfer, 123, 130, (2001) Received: October, 2012