1 Hot Electron Modeling I: Extended Drift Diffusion Models

Transcription

1 ECE539 - Advanced Theory of Semiconductors and Semiconductor Devices Numerical Methods and Simulation / Umberto Ravaioli Review of Conventional Semiconductor Device Models Based on Partial Differential Equations - II 1 Hot Electron Modeling I: Extended Drift Diffusion Models Due to the relative simplicity of the drift diffusion equations, it would be very appealing, to extend the validity of drift diffusion like models well into the hot electron regime. We have seen that the simplest attempt to include high field effects is to make the mobility and the diffusion coefficient field dependent. The current in 1 D is (x, t) J(x, t) = qn(x, t)µ(e)e + qd(e) (1) Here µ(e)e = v(e) is the (steady state) drift velocity for the case of homogeneous field E. The field is also space and time dependent, i.e. E = E(x, t). Mobility and diffusion coefficients are steady state quantities, but the carrier velocity may differ considerably from the steady state value v(e), due to abrupt space or time changes of the electric field. v(e) can be considered accurate only if space or time variations of the field are very smooth (adiabatic). Velocity overshoot occurs when the average electron velocity exceeds the steady state (bulk) velocity. A modified version of the drift diffusion equation to include velocity overshoot was proposed by Thornber. [ J(x, t) = qn(x, t) v(e) + W (E) E +qd(e) (x, t) + qa(e) ] E + B(E) (x, t) where three new terms have been added. The term with W (E) contains the field gradient and corrects the local drift velocity for spatial velocity overshoot effects. The term with B(E) contains the time derivative of the field and corrects for temporal velocity overshoot. The last term with A(E) preserves the invariances of the total current (note that this term does not represent generation, recombination, trapping, etc.), effects which may be incorporated with an addition term premultiplied by A(E) also. The quantities W (E), B(E), and A(E) must be tabulated from detailed transport calculations. In steady state the current equation simply becomes [ J(x) = qn(x) and the resulting continuity equation is (x) v(e) + W (E) E = [ n(x)v(e) + n(x)w (E) E ] + qd(e) ] + D(E) Equation (4) does not describe a real transient, since the time derivatives in the current equation have been neglected. Therefore (4) is valid in the steady state limit / = 0, i.e. t. The equation may be used, however, to solve the pseudo time dependent problem until steady state is () (3) (4) 1

2 achieved. Since the Poisson equation is solved at each time step, the fields and the related variable are continuously updated in space. Alternatively, one may solve the steady state equation obtained with / = 0, using Newton s method. Developing the space derivatives, (4) becomes = v(e) v(e) + n + +D(E) n + D(E) = ( v(e) W (E)E x + n [W (E) E ] + [W (E) E ])n + [v(e) + W (E) E Finally, the pseudo time dependent equation has form + D(E) ] + D(E) n (5) = a(x, n t) + b(x, t) + c(x, t)n (6) Since from Gauss law frac E = ρ/ɛ and ρ depends on the carrier density n(x), the coefficients b(x, t) and c(x, t) are also functions of n(x) and eq. is nonlinear. Often the overshoot parameter W (E) is rewritten in terms of mobility, as W (E) = µ(e)l(e), where L(E) is called length coefficient. Monte Carlo calculations as well as analytical models for the length coefficient have been presented in the literature. Extension to D is not trivial when confining fields (barriers) besides accelerating fields are present. An approximate approach allows the extension to D by using the gradient of the quasi Fermi levels (very small inside barrier regions and equal to the field in accelerating regions) as the functional parameter for the length coefficient. Hot Electron Modeling II: Hydrodynamic Model Transport equations which express conservation of particles, momentum, and energy may be obtained from the moments of the Boltzmann equation ( ) + (vn) = ( ) P P + (vp) = qne (nk BT) + ( W W + (vw ) = qnv E (nk BTv) Q + v is the average velocity, P is the momentum density, W is the kinetic energy density, T is the carrier temperature tensor, Q is the heat flow vector. No assumption has been made on the nature of the ision terms, in the above equations. One set of equations can be written for each valley in the conduction band of the semiconductor considered, with different set of variables n, v, P, Q, W, and T. Equation (7) is a form of the carrier continuity equation, stating that the local increase of carrier density plus the outflow of carriers is equal to the density increase due to isions. The momentum balance equation (8) states that the rate of change plus the outflow of the momentum density equals the change due to the electric forces and to the electric pressure nk B T plus the ) (7) (8) (9)

3 momentum variation caused by isions. The energy balance equation (9) similarly states that the rate of change plus the outflow of kinetic energy density W equals the sum of the energy supplied by the electric field, plus the work performed by the electric pressure, the divergence of the heat flow Q and the rate of kinetic energy density change due to isions. In terms of the electron distribution function, the various variables are defined as n = f(r, k, t)d 3 k (10) nv = v(k)f(r, k, t)d 3 k (11) W = m v (k)f(r, k, t)d 3 k (1) nk B T ij = m [v i (k) v i ][v j (k) v j ]f(r, k, t)d 3 k (13) Q i = m n [v i (k) v i ][v(k) v] f(r, k, t)d 3 k (14) The subscripts i and j indicate the conduction band valleys. The number of unknowns may be reduced to simplfy the equations. Heat conduction has been traditionally assumed to obey the classical law Q = κ T (15) where κ is the heat conductivity of the electron gas. However, it has been recognized in recent times that this approach is not correct for semiconductors, particularly in correspondence of junctions, where the heat flow is much more complicated. The temperature is a measure of the average thermal energy per electron, and can be related to the average total kinetic energy w as w = W n = 3 k BT + m v The momentum density is also simply related to the drift velocity according to (16) P = m nv (17) (16) assumes that T is a local property of the valley rather than a tensor, and therefore in (13) only i = j is considered. The assessment of the ision terms is a rather difficult task. These terms are responsible for momentum and energy relaxation and for intervalley transfer. The simplest way is to adopt a phenomenological approach which introduces relaxation times ( ) i ( ) Pi ( ) Wi = = n i τ ij + n j τ ji (18) = P i (19) (τ m ) i [ ] [ ] ni w i nj w j w i 3k B T o / = + n i (0) (τ e ) ij (τ e ) ji (τ e ) ij T o is the lattice temperature. If the average total kinetic energy w is used, then 3

4 ( ) wi = 1 ( ) Wi w i n i n i and similarly, if the drift velocity replaces the momentum, ( ) vi = 1 m i n i ( ) Pi v i n i ( ) i ( ) i The particle relaxation time for scattering from the i th to the j th valley depends on the average kinetic energy of carriers on the i th valley. The momentum relaxation time includes both intravalley and intervalley scattering, therefore one can write (1) () 1 = (3) (τ m ) i (τ m ) ii (τ m ) ij The first two terms in (0) express the energy relaxation due to intervalley transfer out of and into the i th valley, while the third term describes the relaxation towards the equilibrium with lattice temperature T o. This last term is the only one to be considered in a single valley semiconductor. It is interesting now to look at the simplest form of momentum and energy balance equation, valid for a uniform 1 dimensional, single band semiconductor d[m (w)v] dt dw dt = qe m (w)v τ m (w) = qev w w o τ e (w) (4) (5) Note that the effective mass is a function of energy also, to account for a non parabolic band, and w o is the average equilibrium electron energy 3k BT. Equation (4) and (5) can be used to study bulk properties, like velocity overshoot, as done for instance by Nougier et al. This simple set of equations has also been used to study multivalley semiconductors, using special average effective mass in different valleys, which takes into account the fraction β i of the total electrons residing in each valley m av = i β i m i (6) This approach is not too accurate, and tends to overestimate the average mass (10 to 0% too high for average fields in GaAs). A better way is to treat separately the valleys and obtain the coupling through the intervalley particle relaxation in the continuity equations. If one really wants to use an average mass to account for a multivalley situation, a Monte Carlo experiment should be run for each value of the field, relating effective mass to field with the elementary formula m i v i (n) = qe t i (n) (7) n n where n indicates, v i (n) is the velocity change during the flight, t i (n) is the time of flight. If a simulation is run for a time t tot = i t i (n) (8) n recording the evolution of one electron, the average effective mass is found as 4

5 qet tot i m av = n t i(n) and the fraction of electrons and the average mass in the i th valley are (9) β i = t i (n) (30) n t tot m n i = qe t i(n) n v (31) i(n) therefore m i av = β i m (3) i To include non parabolicity, the energy momentum relation becomes h k m o = w(a + αw) (33) where α is the non parabolicity factor and m o is the effective mass at the energy minimum. The energy dependent effective mass for motion parallel to to the E field is m = ( 1 ) 1 w h k = m o (1 + αw) 3 (34) To solve a system of balance equations for device simulation, very frequently assumptions are made on the shape of the distribution function in momentum (velocity) space. A displaced Maxwellian is often postulated, or simply a symmetric distribution function f(r, v, t) = f(r, v v(r), t) (35) One should note that normally a hot carrier distribution is likely to approach a displaced Maxwellian if considerable carrier carrier scattering is present, i.e. the carrier concentration is higher than a given critical concentration. This is therefore very questionable in the case of pinch off regions. Because of the symmetry of the distribution, one can say that the average kinetic energy is dominated by carrier temperature effects, rather than drift, i.e. w 3 k BT e 1 m v (36) where T e refers to an electron distribution. In steady state we may write the simplified system of equations (nv) = 0 (37) v = τ [ m(w) m qe w 3 w ] 3n n (38) [ v w + w ] = qv E w w o (39) 3 τ w (w) E = q(n N D) (40) ɛ 5

6 Although considerably simplified, the equations are still fairly difficult to solve, particularly if one wants to use the method for engineering applications, involving intensive computations. Further approximations are possible if the transport properties of a particular material (e.g. Si or GaAs) are well known, or if a special geometry allows one to make assumptions on the electric fields or on depletion regions. An example for MESFET simulation may be found in the mentioned paper by Cook and Frey. A general solution of the original equations (7) (9) is in principle possible if a real accurate answer is actually sought. However, it is very difficult to find a solution without assuming a particular form of the distribution function, unless one has a way to calculate the auxilliary equations (10) (14) and the ision terms accurately. A possible approach is to use Monte Carlo methods to generate extensive tables of the various parameters, for different fields and concentrations. Since the Monte Carlo method is stochastic in nature, no assumptions need to be made on the distribution function, and the scattering rates may be input directly without assuming any relaxation time. Even if all the needed parameters may be evaluated accurately, a complete solution of the (7) (9) plus Poisson s equation for a general device simulation remains a formidable task. If the three conduction valleys of GaAs are included, for instance, one is left with 9 nonlinear partial differential equations, with strong hyperbolic nature, coupled to Poisson s equation. 3 Hot Electron Modeling III: Energy Transport Models The energy transport models offer an alternative to hydrodynamic approaches. Instead of taking the moments of Boltzmann equation and then approximating the ision integrals by using average relaxation times for momentum and energy, in this approach a macroscopic relaxation time is assumed in the Boltzmann equation. In steady state for electrons in a single energy valley, the Boltzmann equation becomes For electrons, the zeroth and second order moments of the steady-state Boltzmann transport equation (BTE) give the carrier continuity and energy balance equations: J = q(g R) ; (41) S = F J n < E > (4) where the current J and the energy flow S are defined by J = qn < v >= q d 3 kvf and S = n < Ev >= d 3 k Evf. Here, G R stands for the carrier generation-recombination rate, F is the electric field, q the elementary charge, n the carrier concentration, v the carrier velocity, k the crystal momentum, f the carrier distribution function and E the carrier energy. The angular brackets denote averages over the distribution function, i.e., < A > = d 3 kaf/ d 3 kf. The last term in (4) is the energy loss rate due to isions. By decomposing f into its even and odd parts, f 0 and f 1, and using the microscopic relaxation time approximation ( f 1 /) = f 1 /τ [?], the odd part of the BTE becomes f 1 = qτf h kf 0 τv r f 0. (43) Substitution of (43) into the expressions for J and S yields J = q ( d 3 kvf 1 = q nˆµf + (n ˆD) ) ; (44) S = d 3 k Evf 1 = nˆµ E F (n ˆD E ). (45) 6

7 where all the transport coefficients ˆµ, ˆD, ˆµ E, and ˆD E are tensors and their components are defined by µ i,j = q h < τv i f 0 f 0 k j > 0 ; D i,j =< τv i v j > 0 ; (46) µ E i,j = q h < τev i f 0 f 0 k j > 0 ; D E i,j =< τev i v j > 0. (47) Here, < > 0 denotes average over f 0. The divergence of the tensors in (44) and (45) means to take the divergence of each row of the tensor as a vector element. It is worth noticing that the constraint f 1 f 0, which is the basis of the Legendre polynomial expansion approach in [?] and the perturbed distribution approach in [?], is not necessary in this model. For materials like Si, where the mean kinetic energy m < v > / is much smaller than the mean thermal energy m < v > /, the correction from the field-induced anisotropy is generally not important [?]. Therefore, f 0 can be approximated by an isotropic distribution f 0 (E) and ˆµ, ˆD, ˆµ E, and ˆD E become scalar functions of the mean carrier energy < E >. We assume in the following τ = τ(e) E p, where p is determined by the dominant scattering mechanism and ranges from 0.5 to 1 according to our preliminary bulk MC simulations. Additional hot carrier effects are included by incorporating the following first order approximations: (a) non parabolic bands: h k /m = E(1 + αe); (b) non-maxwellian effects: f 0 = (1 + γe/k B T e )f m (E). Here, f m (E) is the Maxwellian distribution at an elevated temperature T e and γ is the non-maxwellian parameter. With these assumptions and from Eqs. (46) and (47), the modified Einstein relations are valid not only for µ and D but also for µ E and D E, i.e., D = k B T m µ/q and D E = k B T m µ E /q, where the equivalent carrier temperature T m = (1 + γ)t e is related to the mean carrier energy by < E > = (1 + γ + 5 αk BT e ) 3 k BT e (1 + 5 αk BT m ) 3 k BT m. (48) Then, we can recast (44) and (45) into J = q( µn ψ + k B (nµt m ) ; (49) S = C e ( µnk B T m ψ + k B q (nµt m ) ; (50) where ψ is the electrical potential and C e = µ E /(k B T m µ) = (5/ p)(1 αk B T m /) is the energy flow factor C e. C e is in general a smoothly varying function of the mean carrier energy and for nearly parabolic bands C e 5/ p. It is important to notice that in our approach the Wiedmann- Franz law for heat flow is never invoked, as it is usually necessary in the conventional HD method to obtain a closure of the moment equations. Based on the similarity of the functional expressions in (49) and (50), an extended Scharfetter-Gummel method [?] was developed for self-consistent discretization of both carrier continuity equation and energy balance equation. By assuming that along each mesh line J, S, and (1/T )(dψ/dl) are constant, we can solve 1-D forms of both (49) and (50) on the mesh lines and perform the standard box-integration around each node [?]. With the following normalized notation: J = J/q, S = S/qC e, Ñ = nµ, T = k B T m /q, Eqs. (49) and (50) are evaluated on the mesh line connecting the nodes i and j by J l = 1 L ij (B(ũ ij )N j Tj B( ũ ij )N i Ti ) (51) S l = 1 L ij (B(ũ ij )N j Tj B( ũij )N i Ti ). (5) 7

8 L ij is the distance between nodes i and j, B(x) = x/(exp(x) 1) is the Bernoulli function, and ũ ij = (ψ j ψ i )/ T, with T = xj xi T dx/l ij. Usually, the carrier temperature varies much more slowly than the potential does. Therefore, the linear approximation T = ( T i + T j )/, is reasonable in practice. This also implies a quadratic approximation in space for the potential according to the above discretization assumptions. Since F = ψ, by substituting F J with (ψj) + ψ J and employing a macroscopic relaxation time approximation for ( E/), we can rewrite (4) as H = ψ (G R) n < E > E 0 τ ɛ (53) where E 0 = 3k B T 0 /, with T 0 being the lattice temperature, τ ɛ is the energy relaxation time, and H = S + ψj is the total energy flow, consisting of both the thermal energy flow S and the electrical energy flow ψj. From Eqs. (51) and (5), and using ψ = (ψ i + ψ j )/, we can express the total energy flow on the mesh line as H = q(ψ J l C e Sl ) q = (B(ũ ij )N j Tj ( L ψ C e Tj ) B( ũ ij )N i Ti ( ψ C e Ti )). (54) ij In this improved ET model, the necessary transport parameters µ(t m ), C e (T m ) and τ ɛ (T m ) can all be determined, for instance, from bulk MC simulations by evaluating the following ensemble averages µ = < v > ; C e = 3 < Ev > F < E >< v > ; τ ɛ = < E > E 0 qf < v >. (55) The energy transport approach is appealing for practical simulation applications, because the numerical procedure developed for drift diffusion problems can be easily adapted. The numerical difficulties encountered with hydrodynamic approaches are not present here, because instead of hyperbolic nature, the energy transport equations have parabolic nature. References K. Blotekjaer, Transport equations for electrons in two valley semiconductors, IEEE Transactions on Electron Devices, vol. ED-17, pp , K. Blotekjaer and E. B. Lunde, Collision integral for displaced Maxwellian distributions, Physical Status Solidi, vol. 35, pp , R. Stratton, Diffusion of hot and cold electrons in semiconductor barriers, Physical Review, vol. 16, pp , 196. R. Cook and J. Frey, Two dimensional numerical simulation of energy transport effects in Si and GaAs MESFETs, IEEE Transactions on Electron Devices, volt ED-9, pp , 198. J. P. Nougier et al., Determination of the transient regime in semiconductor devices using relaxation time approximation, Journal of Applied Physics, vol. 5, pp.85 83, K. K. Thornber, Current equations for velocity overshoot, IEEE Electron Device Letters, volt. EDL-3, pp ,