CN106156388A - MESFET electric heating Method In The Whole-process Analysis under high-power electromagnetic impulse action - Google Patents

Abstract

The invention discloses MESFET electric heating Method In The Whole-process Analysis under a kind of high-power electromagnetic impulse action.The method is with electron quasi-Fermi gesture, the quasi-Fermi potential in hole and electromotive force as variable, time domain spectral element method is used to solve drift-diffusion equation group, obtain metal-semiconductor field effect pipe (MESFET) quasi-Fermi potential instantaneous under high power pulse effect and electromotive force, and then obtain electric field intensity and the electric current density of current time.Under joule Source, it is considered to ambient temperature and the impact of thermal convection current, current time Temperature Distribution everywhere can be obtained.Update carrier mobility according to variations in temperature, generation recombination rate recalculates Electric Field Distribution, so iterative cycles, until drift-diffusion equation group meets convergence precision, Electric Field Distribution and heat distribution now are exactly the distribution of the electric heating within current time MESFET that should ask.The method, based on MESFET physical model, can clearly obtain under the effect of high-power electromagnetic pulse, the time dependent distribution situation such as device inside electric field and temperature.

Description

MESFET electric heating Method In The Whole-process Analysis under high-power electromagnetic impulse action

Technical field

The invention belongs to the electrocaloric effect analysis of the transient state of semiconductor device, a kind of numerical analysis method for MESFET design.

Background technology

Semiconductor device art develops rapidly, the research of semiconductor device is constantly deepened by people, after silicon materials, compound semiconductor materials GaAs (GaAs) and indium phosphide (InP) become important basic electronic material with its excellent characteristic, and particularly GaAs has become as one of most important microelectronic material.Except the continuous renewal of new semi-conducting material emerges in large numbers, the Technology of quasiconductor the most constantly improves, and the metal-semiconductor field effect pipe (MESFET) with excellent electric property is widely used in engineering practice.

Electromagnetic pulse is a kind of transient electromagnetic phenomenon.High-power electromagnetic impulses injection is to after integrated circuit, and the particularly MESFET of the active component in circuit easily absorbs the electromagnetic energy of radiation, is easily subject to the impact of electric stress, makes device inside electric current acutely increase, and high temperature raises, thus loses efficacy, and even damages.In order to adopt an effective measure to electronic equipment or electronic system from the harm of high-power electromagnetic pulse, just there are important theory significance and practical value by the software emulation prediction the most widely used field effect transistor of semiconductor device.

Numerical simulation to MESFET physical model can accurately emulate the Electric Field Distribution within MESFET and heat distribution, provides for electromagnetic protection and effectively instructs.For the emulation of semiconductor device with model partition, mainly have classical model, semi-classical model and quantum model (He Ye, Wei Tongli. the computer simulation method [M] of semiconductor device. Beijing: Science Press, 1989.12).Classical model is just to solve for drift-diffusion equation group, it is contemplated that in electromagnetic pulse, electrical quantity is time-varying function and the feature of thermal model time duration, uses time domain approach more particularly suitable, and general FD TD, FEM more often uses.Yee grid characteristics yet with FDTD is restricted when the model that model configuration is complicated.FEM when being applied to time domain each time step be directed to system of linear equations is solved, amount of calculation is the hugest, loses time very much.

Summary of the invention

It is an object of the invention to provide MESFET electric heating Method In The Whole-process Analysis under a kind of high-power electromagnetic impulse action, it is achieved the method quickly obtaining device inside Electric Field Distribution and Temperature Distribution.

The technical solution realizing the object of the invention is: MESFET electric heating Method In The Whole-process Analysis under a kind of high-power electromagnetic impulse action, and step is expressed as follows:

The first step, sets up solving model and the mesh generation of MESFET.And use bent hexahedron that model carries out overall subdivision, obtain the structural information of model, including the node numbering of each bent hexahedral element and coordinate etc..The size of subdivision grid is more than meeting the subdivision size needed for precision；

Second step, from equation of current density, Current continuity equation and Poisson's equation, the electron quasi-Fermi gesture, the quasi-Fermi potential in hole and the electromotive force that solve with applicable unity couping are as variable, and peer-to-peer uses Galerkin method test, imposed boundary con ditions, and then solve electric field and the CURRENT DISTRIBUTION obtaining each node；

3rd step, above step the electric field obtained and CURRENT DISTRIBUTION draw the power density of each point；

4th step, sets up the equation of heat conduction of MESFET, power density is substituted in the equation as thermal source item, solves and obtains the distribution of each node temperature；

5th step, the temperature obtained by upper step updates the carrier mobility in electric field equation, produces compound term parameter, repeat step 2, three, four, five steps, so iterative cycles, until electric field equation reaches the condition of convergence, electric field, CURRENT DISTRIBUTION and Temperature Distribution now is exactly the electric heating distribution results of current time.

In the first step, the model of MESFET is physical model, with ANSYS, model is carried out subdivision.

In second step, with electron quasi-Fermi gesture φ_n, hole quasi-Fermi potential φ_pAnd electromotive forceFor variable, grid is Schottky contacts boundary condition.

For the model equation of semiconductor device, there is a method of taking of three kinds of variablees:

The first is with electronics Fermi potential exponential termAnd electromotive forceFor variable；

Electronics Fermi potential exponential term:

Hole Fermi potential exponential term:

In the steady-state characteristic of semiconductor device is simulated, owing to hole is the least on analog result impact, hole current equation of continuity can be ignored, it is believed that hole current is zero.But during the transient simulation to semiconductor device, the effect in hole can not be ignored, and hole Fermi potential exponential termIn the simulation of bipolarity collective pipe, the situation that numerical value is excessive and overflows easily occurs, thus this variable follow the example of the transient simulation process being not suitable for semiconductor device.

The second is for variable with electron concentration, hole concentration and electromotive force；

The order of magnitude of electron concentration and hole concentration is bigger, and during iterative equation, the requirement to error is relatively low, particularly when device model is fairly simple, and the PN junction of such as Uniform Doped, PIN pipe etc., solve and be easier to convergence.But when the model of device is more complicated, such as MESFET etc., when simulating breakdown characteristics, if subdivision grid is the closeest, electron concentration or hole concentration there will be non-physical concussion (He Ye, Cao Guoxiang, Wang Yuanming. a kind of theory and the method that suppress drift-expansion calculation equation numerical value to connect vibration. the Science Bulletin .1991.12 phase), there is a large amount of negative value, cause solving and dissipate, this problem can be improved by encryption subdivision grid, but the closeest subdivision grid can make again unknown quantity significantly increase, increasing the consumption of internal memory and solve the time, efficiency reduces.

The third is with electron quasi-Fermi gesture φ_n, hole quasi-Fermi potential φ_pAnd electromotive forceAs variable.

Electron quasi-Fermi gesture φ_nFermi potential φ quasi-with hole_pNumerical value typically 0～10³In the range of, numeric distribution span is little, will not occur that non-physical is shaken as making variable with carrier concentration, is not encrypting subdivision grid, it is ensured that suitably on the premise of unknown quantity, be easier convergence.Further, electron quasi-Fermi gesture φ during Newton iteration_nFermi potential φ quasi-with hole_pFront and back the relative error of twice iteration is 1 × 10^-3Ensure that the precision of device C-V characteristic.

After model equation normalization as follows:

Poisson's equation:

Electron current density equation:

In above formula (1.2), J_nFor electron current density, μ_nFor electron mobility；

Hole current density equation:

In above formula (1.3), J_pFor electron current density, μ_pFor electron mobility；

Electronic current equation of continuity: $\frac{\∂n}{\∂t}=\▿\·J_n+G-R---\left(1.4\right)$

Hole current equation of continuity: $\frac{\∂p}{\∂t}=\▿\·J_p+G-R---\left(1.5\right)$

In formula (1.4) and formula (1.5), G is that snowslide produces item, use Okuto-Crowell model (Y.Okuto and C.R.Crowell, " Threshold Energy Effect on Avalanche Breakdown in Semiconductor Junctions ", Solid-State Electronics, vol.18, pp.161-168,1975), R is Carrier recombination rate (He Ye, Wei Tongli. the computer simulation method [M] of semiconductor device. Beijing: Science Press, 1989.12).

With backward Euler's method, formula (1.4) and (1.5) are carried out time difference, obtain:

In formula (1.6) and (1.7), n^m,p^mElectronics and the concentration value in hole, n for current time^m-1,p^m-1For electronics and the concentration value in hole of previous moment,Electromotive force for current time.

WillWithSubstitute in above formula (1.6) and (1.7), respectively electronic current equation of continuity (formula 1.6), hole current equation of continuity (formula 1.7) and Poisson's equation (formula 1.1) are carried out a series of conversion such as gal the Liao Dynasty gold test, can be to obtain following form:

In above formula (1.8) and formula (1.9), coefficient A is only just 1 in Schottky boundary face, and other faces are 0.

Formula (1.8), formula (1.9) and formula (1.10) are carried out Taylor expansion by the form of formula (1.11) and remove non-linear and coupling processing:

Derive through above, obtained being easily programmed the form of equation realizing solving:

Solve formula (1.12) and the electronics of current time, the quasi-Fermi potential in hole and electromotive force can be obtained.

In 3rd step, the electronics of second step, the quasi-Fermi potential in hole and electromotive force can obtain the electric field intensity of the internal every bit of MESFETAnd electric current densityPower density

In 4th step, do not consider the impact of cooling stream, just ignoring hot repair that the exchanged heat in electronics, hole generation-recombination process, temperature contrast and broadband difference produce, only using power density as thermal source, substitute into formula:

$\frac{\∂T}{\∂t}=D_t({\▿}^{2}T+\frac{p_d}{k_t})---\left(1.13\right)$

Solving equation 1.13 just can get the temperature of the internal every bit of MESFET.

Compared with prior art, its remarkable advantage: (1) SETD uses bent hexahedron subdivision, flexibly, subdivision is convenient in modeling, uses with specific orthogonal polynomial as basic function, and along with the raising of polynomial order, calculating error will exponentially decline for the present invention.(2) model equation is with electron quasi-Fermi gesture, the quasi-Fermi potential in hole and electromotive force as variable, overcome the non-physical reforming phenomena being easily generated when making variable with carrier concentration, save unknown quantity, and then save the time of solving, solve the more efficient when puncturing of MESFET.(3) by the electrical characteristics of MESFET and thermal characteristics integrated analysis, do not isolated and come.And influencing each other between electric heating can be connected by next step.

Accompanying drawing explanation

Fig. 1 is the two-dimensional cross section of GaAs MESFET.

Fig. 2 is the output characteristic curve that the present invention is calculated that GaAs MESFET grid voltage increases linearly over time.

Fig. 3 is that the present invention is calculated the internal transient temperature scattergram of GaAs MESFET.

Detailed description of the invention

MESFET electric heating Method In The Whole-process Analysis under one high-power electromagnetic impulse action of the present invention.The method is with electron quasi-Fermi gesture, the quasi-Fermi potential in hole and electromotive force as variable, time domain spectral element method is used to solve drift-diffusion equation group, obtain metal-semiconductor field effect pipe (MESFET) quasi-Fermi potential instantaneous under high power pulse effect and electromotive force, and then obtain electric field intensity and the electric current density of current time.Under joule Source, it is considered to ambient temperature and the impact of thermal convection current, current time Temperature Distribution everywhere can be obtained.Update carrier mobility according to variations in temperature, generation recombination rate recalculates Electric Field Distribution, so iterative cycles, until drift-diffusion equation group meets convergence precision, Electric Field Distribution and heat distribution now are exactly the distribution of the electric heating within current time MESFET that should ask.This analysis method is based on MESFET physical model, can clearly obtain under the effect of high-power electromagnetic pulse, the time dependent distribution situations such as device inside electric field and temperature, destroy anti-high powers of semiconductor device such as research MESFET and have extremely important realistic meaning.Electric heating solves analysis and all uses spectral element method, uses identical grid discrete, and flexibly, subdivision is convenient in modeling, and the matrix of formation has good openness, and solution efficiency is higher.

Below in conjunction with the accompanying drawings the present invention is described in further detail.

One, the spectral element method of model equation is derived

Solve transient state drift-diffusion equation with coupling process, will Poisson's equation and Current continuity equation solve, with electron quasi-Fermi gesture φ simultaneously_n, hole quasi-Fermi potential φ_pAnd electromotive forceAs variable.

The transient model equation of MESFET includes:

Normalized Poisson's equation:

Normalized electron current density equation:

Normalized hole current density equation:

Normalized electronic current equation of continuity: $\frac{\∂n}{\∂t}=\▿\·J_n+G-R---\left(\mathrm{5.1.4}\right)$

Normalized hole current equation of continuity: $\frac{\∂p}{\∂t}=-\▿\·J_p+G-R---\left(\mathrm{5.1.5}\right)$

Normalized recombination rate model: $R=\frac{\mathrm{pn}-1}{{\mathrm{\τ}}_n(n+1)+{\mathrm{\τ}}_p(p+1)}---\left(\mathrm{5.1.6}\right)$

As it is shown in figure 1, the boundary condition of MESFET, drain electrode and source electrode are ohmic contact boundaries condition, be parallel to x coordinate axle for floating boundary condition.

Grid is Schottky contacts boundary condition:

Wherein,It is the method phase component outside by quasiconductor, v_n、v_pIt is the recombination rate in electronics and hole respectively, n₀、p₀It is the density in the electronics under quasi-balanced state and hole.n₀、p₀It is given by the following formula:

$\begin{array}{c}{n}_0=N_c\exp (-\frac{{\mathrm{\Φ}}_B}{\mathrm{kT}})\\ p_0=N_v\exp (-\frac{E_g-{\mathrm{\Φ}}_B}{\mathrm{kT}})\end{array}---\left(\mathrm{5.1.8}\right)$

In formula, Φ_BFor the transmitting potential barrier of electronics, it is expressed as:

Φ_B=Φ_m-χ (5.1.9)

Noting, in threedimensional model, front-back is set to floating boundary condition.

Owing to Current continuity equation and Poisson's equation are all nonlinear, by Taylor expansion by equation linearisation.

The method using unity couping solves drift-diffusion equation, and the equation after Taylor expansion being processed is write as the form of formula (5.1.12):

By being suitably derived by final matrix form:

For Drift-diffusion Model, it needs to be noted that snowslide produces the processing method of item.Its expression formula is as shown in (5114):

$G=\frac{1}{q}\left({\mathrm{\α}}_0\right|J_n|+{\mathrm{\α}}_p|J_p\left|\right)---\left(\mathrm{5.1.14}\right)$

In above formula (5.1.14), the ionization coefficient in electronics and hole is:

$\begin{array}{c}{\mathrm{\α}}_n=A_n[1+C_n(T-T_{\mathrm{ref}})]E_{\left|\right|,n}\exp [-{\left(\frac{B_n(1+D_n(T-T_{\mathrm{ref}}))}{E_{\left|\right|,n}}\right)}^2]\\ {\mathrm{\α}}_p=A_p[1+C_p(T-T_{\mathrm{ref}})]E_{\left|\right|,p}\exp [-{\left(\frac{B_p(1+D_p(T-T_{\mathrm{ref}}))}{E_{\left|\right|,p}}\right)}^2]\end{array}$

Wherein, T is the temperature of device inside current time, T_refIt is original ambient temperature, A_n,B_n,C_n,D_nAnd A_p,B_p,C_p,D_pIt it is constant.Owing to snowslide item containing electric current density and electric field intensity, it is carried out the operations such as gal the Liao Dynasty gold test extremely difficult numerous and diverse, so and the thought combining the non-coupled method solving employing Gummel of equation (1.13).

Two, the spectral element method of the equation of heat conduction is derived

In the solution procedure of the internal heat distribution of MESFET, density of material, heat conductivity and specific heat at constant pressure are set to definite value, do not consider the impact of cooling stream, the conduction of heat solution formula that can be simplified:

$\frac{\∂T}{\∂t}=D_t({\▿}^{2}T+\frac{p_d}{k_t})---\left(\mathrm{5.2.2}\right)$

As shown in Figure 1, when solving the equation of heat conduction, the GH face, base (as shown in Figure 1) that ambient temperature is set to 300K, MESFET is set to First Boundary Condition, temperature constant is 300K, and other boundary faces are set to third boundary condition i.e. scattering boundary condition.Through the derivation step similar to first segment, the compact schemes of the spectral element method that finally can get the equation of heat conduction are:

$\left[S\right]T_j+\left[T\right]\frac{\∂T_j}{\∂t}+\left[R\right]T_j=\left[\mathrm{Rq}\right]+\left[F\right]---\left(\mathrm{5.2.3}\right)$

Wherein:

${\left[S\right]}_{\mathrm{ij}}=\frac{k_t}{{\mathrm{\ρ}}_m{c}_m}\∫\∫\∫\▿N_i\·\▿N_j\mathrm{dv}---\left(\mathrm{5.2.4}\right)$

${\left[T\right]}_{\mathrm{ij}}=\∫\∫\∫N_i\·N_j\mathrm{dv}---\left(\mathrm{5.2.5}\right)$

${\left[R\right]}_{\mathrm{ij}}=\frac{h}{{\mathrm{\ρ}}_m{c}_m}\∫\∫N_i\·N_j\mathrm{ds}---\left(\mathrm{5.2.6}\right)$

${\left[\mathrm{Rq}\right]}_i=\frac{{\mathrm{hT}}_{\mathrm{air}}}{{\mathrm{\ρ}}_m{c}_m}\∫\∫N_i\mathrm{ds}---\left(\mathrm{5.2.7}\right)$

${\left[F\right]}_i=\frac{P_d}{{\mathrm{\ρ}}_m{c}_m}\∫\∫\∫N_i\mathrm{dv}---\left(\mathrm{5.2.8}\right)$

Forward difference form is used to obtain:

[T]Tⁿ=([T]-Δ t ([S]+[R])) T^n-1+Δt[Rq]+Δt[F] (5.2.9)

Wherein, matrix T is mass matrix, is diagonal matrix or Block diagonal matrix, and the method that available block diagonal matrix is inverted obtains the inverse of matrix T in advance, and equation to be solved becomes explicit equation, reduces amount of calculation, improves computational efficiency.

Three, the basic theories that MESFET electro thermal coupling is analyzed

The thinking of numerical simulation emulation MESFET physical model is: first, initialization, at t_nMoment provides the initial value of electron concentration, hole concentration and electromotive force respectively, substitute into equation (5.1.13), solve and obtain the internal electron concentration of every bit of MESFET, hole concentration and the value of electromotive force, calculating electric field intensity and the electric current density of each point, power density is exactly the product of electric field intensity and electric current density.Then, the power density obtained is updated in the equation of heat conduction, obtains the temperature of each point.Update the parameters relevant with temperature such as carrier mobility.So iterative cycles, until reaching convergence, the distribution of electric field now, temperature etc. is exactly the result of current time.Identical subdivision grid cell is used in the solution procedure of thermal field and electric field.

As a example by the MESFET that Fig. 1 analyzes: the size of this MESFET pipe is 2.2 × 0.5 × 10 μm, drain electrode, a length of 0.2 μm of source electrode pole plate, a length of 0.6 μm of grid pole plate, grid leak and grid source spacing are 0.6 μm, and doping content is 10¹⁶/cm³.Mobility model uses Caughey-Thomas mobility model.

By Fig. 2, grid voltage being fixed as 0V, adding amplitude in drain electrode is 25V, and rising edge is the step signal of 500ps, obtains drain current and changes over curve, 450ps, and when drain voltage is 22.5V, this MESFET occurs avalanche breakdown, and current spikes increases.Under identical subdivision, if choosing electronics, hole concentration and electromotive force is that unknown quantity system before avalanche breakdown occurs is the most unstable, it is impossible to obtain correct result.

Claims (4)

1. MESFET electric heating Method In The Whole-process Analysis under a high-power electromagnetic impulse action, it is characterised in that step is such as Under:

The first step, sets up the solving model of MESFET, and uses bent hexahedron that model is carried out subdivision, obtain model Structural information, including hexahedral unit information and nodal information；

Second step, from equation of current density, Current continuity equation and Poisson's equation, when first carrying out with backward Euler Between difference, then use Galerkin method to test it, imposed boundary con ditions, solve and obtain the electric field of each node and electric current divides Cloth；

3rd step, is drawn each node power density by electric field and CURRENT DISTRIBUTION；

4th step, sets up the equation of heat conduction of MESFET, power density is substituted in the equation as thermal source item, solves Obtain the distribution of each node temperature；

5th step, upper step the temperature obtained updates carrier mobility in drift-diffusion equation, produces compound term, again Calculate Electric Field Distribution and the CURRENT DISTRIBUTION of each node；Repetition step 2, three, four, five steps, such iterative cycles, until Drift-diffusion equation reaches the condition of convergence, and electric field, CURRENT DISTRIBUTION and Temperature Distribution now is exactly that the electric heating of current time divides Cloth result.

MESFET electric heating Method In The Whole-process Analysis under high-power electromagnetic impulse action the most according to claim 1, It is characterized in that: in the first step, the model of MESFET is physical model, with ANSYS, model is carried out subdivision.

MESFET electric heating Method In The Whole-process Analysis under high-power electromagnetic impulse action the most according to claim 1, It is characterized in that: in second step, with electron quasi-Fermi gesture φ_n, hole quasi-Fermi potential φ_pAnd electromotive forceFor variable, grid is Schottky contacts boundary condition；

Electron quasi-Fermi gesture φ_nFermi potential φ quasi-with hole_pNumerical value 0～10³In the range of,

After model equation normalization as follows:

Poisson's equation:

Electron current density equation:

In above formula (1.2), J_nFor electron current density, μ_nFor electron mobility；

Hole current density equation:

In above formula (1.3), J_pFor electron current density, μ_pFor electron mobility；

Electronic current equation of continuity: $\frac{\∂n}{\∂t}=\▿\·J_n+G-R---\left(1.4\right)$

Hole current equation of continuity: $\frac{\∂p}{\∂t}=-\▿\·J_p+G-R---\left(1.5\right)$

In formula (1.4) and formula (1.5), G is that snowslide produces item, and R is Carrier recombination rate；

With backward Euler's method, formula (1.4) and (1.5) are carried out time difference, obtain:

In formula (1.6) and (1.7), n^m,p^mElectronics and the concentration value in hole, n for current time^m-1,p^m-1For previous moment Electronics and the concentration value in hole,For the electromotive force of current time, Δ t is discrete time step；

WillWithSubstitute in above formula (1.6) and (1.7), respectively to electronic current seriality Equation (formula 1.6), hole current equation of continuity (formula 1.7) and Poisson's equation (formula 1.1) carry out gal the Liao Dynasty gold test change, obtain Following form:

In above formula (1.8) and formula (1.9), coefficient A is only just 1 in Schottky boundary face, and other faces are 0；

Formula (1.8), formula (1.9) and formula (1.10) are carried out Taylor expansion by the form of formula (1.11) and remove non-linear and coupling processing:

Derive through above, the form of the equation obtained:

Solve formula (1.12) and obtain the electronics of current time, the quasi-Fermi potential in hole and electromotive force.

MESFET electric heating Method In The Whole-process Analysis under high-power electromagnetic impulse action the most according to claim 1, It is characterized in that: in the 3rd step, the electronics of second step, the quasi-Fermi potential in hole and electromotive force obtain MESFET internal each The electric field intensity of pointAnd electric current densityPower density