Drift-Diffusion Model: Introduction

Transcription

1 Drift-Diffusio Model: Itroductio Preared by: Dragica Vasileska Associate Professor Arizoa State Uiversity

2 The oular drift-diffusio curret equatios ca be easily derived from the oltzma trasort equatio by cosiderig momets of the TE. Cosider steady state coditios ad, for simlicity, a 1-D geometry. With the use of a relaxatio time aroximatio, the oltzma trasort equatio may be writte [1] ee f f f0 f( v, x) + v = m* v x τ. (0.1) I writig Eq. (3.1) arabolic bads have bee assumed for simlicity, ad the charge e has to be take with the roer sig of the article (ositive for holes ad egative for electros). The geeral defiitio of curret desity is reeated here for comleteess J ( x) = e vf( v, x) dv (0.) where the itegral o the right had side reresets the first momet of the distributio fuctio. This defiitio of curret ca be related to Eq. (3.1) after multilyig both sides of (3.1) by v ad itegratig over v. From the RHS of Eq. (3.1) we get 1 J ( x) vf0dv vf (, v x) dv = τ eτ (0.3) The equilibrium distributio fuctio is symmetric i v, ad hece the first itegral is zero. Therefore, we have eτ f d = m* v dx J ( x) e E v dv eτ v f( v, x) dv (0.4) Itegratig by arts we have f v dv = [ vf ( v, x) ] f ( v, x) dv = ( x) v (0.5) ad we ca write

3 v f (, v x ) dv = () x v (0.6) where v is the average of the square of the velocity. The drift-diffusio equatios are derived eτ µ = itroducig the mobility m* ad relacig 3kT * v kt with its average equilibrium value m * for a 1D case ad m for a 3D case, therefore eglectig thermal effects. The diffusio coefficiet D is also itroduced, ad the resultig drift-diffusio curret exressios for electros ad holes are d J = q( x) µ E( x) + qd dx d J = q( x) µ E( x) qd dx (0.7) resectively, where q is used to idicate the absolute value of the electroic charge. Although o direct assumtios o the o-equilibrium distributio fuctio, f(v,x), were made i the derivatio of Eqs. (3.7), i effect, the choice of equilibrium (thermal) velocity meas that the drift-diffusio equatios are oly valid for small erturbatios of the equilibrium state (low fields). The validity of the drift-diffusio equatios is emirically exteded by itroducig fielddeedet mobility µ ( E) ad diffusio coefficiet D(E), obtaied from emirical models or detailed calculatio to cature effects such as velocity saturatio at high electric fields due to hot carrier effects. Physical Limitatios o Numerical Drift-Diffusio Schemes The comlete drift-diffusio model is based o the followig set of equatios i 1D: 1. Curret equatios d J = q( x) µ E( x) + qd dx d J = q( x) µ E( x) qd dx (0.8)

4 . Cotiuity equatios 1 = J + U t q 1 = J + U t q (0.9) 3. Poisso's equatio + ( D A ) ε V = + N N (0.10) where U ad U are the et geeratio-recombiatio rates. The cotiuity equatios are the coservatio laws for the charge carriers, which may be easily derived takig the zeroth momet of the time deedet TE. A umerical scheme which solves the cotiuity equatios should 1. Coserve the total charge iside the device, as well as that eterig ad leavig.. Resect the local ositive defiite ature of carrier desity. Negative desity is uhysical. 3. Resect mootoicity of the solutio (i.e. it should ot itroduce surious sace oscillatios). Coservative schemes are usually achieved by subdivisio of the comutatioal domai ito atches (boxes) surroudig the mesh oits. The currets are the defied o the boudaries of these elemets, thus eforcig coservatio (the curret exitig oe elemet side is exactly equal to the curret eterig the eighborig elemet through the side i commo). For examle, o a uiform -D grid with mesh size, the electro cotiuity equatio may be discretized i a exlicit form as follows [] x 1 x 1 y 1 x (, ; ) (, ; ) (, ; ) (, 1 i (, jk ; + 1) i (, jk, ) J i+ j k J i j k J i j+ k J i j ; k) = + t q q (0.11)

5 I Eq. (3.11), the idices i,j describe satial discretizatio, k corresods to the time rogressio, ad the suerscrits x ad y deote the x- ad y-coordiate of the curret desity vector. This simle aroach has certai ractical limitatios, but is sufficiet to illustrate the idea behid the coservative scheme. With the reset covetio for ositive ad egative comoets, it is easy to see that i the absece of geeratio-recombiatio terms, the oly cotributios to the overall device curret arise from the cotacts. Remember that, sice electros have egative charge, the article flux is oosite to the curret flux. The actual determiatio of the curret desities aearig i Eq. (3.11) will be discussed later. Whe the equatios are discretized, usig fiite differeces for istace, there are limitatios o the choice of mesh size ad time ste [3]: 1. The mesh size x is limited by the Debye legth.. The time ste is limited by the dielectric relaxatio time. A mesh size must be smaller tha the Debye legth where oe has to resolve charge variatios i sace. A simle examle is the carrier redistributio at a iterface betwee two regios with differet doig levels. Carriers diffuse ito the lower doed regio creatig excess carrier distributio which at equilibrium decays i sace dow to the bulk cocetratio with aroximately exoetial behavior. The satial decay costat is the Debye legth LD = εkt q N (0.1) where N is the doig desity. I GaAs ad Si, at room temerature the Debye legth is aroximately 400 Å whe N 10 cm 16 3 ad decreases to about oly 50 Å whe N cm. The dielectric relaxatio time is the characteristic time for charge fluctuatios to decay uder the ifluece of the field that they roduce. The dielectric relaxatio time may be estimated usig t dr = ε qn µ (0.13)

6 To see the imortace of resectig the limitatios related to the dielectric relaxatio time, imagie we have a satial fluctuatio of the carrier cocetratio, which relaxes to equilibrium accordig to the rate equatio ( t = 0) = t t dr (0.14) A fiite differece discretizatio of this equatio gives at the first time ste t (0) ( t) = (0) t dr (0.15) Clearly, if t > tdr, the value of ( t) is egative, which meas that the actual cocetratio is evaluated to be less tha the equilibrium value, ad it is easy to see that the solutio at higher time stes will decay oscillatig betwee ositive ad egative values of. A excessively large t may lead, therefore, to ohysical results. I the case of high mobility, the dielectric relaxatio time ca be very small. For istace, a samle of GaAs with a mobility of 6000 cm / V s ad doig time ste cm has aroximately t should be chose to be cosiderably smaller. tdr s, ad i a simulatio the Steady State Solutio of iolar Semicoductor Equatios The geeral semicoductor equatios for electros ad holes may be rewritte i 3D as ( ε V) q( N ) = + J = qu(, ) + q t J = qu(, ) + q t kt q J = qµ V + J = qµ V kt q (0.16)

7 with N = NA ND. We ote that the above equatios are valid i the limit of small deviatios from equilibrium, sice the Eistei relatios have bee used for the diffusio coefficiet, ormally valid for low fields or large devices. The geeratio-recombiatio term U will be i geeral a fuctio of the local electro ad hole cocetratios, accordig to ossible differet hysical mechaisms, to be examied later i more detail. We will cosider from ow o steady state, with the time deedet derivatives set to zero. These semicoductor equatios costitute a couled oliear set. It is ot ossible, i geeral, to obtai a solutio directly i oe ste, rather a oliear iteratio method is required. The two more oular methods for solvig the discretized equatios are the Gummel's iteratio method [4] ad the Newto's method [5]. It is very difficult to determie a otimum strategy for the solutio, sice this will deed o a umber of details related to the articular device uder study. There are i geeral three ossible choices of variables. 1. Natural variable formulatio (V,,). Quasi-Fermi level formulatio ( V, φ, ) φ, where the quasi-fermi levels derive from the followig defiitio of carrier cocetratio out of equilibrium (for o-degeerate case) qv ( φ ) = i ex kt q( φ V ) = i ex kt (,, ) 3. Slotboom formulatio V Φ Φ where the Slotboom [6] variables are defied as qφ Φ = iex kt qφ Φ = iex kt The Slotboom variables are, therefore, related to the carrier desity exressios, ad the extesio to degeerate coditios is cumbersome.

8 Normally, there is a referece for the quasi-fermi level formulatio i steady state simulatio, ad for the atural variables ad i trasiet simulatio. Normalizatio ad Scalig For the sake of clarity, all formulae have bee reseted without the use of simlificatios or ormalizatio. It is however commo ractice to erform the actual calculatio usig ormalized uits to make the algorithms more efficiet, ad i cases to avoid umerical overflow ad uderflow. It is advisable to iut the data i M.K.S. or ractical uits (the use of cetimeters is for istace very commo i semicoductor ractice, istead of meters) ad the rovide a coversio block before ad after the comutatio blocks to ormalize ad deormalize the variables. It is advisable to use cosistet scalig, rather tha set certai costats to arbitrary values. The most commo scalig factors for ormalizatio of semicoductor equatios are listed i Table 0-1 [7]. Table 0-1. Scalig factors Variable Scalig Variable Formula Sace Itrisic Debye legth (N= i ) ε kt L = q N Extrisic Debye legth (N=N max ) Potetial Thermal voltage kt V* = q Carrier cocetratio Itrisic cocetratio N= i Diffusio coefficiet Mobility Geeratio- Recombiatio Maximum doig cocetratio Practical uit Maximum diffusio coefficiet N=N max D = 1 cm s D = D max D M = V * DN R = L

9 Time L T = D Refereces 1 S. Selberherr, Simulatio of Semicoductor Devices ad Processes (Sriger-Verlag, Wie New York). 3 K. Tomizawa, Numerical Simulatio of Submicro Semicoductor Devices (The Artech House Materials Sciece Library). 4 H. K. Gummel, A self-cosistet iterative scheme for oe-dimesioal steady state trasistor calculatio, IEEE Trasactios o Electro Devices, Vol. 11, (1964). 5 T. M. Aostol, Calculus, Vol. II, Multi-Variable Calculus ad Liear Algebra (laisdell, Waltham, MA, 1969) ch. 1 6 J.W. Slotboom, Comuter-aided two-dimesioal aalysis of biolar trasistors, IEEE Trasactios o Electro Devices, Vol. 0, (1973). 7 A. DeMari, A accurate umerical steady state oe-dimesioal solutio of the - juctio, Solid-state Electroics, Vol. 11, (1968).