VLSI DESIGN 1998, Vol. 6, Nos. (1--4), pp. 205--208 Reprints available directly from the publisher Photocopying permitted by license only (C) 1998 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint. Printed in India. A Hot-Hole Transport Model Based on Spherical Harmonics Expansion of the Anisotropic Bandstructure H. KOSINA* and M. HARRER Institute :Cbr Microelectronics, TU Vienna, Gusshausstrasse 27-29, A-1040 Vienna, Austria To represent the valence bands of cubic semiconductors a coordinate transformation is pro- posed such that the hole energy becomes an independent variable. This choice considerably simplifies the evaluation of the integrated scattering probability and the choice of the state after scattering in a Monte Carlo procedure. In the new coordinate system, a numerically given band structure is expanded into a series of spherical harmonics. This expansion tech- nique is capable of resolving details of the band structure at the Brillouin zone boundary and hence can span an energy range of several electron-volts. Results of a Monte Carlo simulation employing the new band representation are shown. Keywords: Monte Carlo Method, Hot Carrier, Transport, Semiconductor, Valence Band, Modeling 1. INTRODUCTION Efforts on numerical modeling of hot carrier transport published to date deal mainly with hot electrons. One reason might be that for electrons some important transport properties are readily revealed by assuming simple effective-mass band models. For holes, how- ever, an effective mass approximation is poor even very close to the F-point. Non-parabolicity is very pronounced and cannot be described by simple ana- lytic expressions. The warped-band model [1 ], which is essentially parabolic, cannot be implemented in the Monte Carlo technique without additional simplifica- tions [2]. The representation of the valence bands we present is specifically tailored to the needs of Monte Carlo transport calculations. These needs include efficient calculation of the scattering integrals and a straight- forward algorithm for the choice of the state after scattering. 2. REPRESENTATION OF THE BANDSTRUCTURE To obtain the total scattering rate the transition proba- bility given by Fermi’s Golden rule has to be inte- grated in the three-dimensional k-space. Because of the energy-conserving 8-function in the transition probability a coordinate transformation is desirable such that energy becomes one of the integration varia- bles. Assume that the band structure is given in polar coordinates" e %(k, f2). We now introduce a coordi- nate transformation (k, if2) ---) (e, f2) by inverting the function %(k, f2) with respect to k. The result of such an inversion is a function describing equi-energy Corresponding author. Tel: +43 58801-3719. Fax: +431 5059224. E-mail: [email protected]205
Transcript
VLSI DESIGN1998, Vol. 6, Nos. (1--4), pp. 205--208Reprints available directly from the publisherPhotocopying permitted by license only
(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under
the Gordon and Breach Science
Publishers imprint.Printed in India.
A Hot-Hole Transport Model Based on SphericalHarmonics Expansion of the Anisotropic Bandstructure
H. KOSINA* and M. HARRER
Institute:Cbr Microelectronics, TU Vienna, Gusshausstrasse 27-29, A-1040 Vienna, Austria
To represent the valence bands of cubic semiconductors a coordinate transformation is pro-posed such that the hole energy becomes an independent variable. This choice considerablysimplifies the evaluation of the integrated scattering probability and the choice of the stateafter scattering in a Monte Carlo procedure. In the new coordinate system, a numericallygiven band structure is expanded into a series of spherical harmonics. This expansion tech-nique is capable of resolving details of the band structure at the Brillouin zone boundary andhence can span an energy range of several electron-volts. Results of a Monte Carlo simulationemploying the new band representation are shown.
Keywords: Monte Carlo Method, Hot Carrier, Transport, Semiconductor, Valence Band, Modeling
1. INTRODUCTION
Efforts on numerical modeling of hot carrier transportpublished to date deal mainly with hot electrons. Onereason might be that for electrons some importanttransport properties are readily revealed by assumingsimple effective-mass band models. For holes, how-ever, an effective mass approximation is poor even
very close to the F-point. Non-parabolicity is verypronounced and cannot be described by simple ana-lytic expressions. The warped-band model [1 ], whichis essentially parabolic, cannot be implemented in theMonte Carlo technique without additional simplifica-tions [2].The representation of the valence bands we present
is specifically tailored to the needs of Monte Carlotransport calculations. These needs include efficientcalculation of the scattering integrals and a straight-
forward algorithm for the choice of the state afterscattering.
2. REPRESENTATION OF THEBANDSTRUCTURE
To obtain the total scattering rate the transition proba-bility given by Fermi’s Golden rule has to be inte-
grated in the three-dimensional k-space. Because ofthe energy-conserving 8-function in the transitionprobability a coordinate transformation is desirablesuch that energy becomes one of the integration varia-
bles. Assume that the band structure is given in polarcoordinates" e %(k, f2). We now introduce a coordi-nate transformation (k, if2) ---) (e, f2) by inverting thefunction %(k, f2) with respect to k. The result of suchan inversion is a function describing equi-energy
surfaces in k-space k K(v-, f2). Inversion of a func-tion is possible only in an interval where the functionis monotonous. By inspection of the full band struc-
ture one finds that both the heavy hole and split-offbands can entirely be represented by such functions KAbove a hole energy of Ex (3.04eV) inversion of thelight hole band is no longer unique.
In this work, we represent the function as a seriesof spherical harmonics.
Kb(’’2)3 Z Z ab,lm(E)e[nl(cSO)cOSm’/=0m=0
b--H,L, SO (1)
Derivation of the scattering rates is considerablyeased by taking the third power of as the function to
be expanded. For symmetry reasons non-vanishingcoefficients only exist for even values of and for rn
being a multiple of 4. With (1) a set of functions
ab,tm(e) contains the whole band structure informa-tion.The density of states of a band represented by (1) is
solely determined by the zero order coefficient.
dgb(t) 4/1:3 dv.a6,oo(v.), b- H,L, SO (2)
All these mechanisms induce both intraband andinterband transitions. Other than for electrons, over-
lap integrals cannot be neglected for holes. The used
approximations are of the form Gii ( + 3COS2[)
Gij-----]sin2[3. The Coulomb scattering rate,
which additionally depends on the solid angle ofthe wave vector, is expressed as a series of spheri-cal harmonics. In Eq. (5), (kj) denotes an averagevalue over the solid angle, which is defined as
(3 a 1./3
(kj) - j, 00(e) The coefficients hjJ(e) being
a result of integration can be expressed in terms of
Legendre functions of the second kind.
The distribution functions of the solid angle afterscattering are given as spherical harmonics series. In a
Monte Carlo procedure, the after scattering state can
be chosen according to these distributions by a simplerejection technique.
4. RESULTS AND DISCUSSION
3. SCATTERING RATES
Wihthin this framework, we derived the scatteringrates for acoustic deformation potential (ADP) scat-
tering in the elastic approximation, optic deformationpotential (ODP) scattering and ionized impurity scat-
tering (ION) in the Brooks and Herring formalism.
8rc2hOv2 -aj,oo(e) (3/
xDP(e)-- 3D ( NP ) d
242pOP Nop +-a2,oo(e + ho)
(4)
(5)
In this work, we use the series expansion (1) to repre-sent the heavy and light hole bands up to eole3.04eV, which is the band-energy at the X-points. Thenumerical band structure has been computed by a
nonlocal empirical pseudopotential method.
The functions ab,tm(e) are represented numericallyby means of a finite element method. To ensure con-
tinuous derivatives shape functions of third orderhave been chosen. The unknowns associated with thenodes of the energy grid have been determined by a
variational approach [3]. From numerical band datathe functions ab,lm(E can well be computed for non-
vanishing hole energies, but not for an energy of zero.
To obtain the ab,tm(O) we expand the expression forthe warped band approximation. In this way, our bandmodel combines the warped band approximation in
the vicinity of the F-point where not enough numeri-
cal data points are available, and the numerical bandstructure for higher hole energies.
A HOT-HOLE TRANSPORT MODEL BASED ON SPHERICAL HARMONICS EXPANSION 207
-0.5
-1
-1.5
g -2
-3.5L O X UK G
FIGURE Comparison of numerical band structure (symbols)and the spherical harmonics expansion (lines) for the heavy holeband
Figure shows the band diagram for the heavy holeband of silicon. Symbols refer to the data points of thenumerical band structure, solid lines to the series
expansion. Equi-energy lines in k-space are plotted in
Figure 2. It turned out that at low energies less har-monics are required than at high energies. Therefore,we make the number of harmonics a function ofenergy. For instance, for the light hole band lmax 20at 0.5eV, and lmax 60 at 3.0eV. The weak ripples at
3.0eV indicate that some higher order harmonics are
still missing. In general, the higher the number of har-moncis, the better the details of the band structure can
be resolved at the boundary of the Brillouin zone. Onthe other hand, for hole energies below Et (1.27eV),where the band structure does not yet touch the zone
boundary, a lower value of Imax is sufficient (typically
lmax < 28).As can be seen in Figure 2 the series representation
provides states outside the first Brillouin zone whichdo not exist in reality. These artificial states yield an
increased density of states and hence increased scat-
tering rates. In the Monte Carlo procedure, scatteringevents to such artificial states outside the Brillouinzone are rejected and self-scattering is performedinstead.In Figure 3 the simulated drift velocity is compared to
measured data [4]. The split-off band has been
neglected in this simulation.
le+07
.x.
le+06 (1- measured (100)> Monte Carlo (I I)- measured (! I)
100(X1000 10000 100000
Electric Field (V/cm)
FIGURE 3 Comparison of simulated and measured [11 hole driftvelocities as function of the electric field at 300K
///
FIGURE 2 Cross section through the heavy hole (left) and light hole (right) bands from 0.5eV to 3.0eV in 0.5eV steps.The surrounding octagon indicates the boundary of the Briliouin zone
208 H. KOSINA et aL
5. CONCLUSION
A new method to represent numerical valence banddata for Monte Carlo transport calculations has beendeveloped. A function basically describing equi-energy surfaces in k-space is expanded into a series ofspherical harmonics. Depending on the energy rangeaccounted for and the number of harmonics invokedthe model can be considered either as an improvedanalytical band model or as a full-band model. In thiswork we demonstrated the full-band capabilities forhole energies up to EX (3.04eV).
[2] C. Jacoboni and L. Reggiani, "The Monte Carlo Method forthe Solution of Charge Transport in Semiconductors withApplications to Covalent Materials," Rev.Mod.Phys., vol. 55,no. 3, pp. 645-705, 1983.
[3] W. Vetterling and S. Teukoisky, Numerical Recipes. Cam-bridge University Press, 1986.
[4] C. Canali, G. Ottaviani, and A. Quaranta, "Drift Velocity ofElectrons and Holes and Associated Anisotropic Effects inSilicon," J.Phys.Chem.Solids, vol. 32, no. 8, pp. 1707-1720,1971.
