A the Valence Bands up to High
4
A Full-Band Spherical Harmonics Expansion of the Valence Bands up to High Energies A. T. Phama, C. Jungemannb. and B. Meinerzhagena a BST, TU Braunschweig, Postfach 33 29, 38023 Braunschweig, Germany Email: pham@nst. ing. tu-bs . de Telephone: +49 531 391 7731 Fax: +49 531 391 8189 b EIT4, 85577 Neubiberg, Universitat der Bundeswehr Miinchen, Germany Abstract- The valence bands of silicon can be expanded exactly by spherical harmonics only up to the energy, where the bands reach the surface of the Brillouin zone (BZ). For higher energies an approximation is required. Therefore an anisotropic extension of the band structure is presented which is determined by matching the density of states (DOS) and moments of the inverse group velocity of the exact full-band (FB) structure. The Boltzmann equation (BE) based on the exact FB at low energies and the approximation at high energies is solved by the spherical harmonic expansion (SHE) method for bulk silicon including impact ionization. The results are compared to Monte Carlo (MC) data based on the exact FB structure. both spaces. This is only possible as long as the mapping is unique in both directions [1]. In the case of the FB structure for heavy holes in unstrained Si, this is possible up to 1.27eV (light holes: 1.31eV, split off: 7.09eV). For higher energies, the bands are constructed in such a way that, on the one hand, the dispersion relation k' (E 0, q) of band v depends monotonic on E for all angles (0, 5), and on the other hand, the moments (1) I. INTRODUCTION In Ref. [1] a SHE of the valence band structure has been presented. This expansion is only possible as long as the energy depends in a monotonic way on the wave vector. For heavy holes in unstrained silicon this is the case up to about 1.27eV limiting the usefulness of this approach. We present an approximation with which this approach can be extended to higher energies. In contrast to previous work (e.g., [2]), our approach is anisotropic and the DOS and the group velocity are consistent with the band structure, which is a prerequisite for a stable discretization of the BE in devices [3]. In section 2, the theory for the anisotropic extension of the valence band is described. The DOS and moments of the inverse group velocity are matched to those of the exact FB structure [4]. With the valence bands extended up to high energies meaningful hot hole simulations are possible, and the BE is solved at high electric fields by the SHE method for bulk silicon including impact ionization. The results, which are shown in section 3, are compared to MC data based on the exact FB structure [5]. II. THEORY The SHE of the distribution function on equienergy surfaces has many advantages over an expansion with respect to the modulus of the wave vector [3]. The BE is therefore projected onto SHs for constant energies. This requires the mapping of the spherical coordinates of the k space (k, 0, ¢b) onto an energy space (£, 0, r), where the angles are the same in are reproduced as good as possible. v>'(k) is the group velocity in k space, n an integer number and the integral is taken on the equienergy surface £>(k) = E within the first BZ. With the methods presented in Ref. [5] the moments V, are calculated by numerical integration over the first BZ of the FB structure [4]. However for evaluating the integral V, in spherical coordinates (£, 0, r), k and its derivatives with respect to energy and angles are required for all (0, r). The construction for k at high energies is based on a recursive calculation of kt'(E 0, b) from its value k'( -d ,0, b) at the energy £-0 , where 6E denotes the energy spacing. The initial value of k is given by the value at the highest energy for which the exact FB can be still expanded. The approximative SHE of k'(£F 0, Q) is obtained by adding a few low order SH terms to the previous k' ( - ,0, r). In addition, the SHE of k contains only even harmonics and has a fourfold symmetry in q due to the symmetry of the band structure [3]. Adding even SHs up to the fourth order to kt'( -d 0, b) yields k (£, 0, 0) = k(E (£- , 0, 0) + min kV(E- 6E£ 0~o) minF'(E, 0, )] (2) with r (£, 0, ) = j' (£) Cos 0 + _ (E) COS4 0 +-K (£) sin4 0 cos 4q (3) 1-4244-0404-5/06/$20.00 © 2006 IEEE 1 6(. Vn (--) = 1: - n v z i.v (k) x [-.O' (E -) I' + r, (--, 0, 0) SISPAD 2006 361
Transcript of A the Valence Bands up to High
A Full-Band Spherical Harmonics Expansion of theValence Bands up to High Energies
A. T. Phama, C. Jungemannb. and B. Meinerzhagena
a BST, TU Braunschweig, Postfach 33 29, 38023 Braunschweig, GermanyEmail: pham@nst. ing. tu-bs . de
Telephone: +49 531 391 7731Fax: +49 531 391 8189
b EIT4, 85577 Neubiberg, Universitat der Bundeswehr Miinchen, Germany
Abstract- The valence bands of silicon can be expandedexactly by spherical harmonics only up to the energy, where thebands reach the surface of the Brillouin zone (BZ). For higherenergies an approximation is required. Therefore an anisotropicextension of the band structure is presented which is determinedby matching the density of states (DOS) and moments of theinverse group velocity of the exact full-band (FB) structure. TheBoltzmann equation (BE) based on the exact FB at low energiesand the approximation at high energies is solved by the sphericalharmonic expansion (SHE) method for bulk silicon includingimpact ionization. The results are compared to Monte Carlo (MC)data based on the exact FB structure.
both spaces. This is only possible as long as the mapping isunique in both directions [1]. In the case of the FB structurefor heavy holes in unstrained Si, this is possible up to 1.27eV(light holes: 1.31eV, split off: 7.09eV). For higher energies, thebands are constructed in such a way that, on the one hand, thedispersion relation k' (E 0, q) of band v depends monotonicon E for all angles (0, 5), and on the other hand, the moments
(1)
I. INTRODUCTION
In Ref. [1] a SHE of the valence band structure has beenpresented. This expansion is only possible as long as theenergy depends in a monotonic way on the wave vector. Forheavy holes in unstrained silicon this is the case up to about1.27eV limiting the usefulness of this approach. We presentan approximation with which this approach can be extendedto higher energies. In contrast to previous work (e.g., [2]), ourapproach is anisotropic and the DOS and the group velocityare consistent with the band structure, which is a prerequisitefor a stable discretization of the BE in devices [3].
In section 2, the theory for the anisotropic extension ofthe valence band is described. The DOS and moments ofthe inverse group velocity are matched to those of the exactFB structure [4]. With the valence bands extended up to highenergies meaningful hot hole simulations are possible, and theBE is solved at high electric fields by the SHE method forbulk silicon including impact ionization. The results, whichare shown in section 3, are compared to MC data based onthe exact FB structure [5].
II. THEORY
The SHE of the distribution function on equienergy surfaceshas many advantages over an expansion with respect to themodulus of the wave vector [3]. The BE is therefore projectedonto SHs for constant energies. This requires the mappingof the spherical coordinates of the k space (k, 0, ¢b) ontoan energy space (£, 0, r), where the angles are the same in
are reproduced as good as possible. v>'(k) is the group velocityin k space, n an integer number and the integral is takenon the equienergy surface £>(k) = E within the first BZ.With the methods presented in Ref. [5] the moments V,are calculated by numerical integration over the first BZ ofthe FB structure [4]. However for evaluating the integral V,in spherical coordinates (£, 0, r), k and its derivatives withrespect to energy and angles are required for all (0, r). Theconstruction for k at high energies is based on a recursivecalculation of kt'(E 0, b) from its value k'( -d ,0, b) atthe energy £-0 , where 6E denotes the energy spacing. Theinitial value of k is given by the value at the highest energy forwhich the exact FB can be still expanded. The approximativeSHE of k'(£F 0, Q) is obtained by adding a few low order SHterms to the previous k' ( - ,0, r). In addition, the SHE ofk contains only even harmonics and has a fourfold symmetryin q due to the symmetry of the band structure [3]. Addingeven SHs up to the fourth order to kt'( -d 0, b) yields
k (£, 0, 0) = k(E(£- , 0, 0) + min kV(E- 6E£ 0~o)
minF'(E, 0, )] (2)
with
r (£, 0, ) = j' (£) Cos 0 + _ (E) COS4 0
+-K (£) sin4 0 cos 4q (3)
1-4244-0404-5/06/$20.00 © 2006 IEEE
1 6(.Vn (--) = 1: -n
v z i.v (k)
x [-.O' (E -) I' + r, (--, 0, 0)
SISPAD 2006 361
where NYo1,2,3 are fitting parameters. With this ansaltz, obvi-ously k"(E 0, ) > k(E -6,O, ). It means that withoutany restriction for the fitting parameters, the monotonicitycondition of k'(E, 0, b) is automatically fulfilled for all angles(0, ). The derivatives of k'(E) with respect to (w.r.t) theangles can be calculated in a straight forward manner. Thederivative of k' w.r.t energy is determined by a left sidedfinite difference approximation:
&k(E 0 (4)(EO, k)-"(E- £E 0:(4)
With (2), (4) and the derivatives of k-'(E) w.r.t angles, V,can be expressed in terms of the fitting parameters and theknown k1, U" at energy £- E. Matching the V, to the exactvalues of the FB structure by the nonlinear least square errorLevenberg-Marquardt method [6], [7] yields the set of tY0 12 3for energy E. As soon as the fitting parameter set has beenobtained for E, the resultant k (E, 0, b) is used as initial valuefor the dispersion relation of the next energy level E + 6E. Byiteration the fitting parameters are obtained for all energies.
To minimize iteration steps for each energy level, the fittingparameters must be initialized in a proper way. By using smallenergy steps, the resultant fitting parameters at energy E canbe used as first guess for the optimization of the next energylevel E + 6E. For the energy level at the beginning of thewhole optimizing process, all fitting parameters are initializedas zeroes except -y0. With the assumption that the initialized<)0 (= -o,init) are the same for all bands, 'YO,init can be foundby matching V0 with its exact value.
The described method allows to extend the SHE of the FB tothe high energies required for meaningful hot hole simulations.The BE is solved with a deterministic SHE method basedon the phonon system described in Ref. [5] and the impactionization model of Ref. [8].
III. RESULTS
Up to an energy of 7.09eV the band structure extensionis only required for the heavy (HH) and light hole (LH)bands, because the relation between energy and wave vectorremains monotonic for the split-off band in this energy range.For energies below 1.2eV, the exact SHE of the HH andLH bands is used. The optimizing process is performed forenergies above 1.2eV with an energy step of ImeV. The bandindex in definition (1) for moments V, represents HH andLH bands. Fig. 1 - 5 show that V0 (-DOS) and V1-V5 basedon the approximated band structure fit well the exact FBdata. The fitting parameters as functions of energy are shownfor HH (Fig. 7) and LH band (Fig. 8). It can be seen thatN'o, -Yi, -Y2 :t 0 for both HH and LH bands. This means thatthe extension of HH and LH bands is anisotropic. Moreover,1-Y3 «< 1 Y2 1, | -fti| implies that the contribution of the harmonicY4,4 (- sin4 0 cos 4O) to the SHEs of the HH and LH bandscan be neglected.The hole drift velocity calculated by SHE for different direc-
tions of the electric field is shown together with MC data basedon the exact FB in Fig. 9. Good agreement of both methods
is obtained and the full anisotropy is reproduced by SHE. Inthe nonlinear regime, if the applied electric field is rotatedby 450 from (110) to (100) direction, the hole drift velocityincreases significantly. The energy distribution function (e.d.f.)is plotted in Fig. 10. The SHE results reproduce accuratelythe MC results, even for E = 300kV/cm. Fig. 11 shows theII coefficient versus inverse electric field. Again the deviationfrom MC is very small. For E = 200kV/cm, it takes six daysto obtain the II coefficient with 0.5% error by MC, whereasthe CPU time required by SHE is only about eight minutes.
IV. CONCLUSIONSWe have developed for the first time an anisotropic method
to extend the SHE of the valence bands to high energies andthe results agree well with simulations based on the exact FBstructure. Since the expansion is exact below 1.2eV, all theimportant magnetotransport and transport effects in confiningfields, which are due to the anisotropic band structure, aredescribed correctly in contrast to previous approaches, whichwere based on an isotropic approximation of the band structure(e.g., [2]). In addition, our approach avoids the inconsistencyof the group velocity and density of states inherent to thoseschemes.
REFERENCES
[1] H. Kosina, M. Harrer, P. Vogl, and S. Selberherr, "A Monte Carlo transportmodel based on spherical harmonics expansion of the valence bands," inProc. SISPAD, pp. 396-399, 1995.
[2] M. C. Vecchi and M. Rudan, "Modeling electron and hole transport withfull-band structure effects by means of the spherical-harmonics expansionof the BTE," IEEE Trans. Electron Devices, vol. 45, no. 1, pp. 230-238,1998.
[3] C. Jungemann, A. T. Pham, B. Meinerzhagen, C. Ringhofer, andM. Bollh6fer, "Stable discretization of the boltzmann equation based onspherical harmonics, box integration, and a maximum entropy dissipationprinciple," J. Appl. Phys., vol. 99, 2006.
[4] M. M. Rieger and P. Vogl, "Electronic-band parameters in strainedSil- Ge1 alloys on SilyGey substrates," Phys. Rev. B, vol. 48,pp. 14276-14287, 1993.
[5] C. Jungemann and B. Meinerzhagen, Hierarchical Device Simulation:The Monte-Carlo Perspective. Computational Microelectronics, Wien,New York: Springer, 2003.
[6] K. Levenberg, "A Method for the Solution of Certain Problems in LeastSquares," Quart. Appl. Math., vol. 2, pp. 164-168, 1944.
[7] D. Marquardt, "An Algorithm for Least-Squares Estimation of NonlinearParameters," SIAM J. Appl. Math., vol. 11, pp. 431-441, 1963.
[8] Y Kamakura, I. Kawashima, K. Deguchi, and K. Taniguchi, "Verificationof hot hole scattering rates in silicon by quantum-yield experiment,"J. Appl. Phys., vol. 88, no. 10, pp. 5802-5809, 2000.
1-4244-0404-5/06/$20.00 © 2006 IEEESISPAD 2006 362
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Fig. 5. V4 contributing by HH and LH bands
V5
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1-4244-0404-5/06/$20.00 2006 IEEE
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SISPAD 2006 363
HH
X..i ...............
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1.5 2 2.5 3 3.5 4Energy [eV]
Fig. 7. Fitting parameters of HH band
LH
Y" ...........
.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Y +.
.,~~~~~~~~~~ ~ ~ ~~~~~~~~~~~~~~...,........
1 1.5 2 2.5 3Energy [eV]
Fig. 8. Fitting parameters of LH band
3.5 4
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Fig. 10. Hole energy distribution function without II (applied electric fieldsin (110) direction)
1000
100
10
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0.001
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Impact ionization coefficient (applied electric
8 9 10
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0 50 100 150 200 250 300Electric field [kWcm]
Fig. 9. Hole drift velocity v.s electric fields applied on different directions,without II
1-4244-0404-5/06/$20.00 2006 IEEE
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