J Comput Electron (2010) 9: 160–172DOI 10.1007/s10825-010-0332-9

Modified valence force field approach for phonon dispersion:from zinc-blende bulk to nanowires

Methodology and computational details

Abhijeet Paul · Mathieu Luisier · Gerhard Klimeck

Published online: 23 October 2010© Springer Science+Business Media LLC 2010

Abstract The correct estimation of the thermal propertiesof ultra-scaled CMOS and thermoelectric semiconductordevices demands for accurate phonon modeling in suchstructures. This work provides a detailed description ofthe modified valence force field (MVFF) method to obtainthe phonon dispersion in zinc-blende semiconductors. Themodel is extended from bulk to nanowires after incorpo-rating proper boundary conditions. The computational de-mands by the phonon calculation increase rapidly as thewire cross-section size increases. It is shown that nanowirephonon spectra differ considerably from the bulk disper-sions. This manifests itself in the form of different physi-cal and thermal properties in these wires. We believe thatthis model and approach will prove beneficial in the under-standing of the lattice dynamics in the next generation ultra-scaled semiconductor devices.

Keywords Dynamical matrix · Nanowire · Phonons ·Valence Force Field

1 Introduction

The lattice vibration modes known as ‘phonons’ determinemany important properties in semiconductors like, (i) thephonon limited low field carrier mobility in mosFETs [1, 2],(ii) the lattice thermal conductivity in semiconductors whichplays an important role in thermoelectric design [3–5],

A. Paul (�) · M. Luisier · G. KlimeckSchool of Electrical and Computer Engineering and Network forComputational Nanotechnology, Purdue University, 47907 WestLafayette, USAe-mail: abhijeet.rama@gmail.com

and (iii) the structural stability of ultra-thin semiconduc-tor nanowires [6]. A physics-based method to calculate thephonon dispersion in semiconductors is required to under-stand and link all these issues. As the device size approachesthe nanometer scale and as the number of atoms in thestructure becomes countable, a continuum material descrip-tion is no longer accurate. This work provides a completeand elaborate description of an atomistic phonon calcula-tion method based on a ‘Valence Force Field’ (VFF) model[7–9], within a frozen phonon approach, with application tobulk and nanowire structures.

A variety of methods have been reported in the litera-ture for the calculation of the phonon spectrum such as theValence Force Field (VFF) method and its variants [7–10],Bond Charge Model (BCM) [11, 12], Density FunctionalMethod [6, 13], etc. We focus on VFF methods in this work.There are multiple reasons for using a VFF based model:(a) in covalent bonded crystals, like Si, Ge, GaAs, sim-ple VFF potentials are sufficient to match the experimentaldata [10], (b) valence coordinates and hence the potential en-ergy (U) depend only on the relative positions of the atomsand are independent of rigid translations and rotations of thesolid, and (c) it is easy to extend the model to confined ultra-scaled structures made of few atoms since the interactionsare at the atomic level.

The original Keating VFF model [7] describes the LA,LO and TO phonons reasonably well in zinc-blende mate-rials, however, it does not produce the flatness in the TAbranch in Si and Ge [8, 11]. The inability of the model tocorrectly describe the elastic constants (C11, C12, C44) inthese materials also limits its use [8]. In order to extend theavailable VFF models to nanostructures, we need to identifyan approach that can correctly describe the phonons in theentire Brillouin zone (BZ) in zinc-blende semiconductors. Inolder works as many as six parameters [14] have been used

J Comput Electron (2010) 9: 160–172 161

in VFF to obtain the correct phonon dispersions. However,the goal here is to obtain a VFF model which (i) can cap-ture the correct physics and (ii) is computationally not toointensive. Hence such a model can be extended to nanos-tructures like nanowires, ultra-thin-bodies, etc. To this endwe have identified two VFF models satisfying the require-ments. The modified VFF (MVFF) model presented herecombines these two following models, (i) the VFF modelfrom Sui et al. [8] which is suitable for non-polar materialslike Si and Ge and (ii) the VFF model from Zunger et al. [9]which is suitable for treating polar materials like GaP, GaAs,etc. This extended model is called the ‘MVFF model’ in thisstudy.

The main focus of this work is to show the implementa-tion of VFF models for phonon calculation in zinc-blende(diamond) lattices. We present the details on the atomicgroups which make up the interactions, the application ofboundary conditions in the nanostructures, the eigen valueproblem, the computational requirements, and the evalua-tion of lattice properties. We benchmark the model for vari-ety of zinc-blende materials like Si, Ge, GaP, GaAs, etc., andpresent the results using Si (sometimes Ge too) as a specificexample. We also present a comparison between the stan-dard Keating VFF model [7] and the present MVFF modelfor Si to elucidate the differences in physical results andtheir computational requirements.

Previous theoretical works have reported the calculationof phonon dispersions in SiNWs using a continuum elasticmodel and Boltzmann transport equation [15], atomistic firstprinciple methods like DFPT (Density Functional Perturba-tion Theory) [6, 13, 16, 17] and atomistic frozen phononapproaches like Keating-VFF (KVFF) [3, 18]. Thermal con-ductivity in SiNWs has been studied previously using theKVFF model [4, 19].

This paper has been arranged in the following sections.The MVFF theory (a ‘frozen phonon’ method) for thephonon dispersion in zinc-blende semiconductors is re-viewed in Sect. 2. It provides details about the total potentialenergy (U ) of the crystals in the MVFF model (Sect. 2.1),construction of the dynamical matrix (DM) (Sect. 2.2), ap-plication of boundary conditions to the DM (Sect. 2.3), so-lution of the resulting eigen value problem (Sect. 2.4), andcalculation of sound velocity (Vsnd) (Sect. 2.5), lattice ther-mal conductance (σl) (Sect. 2.6) and the mode Grüneisenparameters (Sect. 2.7) using the phonon spectrum. The com-putational details for the calculation of phonon dispersionare presented in Sect. 3. Different aspects of the dynamicalmatrix like the size, fill-factor, sparsity pattern, etc., are pro-vided in Sect. 3.1. Timing analysis for the assembly of theDM are shown in Sect. 3.2. Section 4 presents, a benchmarkof the MVFF results against experimental data for differentsemiconductors (Sect. 4.1), a comparison of the MVFF andKeating-VFF (KVFF) models (Sect. 4.2), phonon spectrum

in Si nanowires (SiNW) with free and clamped boundaryconditions (Sect. 4.3) and lattice thermal conductance usingSiNW phonon spectrum (Sect. 4.4). Conclusions are givenin Sect. 5.

2 Theory, Approach and Parameters

In a given system, the phonons are modeled by solving theequations of motion of its atomic vibrations. Since VFF is acrystalline model, the dynamical equation for each atom ‘i’can be written as,

mi

∂2

∂t2(�Ri) = Fi = − ∂U

∂(�Ri), (1)

where, �Ri , Fi and U are the vibration vector of atom ‘i’,the total force on atom ‘i’ in the crystal, and the potential en-ergy of the crystal, respectively. Equation (1) indicates thatthe calculation of the vibrational frequencies requires a goodestimation of the potential energy of the system. The nextpart discusses the calculation of U within the MVFF model.

2.1 Crystal Potential Energy (U )

The MVFF method [7–9] approximates the potential en-ergy U , for a zinc-blende (or diamond) crystal, based on thenearby atomic interactions (short-range) [8, 9] as,

U ≈ 1

2

∑

i∈NA

[∑

j∈nn(i)

Uij

bs +j �=k∑

j,k∈nn(i)

(U

jik

bb

+Ujik

bs−bs + Ujik

bs−bb

) +j �=k �=l∑

j,k,l∈COPi

Ujikl

bb−bb

], (2)

where NA, nn(i), and COPi represent the total number ofatoms in one unitcell, the number of nearest neighbors foratom ‘i’, and the coplanar atom groups for atom ‘i’, re-spectively. The first two terms in (2) are from the originalKVFF model [7]. The other interaction terms are needed toaccurately describe the phonon dispersion in the entire Bril-lioun Zone (BZ). The terms U

ij

bs and Ujik

bb represent the elas-tic energy from bond stretching and bond-bending betweenthe atoms connected to each other (Figs. 1a, 1b). The termsU

jik

bs−bs, Ujik

bs−bb and Ujikl

bb−bb represent the cross bond stretch-ing [8, 9], cross bond bending-stretching [9], and coplanarbond bending [8] interactions, respectively (Figs. 1c, 1d, 1e).The functional dependence of each interaction term on theatomic positions is given by,

Uij

bs = 3

8αij

(r2ij − d2

ij,0)2

‖d2ij,0‖

, (3)

162 J Comput Electron (2010) 9: 160–172

Fig. 1 The short range interactions used for the calculation ofphonon dispersion in zinc-blende semiconductors. (a) Bond stretch-ing (b) Bond bending (c) cross bond stretching (d) cross bond bend-ing-stretching and (e) coplanar bond bending interaction

Ujik

bb = 3

8βjik

(�θjik)2

‖dij,0‖‖dik,0‖ , (4)

Ujik

bs−bs = 3

8δjik

(r2ij − d2

ij,0)(r2ik − d2

ik,0)

‖dij,0‖‖dik,0‖ , (5)

Ujik

bs−bb = 3

8γjik

(r2ij − d2

ij,0)(�θjik)

‖dij,0‖‖dik,0‖ , (6)

Ujikl

bb−bb = 3

8

√(νjikνikl)

(�θjik)(�θikl)√‖dij,0‖‖d2

ik,0‖‖dkl,0‖, (7)

where �θjik = rij · rik − dij,0 · dik,0, is the angle deviationof the bond between atom ‘i’ and ’j ’ and bond betweenatom ‘i’ and ‘k’. The term rij (dij,0) is the non-ideal (ideal)bond vector from atom ‘i’ to ‘j ’. The coefficients α, β , δ,γ , and ν determine the strength of the interactions used inthe MVFF model (like spring constants). They are used asfitting parameters to reproduce the bulk phonon dispersion[8, 9]. The unit of these fitting parameters are in force perunit length (N/m). The value of these strength parameterschanges according to the deviation of the bond length andbond angle from their ideal values. This enables the inclu-sion of the anharmonic properties of the lattice vibrations[20] in this model. Hence, MVFF is sometimes referred toas ‘quasi-anharmonic’ model.

Interaction terms: The primitive bulk unitcell used forphonon calculation is made of two atoms (anion-cation pairfor zinc-blende and 2 similar atoms for diamond). The blackdotted box with atom 1 and 2 represents the bulk primi-tive unitcell in Fig. 2. The total number of terms in eachinteraction in (3)–(7) for a bulk unitcell are provided inTable 1. Apart from the coplanar bond bending interac-tion [8] all the other terms involve nearest neighbor inter-actions. There are 21 coplanar (COP) groups in a bulk zinc-blende unitcell which are needed for the calculation of thephonon dispersion. For clarity some of these COP groupsare shown using the number combinations in the caption of

Fig. 2 (Color online) Threeco-planar atom groups (out of21) shown in a bulk zinc-blendeunitcell. The groups are (i)1-2-3-4, (ii) 1-2-5-6 and (iii)1-2-7-8. Atoms 1 and 2 form thebulk unitcell used in thecalculations. Red (black) atomsare cations (anions)

Table 1 Number of terms in different interactions of the MVFF modelin a bulk zinc-blende unitcell (anion-cation pair)

Interaction type Total terms (anion + cation)

Bond stretching (bs) 8

Bond bending (bb) 12

Cross bond stretching (bs-bs) 12

Cross bond stretch-bend (bs-bb) 12

Coplanar bond bending (bb-bb) 21

Fig. 2. Each group consists of 4 atom arranged as anion(A)-cation(C)-anion(A)-cation(C) (e.g. 1(A)-2(C)-3(A)-4(C) inFig. 2). Details about the coplanar interaction groups areprovided in Appendix A.

2.2 Dynamical matrix (DM)

The dynamical matrix captures the motion of the atoms un-der small restoring force in a given system. In this sectionwe discuss the structure of this matrix. The derivation of theDM from the equation of motion is given in Appendix B.The DM calculation is based on the harmonic approximation(see Appendix B). For the interaction between two atoms ‘i’and ‘j ’, the DM component at atom ‘i’ is given by,

D(ij) =⎡

⎢⎣D

ijxx D

ijxy D

ijxz

Dijyx D

ijyy D

ijyz

Dijzx D

ijzy D

ijzz

⎤

⎥⎦ . (8)

The 9 components of D(ij) are defined as,

Dijmn = ∂2Uelastic

∂rim∂r

jn

, (9)

i, j ∈ NA and m,n ∈ [x, y, z],

where NA is the total number of atoms in the unitcell. Foreach atom the size of D(ij) is fixed to 3 × 3. For NA

atoms in the unitcell the size of the dynamical matrix is

J Comput Electron (2010) 9: 160–172 163

3NA ×3NA. However, the matrix is mostly sparse. The spar-sity pattern, fill factor, and other related properties of theDM are discussed in Sect. 3.

Symmetry considerations in the DM: Under the har-monic approximation the dynamical matrix exhibits sym-metry properties that can be readily utilized to reduce itsassembly time. From software development point of viewthis is crucial in optimizing matrix construction time, stor-age and compute times. Due to the continuous nature of thepotential energy U , we have

Dijmn = ∂2Uelastic

∂rim∂r

jn

= ∂2Uelastic

∂rjn∂ri

m

= Djinm. (10)

A closer look at (10) shows the following symmetry relation

D(ij) = D(ji)′, ∀i �= j. (11)

This reduces the total number of calculations required toconstruct the dynamical matrix and speeds up the calcula-tions. Also if the matrix is stored for repetitive use, thenonly one of the symmetry blocks needs to be stored. Thisreduces the memory requirement in the software by a factorof 2. Further reduction in the construction time of DM can beachieved depending on the type of interaction, the symmetryof the crystal, and some implementation tricks (not coveredin this work, see Ref. [21] for more discussion). Also theknowledge about the underlying symmetry of the matrix canhelp in the selection of linear algebra approaches which canreduce the final solution time (not covered in this work).

2.3 Boundary conditions (BC)

To calculate the eigenmodes of the lattice vibration, it is im-portant to apply appropriate boundary conditions to the DM.In the case of bulk material, the unitcell has periodic (Born-Von Karman) boundary conditions along all the directions(x, y, z) [8, 11] since the material is assumed to have an in-finite extent in each direction. However, for nanostructuresthe boundary conditions are different due to the finite extentof the material along certain directions. The boundary con-ditions vary depending on the dimensionality of the struc-ture (1D, 2D or 3D, see Table 2) for which the dynamicalmatrix is constructed. There are 2 types of boundary con-ditions; (i) Periodic Boundary Condition (PBC) which as-sumes infinite material extent in a particular direction and(ii) Finite Edge Boundary Conditions (FEBC) such as openor clamped, which assumes finite material extent in a partic-ular direction. Table 2 provides the boundary condition de-tails depending on the dimensionality of the structure usedfor phonon calculation.

The use of PBC has been discussed in many papers like[8, 9, 11]. In this work we consider the boundary conditionsassociated with geometrically confined nanostructures. The

Table 2 Boundary conditions (BC) in DM based on the dimensional-ity of the structure

Dimensionality Periodic BC Finite edge BC

Bulk (3D) 3 0

Thin Film (2D) 2 1

Wire (1D) 1 2

Quantum Dot (0D) 0 3

Fig. 3 Projected unitcell of a 〈100〉 oriented rectangular SiNW shownwith surface (hollow) and inner (gray filled) atoms

vibrations of the surface atoms can vary from completelyfree (free BC) to damped oscillations (damped BC). It isshown next that all these cases can be handled within onesingle boundary condition.

Boundary conditions for nanostructures: The surfaceatoms (Fig. 3, hollow atoms) of the nanostructures can vi-brate in a very different manner compared to the inner atoms(Fig. 3, filled atoms) since the surface atoms have differentnumber of neighbors and ambient environment compared tothe inner atoms. The degree of freedom of the surface atomscan be represented by a direction dependent damping ma-trix Ξ , defined in Appendix C. In such a case the dynamicalmatrix component between atom ‘i’ and ‘j ’ (D(ij)) is mod-ified to,

D(ij) = ΞiD(ij)Ξj . (12)

2.4 Diagonalization of the dynamical matrix

After setting up the dynamical matrix with appropriate BCsthe following eigen-value problem must be solved,

DQ(λ,q) = Mω2(λ, q)Q(λ,q), (13)

where M is the atomic mass matrix. λ and q are the phononpolarization and momentum vector, respectively. The termQ(λ,q) is a column vector containing all the phonon eigen

164 J Comput Electron (2010) 9: 160–172

displacement modes u(λ, q) associated with the polarizationλ and momentum q . For simplified numerical calculation aslight modification of (13) leads to,

DQ(λ,q) = ω2(λ, q)Q(λ,q). (14)

The detail for obtaining D is outlined in Appendix D. Toderive (14) another step is needed. The time dependent vi-bration of each atom (�R(t)) are represented as the linearcombination of phonon eigen modes of vibration u(λ, q) (acomplete basis set) as

�Ri(t) =∑

q,P

uP (λ, q)ei(q·Ri−ωt), (15)

where, P is the size of the basis set and ω the vibration fre-quency of the modes. Using the result of (15) in the LHSof (1) yields,

mi

∂2

∂t2�Ri(t) = −ω2

∑

q,P

uP (λ, q)ei(q·Ri−ωt). (16)

After some mathematical manipulations and using (13) weobtain the final eigen value problem given in (14).

2.5 Sound velocity (Vsnd)

A wealth of information can be extracted from the phononspectrum of solids. One important parameter is the groupvelocity (Vgrp) of the acoustic branches of the phonon dis-persion which gives the velocity of sound (Vsnd) in the solid.Depending on the acoustic phonon branch used for the cal-culation of Vgrp, the sound velocity can be either, (a) lon-gitudinal (Vsnd,l) or (b) transverse (Vsnd,t ). In solids, Vsnd isobtained near the BZ center (for q → 0) where ω ∼ q . Thus,Vsnd is given by

Vsnd = ∂ω(λ, q)

∂q

∣∣∣∣q→0

, (17)

where λ is the either the transverse or longitudinal polariza-tion of the phonon frequency.

2.6 Thermal conductance (σl)

Another important physical property of the semiconductorsthat can be extracted from the phonon spectrum is the lat-tice thermal conductivity. For a small temperature gradient(�T ) at the two ends of a semiconductor, σl is obtained us-ing Landauer approach [22] as outlined in Refs. [3, 4, 13].For 1D nanowires σl at a temperature ‘T ’ is given by [4, 23],

σl,1D = 1

2π

∫ ωf in

0Π(ω)

∂

∂T

[1

e�ω/kBT − 1

]�ωdω, (18)

where Π(w) is the transmission of a phonon branch atfrequency ω, � and kB are the reduced Planck’s constant,and Boltzmann constants, respectively. Equation (18) is ofgeneral validity and involves a low temperature approxi-mation. Scattering causes the conductance to vary with thenanowires length (L). In the case of ballistic thermal trans-port the transmission Π(ω) is always 1 for all the eigen fre-quencies.

2.7 Mode Grüneisen parameter (γi )

One of the advantages of using the MVFF model is its abilityto keep track of the phonon frequency shift under crystalstress. Under the action of hydrostatic strain the crystal iscompressed without changing its symmetry. With pressure(P ) the phonon frequency shifts, which is measured by aunitless parameter called the mode ‘Grüneisen parameter’given as

γi,q = −∂(ln(ωi,q)

∂ ln(V ), (19)

= B

ωi,q

∂ωi,q

∂P, (20)

where, ωi,q is the eigen frequency for the ith branch at amomentum q . The terms B , P , and V are the volume com-pressibility factor, pressure on the system, and volume of thecrystal, respectively. This parameter is extracted by calculat-ing the eigen frequencies at ambient conditions (P = 0) andat a small hydrostatic pressure (ε = ±0.02) and then tak-ing the difference in the calculated frequencies. The mod-ification of the force constants under hydrostatic pressureis outlined in Ref. [8]. The value of this parameter at highsymmetry points (Γ , X, etc.) in the BZ can be measuredexperimentally by Raman scattering spectroscopy [24].

In the remaining sections we will provide computationaldetails, show results on phonon dispersion in bulk andnanowires and give some results on Vsnd , γi , and ballisticσl in semiconductor nanowires.

3 Computational details

This section provides the computational details to obtainthe phonon dispersion in semiconductor structures. Detailsabout bulk and nanowire (NW) structures are provided.

3.1 Dynamical matrix details

A primitive bulk zinc-blende unitcell has 2 atoms. Thisfixes the size of the DM for the bulk structure to 6 ×6 (3NA × 3NA) (see Sect. 2.2). However, for the case ofnanowires, NA varies with th shape, size, and orientation

J Comput Electron (2010) 9: 160–172 165

Fig. 4 Number of atoms per unitcell (NA) with width (W ) of 〈100〉oriented square SiNW

Fig. 5 Sparsity pattern of the dynamical matrix used in (a) KeatingVFF model and (b) MVFF model. SiNW has W = H = 2 nm with 113atoms in the unitcell

of the wire [13]. In this paper, all the results are for squareSiNW with 〈100〉 orientation. Figure 4 shows the variationof NA with respect to the width (W ) of Silicon NW (SiNW).The number of atoms increase quadratically with W . Fora 6 nm × 6 nm SiNW, NA is 1,013 which means the sizeof DM is 3,039 × 3,039. Extrapolating the NA data givesaround 7,128 atoms for a 16 nm × 16 nm SiNW resultingin a DM of size 21,384 × 21,384 (details in Appendix E).So, the dynamical matrix size increases rapidly with the NWwidth.

The increase of the DM size with wire cross-section im-poses constraint on the structure size which can be solvedusing the atomistic MVFF method. However, the entire ma-trix is quite sparse which is useful to expand the physicalsize of the system that can be simulated. The trick is to usecompressed matrix storage methods. The qualitative ideaabout the filling can be observed from the sparsity pattern fora 2 nm × 2 nm SiNW dynamical matrix as shown in Fig. 5.The quantitative analysis of the fill fraction of the DM andthe number of non-zero elements (NZ) in the DM are shownin Fig. 6. The non-zero elements in the DM increase quadrat-ically with W of SiNW. An estimate for 16 nm × 16 nmSiNW gives about 800,117 non-zero elements (details in

Fig. 6 Non-zero (NZ) elements in the dynamical matrix and fill fac-tor in DM. Fill factor reduces as the wire unitcell size increases eventhough the non-zero elements increase

Appendix E). However, to get an idea about the absolute fill-ing of the DM we define a term called the ‘fill-factor’ givenas,

fill factor = Total nonzero elements/Size of DM

= NZ

(3 × NA)2∝ 1

NA

since NZ ∝ NA

(see Appendix (E.3)). (21)

Thus, the fill factor varies inversely with the number ofatoms in the unitcell. The relation of NZ with NA is givenin Appendix E.

The percentage fill factor of the DM reduces with increas-ing W of SiNW (Fig. 6). This value is ∼0.1% for a SiNWwith W ∼ 25 nm (Appendix E). So even though the non-zero elements increase with W , DM becomes sparser, whichallows to store the DM in special compressed formats likecompressed row/column scheme (CRS/CCS) [25] enablinga better memory utilization.

3.2 Timing analysis for the computation of the DM

The numerical assembly of the DM takes a considerabletime due to the high amount of interactions required by theMVFF model. The assembly time (timeasm) increases as NA

increases. To give an idea about the timing, the dynamicalmatrix for SiNW with different W are constructed on a sin-gle CPU (Intel T6400, 2 GHz processor). The assembly timeis calculated for each width 5 times to obtain a mean valuefor the timeasm. The error bar at each W is the standard devi-ation from the mean timeasm (Fig. 7a). In the present case theassembly of the DM is done atom by atom which is usefulfor distorted materials as well as alloys. The assembly timefor the DM in single materials can be reduced drastically byassuming homogeneous bond lengths and a matrix stampingtechnique [28].

166 J Comput Electron (2010) 9: 160–172

Fig. 7 (a) Time to assemble the DM (timeasm) with width (W ) of〈100〉 oriented SiNW. (b) Time to obtain the eigen solutions per k withW for a 〈100〉 oriented SiNW DM for 100% and 20% of the Eigenspectrum. The timing analysis is done on T6400 Intel processor with

2 GHz speed. Entire Eigen spectrum along with eigen vectors are ob-tained using the ‘eig()’ function in MATLAB [26]. The partial Eigenspectrum is calculated using the ‘eigs()’ function in MATLAB [27]

After the DM is assembled, it is solved to obtain theeigen modes of the phonons. The time needed to diagonal-ize (tdiag) the DM for each momentum point (q), using theMATLAB ‘eig’ function [26] is also extracted (on the sameprocessor). The tdiag value varies as the sixth power of W asshown in Fig. 7b. However, if only 20% percent of the Eigenvalues are calculated the time requirement now goes by thefifth power (shown by the lower line in Fig. 7b). The Eigenvalues in this case are calculated using the ‘eigs’ function inmatlab [27]. The calculation of only 20% of the Eigen spec-trum reduces the per-k energy calculation time by ∼75% fora square SiNW with W = H = 6 nm. However, the possi-bility of using only the partial spectrum to use the importantlattice parameters (like thermal conductance, etc.) is out ofthe scope of the present discussion. For the calculation ofphysical quantities, the complete Eigen spectrum has beenused in this paper.

Extrapolating the data for the computational and tim-ing requirement obtained for the smaller SiNWs, can pro-vide some estimates about the size and time requirementfor larger SiNWs (Table 3). Analytical fits for the varia-tion of the size and time parameters with W are providedin Appendix E. The timing experiments will help us to es-timate the required resources for a future extension of theBandstructure Lab, including the phonons dispersions, onnanoHUB.org [29].

4 Results

In this section we show results for the phonon spectrum inbulk and confined semiconductor structures using both theMVFF and KVFF models. Also some of the physical prop-erties extracted from the phonon dispersions are reported.

Table 3 Resource and timing estimate for larger 〈100〉 SiNW

W NA NZ % fill timeasma Timea

(nm) factor (s) per k (h)

16 7,128 800,117 0.423 238.48 33.2

20 11,120 1,252,490 0.224 370.73 129.5

25 17,346 1.96 × 106 0.101 576.91 505.2

aTime estimates on an Intel T6400, 2 GHz processor

4.1 Experimental benchmarking

The first step to check the correctness of the MVFF modelis to compare the simulated results to experimental data.Figure 8 shows the simulated and experimental [30] bulkphonon dispersion for (a) Silicon and (b) Germanium. Thevalue of the strength parameters are provided in Table 4.A very good agreement between the experimental and sim-ulated data is obtained. To further support the correctness ofthe MVFF model, Vsnd is calculated in bulk Si and Ge alongthe 〈100〉 direction (Table 5). The extracted sound velocityagrees very well with the experimental sound velocity data[31] (max error ≤10%).

The comparison of the second order elastic constants forSi and Ge evaluated using the MVFF model (with the for-mulation provided in Ref. [8]) to experimental data [32] isprovided in Table 6. The MVFF derived values match quitewell to the experimental data [32].

To further test the accuracy of the MVFF model, the cal-culated mode Grüneisen parameters at the high symmetrypoints (Γ and X) are compared to the experimental data (seeTable 7), showing a good agreement.

The advantage of using a higher order phonon model isthat both phonon dispersions as well as the physical parame-ters can be matched to a good accuracy. Hence, the MVFF

J Comput Electron (2010) 9: 160–172 167

Fig. 8 Benchmark of simulatedbulk phonon dispersion withexperimental phonon data for(a) Si and (b) Ge. Experimentaldata is obtained using neutronscattering at 80 K [30]

Table 4 Force constants (N/m) used for phonon dispersion calculation

Material Model α β δ γ ν

Si MVFF [8] 45.1 4.89 1.36 0 9.14

Si KVFF [7] 48.5 13.8 0 0 0

Ge MVFF [8] 37.8 4.24 0.49 0 7.62

Table 5 Sound Velocity in km/s in Si, Ge bulk and square nanowireswith W = H = 6 nm

Material Structure Vsnd calc. Vsnd expt. [31]

Si Bulk Vl [100] 9.09 8.43 (∼8%)

Bulk Vt [100] 5.71 5.84 (∼2%)

NW Vl 6.51 –

NW Vt 4.46 –

Ge Bulk Vl [100] 5.13 4.87 (∼5%)

Bulk Vt [100] 3.36 3.57 (∼6%)

NW Vl 3.70 –

NW Vt 2.61 –

Table 6 Elastic constants (1010 N m−2) obtained from the MVFFmodel compared with experimental data [32] for Si and Ge. The corre-sponding errors in the theoretical values are also shown

Material Model C11 C12 C44

Si MVFF 16.80 6.47 7.63

Si Expt. 16.57 6.39 7.96

Error ∼1.4% ∼1.2% ∼4.14%

Ge MVFF 13.22 4.84 6.29

Ge Expt. 12.40 4.13 6.83

Error ∼6.6% ∼17.2% ∼8%

model captures the experimental phonon dispersion as wellas the elastic properties in bulk zinc-blende material verywell.

Table 7 Comparison of the mode Grüneisen Parameters for bulk Siusing the two phonon models

γi MVFF KVFF Expt./Abinitio Ref.

γ ΓLO,T O 1.05 0.81 0.98 ± 0.06 [24]

γ ΓT A −0.68 −0.43 −0.62 [8]

γ ΓLA 0.95 0.7 0.85 [8]

γ XLO,LA 1.08 0.816 1.03 [33]

γ XT O 1.25 0.83 1.5 ± 0.2 [24]

γ XT A −1.58 −0.33 −1.4 ± 0.3 [24]

4.2 Comparison of VFF models

In this section we compare the original Keating VFF model[7] with the MVFF model to show the need for the moreelaborate MVFF model. The computational requirement andthe physical parameters are compared in this section. Fromcomputational point of view the DM of both models arequite different (Fig. 5). The difference in the sparsity pat-tern arises because of the coplanar interaction present in theMVFF model which takes into account interactions beyondthe nearest neighbors. The KVFF model has fewer non-zeroelements compared to the MVFF model. The increase innumber of NZ elements is faster in the MVFF model thanthe KVFF model (Fig. 9). The MVFF model requires twiceas many matrix elements as compared to the KVFF modelfor a 5 nm × 5 nm SiNW. Thus, the MVFF model demandsmore storage space.

A comparison of the bulk Si phonon dispersion from thetwo models is shown in Fig. 10. The material parametersare reported in Table 4. Qualitatively MVFF shows a bet-ter agreement with the experimental data than the KVFFmodel. There are some important points to note in thebulk phonon dispersion. The KVFF model reproduces theacoustic branches very well near the Brillouin zone (BZ)center but overestimates the values near the zone edge (atthe X and L points in the BZ, Fig. 10). The MVFF modelovercomes this shortcoming and reproduces the acousticbranches very well in the entire BZ. The comparison of thesound velocity along the 〈100〉 direction for bulk Si obtained

168 J Comput Electron (2010) 9: 160–172

from both models show a very good match to the experimen-tal data (Table 8).

The KVFF model overestimates the optical phononbranch frequencies whereas the MVFF model reproducesthe experimental data very well (Fig. 10 and Table 8). Thecomparison of the optical frequency at the Γ point revealsthat the KVFF model overshoots the experimental value by∼7% whereas the MVFF model is higher by only ∼0.6%.

The comparison of the mode Grüneisen parameters forbulk Si using the two models is shown in Table 7. KVFFgives wrong values of these parameters compared to the ex-perimental values. On the other side, the MVFF model isable to reproduce the experimental values very well. Thisshows the importance of using a quasi-anharmonic model tocorrectly obtain the phonon frequency shifts [8, 34]. A sim-ilar failure of the KVFF model for III–V zinc-blende mate-rials has been reported in Ref. [35].

The correct representation of bulk phonons is very impor-tant since this will affect the phonon spectrum in confined

Fig. 9 Matrix size and number of non zero elements required by thetwo models. MVFF has more elements needed for accurate phonondispersion

structures. At the same time the physical properties such aslattice thermal conductivity, phonon density of states (DOS),etc. are also affected. Since the MVFF model matches theexperimental bulk phonon data more accurately, though atthe expense of additional calculations and storage, com-pared to the original KVFF model, we believe that theMVFF model will give better results for phonon dispersionin nanostructures.

4.3 Phonons in nanowires

After benchmarking the bulk phonon dispersion, the sameparameters are used to calculate the phonon spectrum in〈100〉 square SiNW (Fig. 11). The results for a 2 nm × 2 nmfree-standing SiNW are shown in Fig. 11a. Some of the keyfeatures to notice in the phonon dispersion are, (i) the pres-ence of two acoustic branches (ω(q) ∼ q , 1, 2 in Fig. 11a),(ii) the two degenerate modes (3, 4 in Fig. 11a) with ω(q) ∼q2, which are called the ‘flexural modes’, typically observedin free-standing nanowires [6, 13, 18], and (iii) a heavy mix-ing of the higher energy sub-bands leaving no ‘proper’ op-tical mode. These features are quite different from the bulkphonon spectrum and they will strongly affect the physicalproperties of nanowires extracted from the phonon disper-sion.

In Fig. 11b, we explore the effect of a substrate onwhich the nanowire may be mounted. Only the bottom sur-face of the SiNW is clamped whereas the other three sides

Table 8 Comparison of bulk parameters in Si for two models

Model V bulkl,100 V bulk

t,100 ωopt(Γ )

(km/s) (km/s) (THz)

MVFF 9.09 5.71 15.49

KVFF 8.35 5.75 16.46

Expt. 8.43 [31] 5.84 [31] 15.39 [30]

Fig. 10 Comparison of simulated phonon results with experimentaldata (at 80 K from [30]) from the two phonon models (a) Keating VFFand (b) Modified VFF. The KVFF model fails to reproduce many im-portant features in the experimental data as shown by the arrows in (a).

The shortcomings are (i) over estimation of the acoustic mode at X by∼60%, (ii) acoustic branch over estimated at L by ∼95% and failure toreproduce the correct value for the optical branches altogether pointedin (iii) and (iv)

J Comput Electron (2010) 9: 160–172 169

have free boundary condition. Using the boundary condi-tion method discussed in Sect. 2.3, the phonon spectrumin a 2 nm × 2 nm 〈100〉 SiNW are calculated with dif-ferent damping values (Ξ = 1 (free standing) and 0.1).The effect of damping is very prominent at the BZ edgecompared to the zone center. Zone edge frequencies de-crease in energy as the damping increases. A reduction of∼2.11× are observed for the zone edge frequency of the1st branch at Ξ = 0.1 (Fig. 11b) which shows that theNW vibrational energy is decreasing more at higher mo-mentum ‘q’ values. The first four branches are stronglyaffected, while, the higher phonon branches are less af-fected.

4.4 Ballistic lattice thermal conductance (σ ball ) in SiNWs

The ballistic σl of square SiNWs is calculated using theirphonon dispersions. The conductance is calculated us-ing (18) assuming semi-infinite extensions along the wiregrowth axis (X-axis) and CBC on the periphery (Y and Z

Fig. 11 (a) Phonon dispersion in 〈100〉 oriented SiNW withW = H = 2 nm. For clarity only the lowest 40 sub-bands are shown.(b) Dependence of phonon dispersion on the damping of vibration ofthe bottom surface atoms for Ξ = 1 and 0.1. Reduction of phonon en-ergy at the Brillouin zone boundary is stronger compared to the zonecenter

axis) of the wire (Fig. 3). Clamping the bottom surface af-fects the σ bal

l stronger at higher temperature as comparedto lower temperatures (Fig. 12a). Figure 12b shows the σ bal

l

at 300 K. The reduction in σ ball from free-standing wire to

a clamped wire (Ξ = 0.1) is ∼13.1%. Hence, fixing thesurface atoms have a strong impact of the lattice thermalconductance in SiNW.

5 Conclusions

The details for calculating the phonon dispersion in zinc-blende semiconductor structures using a modified ValenceForce Field (MVFF) method have been outlined. The MVFFmethod has been applied to calculate the phonon spectra inconfined nanowire structures with varying boundary con-ditions. The methodology and the computational require-ments of the method have been provided. Comparison ofthe original Keating VFF with the MVFF shows that MVFFprovides an accurate phonon dispersion but at the expenseof higher computational demands. Different VFF modelscan be used to obtain the solution for physical quantitiesdepending on the type of application, size of the struc-ture and the available computational resources. We believethat the MVFF model will provide better phonon dispersionin ultra-scaled nanostructures than the KVFF model. TheMVFF method will be crucial in understanding and mod-eling the thermal properties of ultra-scaled semiconductordevices.

Acknowledgements The authors would like to acknowledge thecomputational resources from nanoHUB.org, an National ScienceFoundation (NSF) funded, NCN project. Financial support from theMSD Focus Center, one of the six research centers funded under theFocus Center Research Program (FCRP), a Semiconductor ResearchCorporation (SRC) entity and by the Nanoelectronics Research Initia-tive (NRI) through the Midwest Institute for Nanoelectronics Discov-ery (MIND) are also acknowledged.

Fig. 12 Ballistic lattice thermalconductivity (σ bal

l for a2 nm × 2 nm 〈100〉 SiNW withdifferent bottom surfacedamping. σ bal

l drops as dampingincreases. Inset shows σ bal

l at100 K, 200 K and 300 K. As thebottom surface changes fromfree-standing to clamped(Ξ = 0.1), σ bal

l reduces by∼12%, ∼12.6% and ∼13.1% at100 K, 200 K and 300 K,respectively

170 J Comput Electron (2010) 9: 160–172

Table 9 Normalized atomic coordinates ([x, y, z] = [x, y, z]/a0) usedfor coplanar interaction calculation

No. x y z No. x y z

1a 0 0 0 14 0 0.50 −0.50

2a 0.25 0.25 0.25 15 −0.50 −0.50 0

3 0.25 −0.25 −0.25 16 −0.50 0 0.50

4 −0.25 0.25 −0.25 17 0 −0.50 0.50

5 −0.25 −0.25 0.25 18 0.25 0.75 0.75

6 0 0.50 0.50 19 −0.25 0.75 0.25

7 0.50 0 0.50 20 −0.25 0.25 0.75

8 0.50 0.50 0 21 0.75 0.25 0.75

9 0 −0.50 −0.50 22 0.75 −0.25 0.25

10 0.50 −0.50 0 23 0.25 −0.25 0.75

11 0.50 0 −0.50 24 0.75 0.75 0.25

12 −0.50 0 −0.50 25 0.75 0.25 −0.25

13 −0.50 0.50 0 26 0.25 0.75 −0.25

aBelong to the main bulk unitcell used for DM calculation

Table 10 Atoms forming the coplanar interaction groups. 4 atoms ineach group

No. Members No. Members

1 2 1 3 9 12 1 2 6 18

2 2 1 4 12 13 1 2 7 21

3 2 1 5 15 14 1 2 8 24

4 3 1 2 6 15 6 2 7 22

5 3 1 4 13 16 6 2 8 25

6 3 1 5 16 17 7 2 6 19

7 4 1 2 7 18 7 2 8 26

8 4 1 3 10 19 8 2 6 20

9 4 1 5 17 20 8 2 7 23

10 5 1 2 8 21 5 1 4 14

11 5 1 3 11

Atom numbers are same as shown in Table 9

Appendix A: Details of bulk zinc-blende coplanarinteraction

The coplanar interactions are important to obtain the flat na-ture of the acoustic phonon branches in Si and Ge [8]. Thereare 21 such interactions in a zinc-blende crystal. The nor-malized locations of all the atoms involved in the coplanarinteractions are shown in Table 9. The corresponding groupsused for bulk phonon dispersion calculations are given in Ta-ble 10.

Appendix B: Derivation of dynamical matrix from theequation of motion

A crystal in equilibrium has zero total force. However, inthe presence of perturbations like lattice vibrations, etc. asmall restoring force works on the system. The total force(Ftotal) under small perturbation is given by the Taylor seriesexpansion as,

Ftotal = −∑

i∈N

∂U

∂�Ri

(= 0 at eqb.)

− 1

2

∑

i,j∈N

∂2U

∂�Ri∂�Rj

· �Rj + · · · , (B.1)

where, N represents all the atoms present in the system andU is the potential energy of the system. In (B.1) first termin RHS is zero under equilibrium. The next non-zero termis the second term in (B.1). Under harmonic approxima-tion, only the second term is considered and the higher order(anharmonic) terms are neglected. Now combining (1) and(B.1) one can obtain the following,

Ftotal =∑

i∈N

mi

∂2

∂t2�Ri

= −1

2

∑

i,j∈N

∂2U

∂�Ri∂�Rj

· �Rj (B.2)

= DR, (B.3)

where, D is called the ‘Dynamical matrix’ and R is a col-umn vector of displacement for each atom given as,

D =

⎡

⎢⎢⎢⎣

D(11) D(12) · · · D(1N)

D(21) D(22) · · · D(2N)...

.... . .

...

D(N1) D(N2) · · · D(NN)

⎤

⎥⎥⎥⎦ , (B.4)

RT = [�R1 �R2 · · · �RN

]. (B.5)

Definition of D(ij) is given in (9).

Appendix C: Treatment of surface atoms

The damped displacement of the surface atom ‘j ’ can berepresented the matrix Ξj given as,

Ξj =⎡

⎢⎣εjx 0 0

0 εjy 0

0 0 εjz

⎤

⎥⎦ . (C.1)

Taking into account the individual components the dis-placement vector for the atom ‘j ’ we obtain,

rjn = ε

jnr

jn , n ∈ [x, y, z]. (C.2)

J Comput Electron (2010) 9: 160–172 171

This modifies (9) as,

Dijmn = ε

jnD

ijmn. (C.3)

Combining (C.1) and (C.3) the dynamical matrix componentbetween atom ‘i’ and ‘j ’ can be represented as,

D(ij) =⎡

⎢⎣εixD

ijxxε

jx εi

xDijxyε

jy εi

xDijxzε

jz

εiyD

ijyxε

jx εi

yDijyyε

jy εi

yDijyzε

jz

εizD

ijzxε

jx εi

zDijzyε

jy εi

zDijzzε

jz

⎤

⎥⎦ (C.4)

which can be written in a compressed form as,

D(ij) = ΞiD(ij)Ξj . (C.5)

The value of εx,y,z ∈ [0,1], where completely free sur-face atoms have value 1 and completely tied atoms havevalue 0.

Appendix D: Inclusion of mass in the dynamical matrix

In (13) the mass of the atoms in on the RHS. It is convenientto include the mass in DM itself. This modifies the LHS ofthe equation. The modified DM component between atom‘i’ and ‘j ’ thus, becomes,

D(ij) =

⎡

⎢⎢⎣

1√mi

Dijxx

1√mj

1√mi

Dijxy

1√mj

1√mi

Dijxz

1√mj

1√mi

Dijyx

1√mj

1√mi

Dijyy

1√mj

1√mi

Dijyz

1√mj

1√mi

Dijzx

1√mj

1√mi

Dijzy

1√mj

1√mi

Dijzz

1√mj

⎤

⎥⎥⎦

(D.1)

here mi and mj are the masses of atom ‘i’ and ‘j ’ respec-tively. Equation (D.1) can be written in a compressed man-ner as,

D(ij) = M−1i D(ij)M−1

j , (D.2)

where Mi is given as,

Mi =⎡

⎣√

mi 0 00

√mi 0

0 0√

mi

⎤

⎦ . (D.3)

Appendix E: Fitted analytical expressions for DMproperties

− Atoms in a 〈100〉 SiNW unitcell: The NA data obtainedfor the square wires till 6 nm × 6 nm can be fitted to aquadratic polynomial given as,

NA(W) = 27.57W 2 + 4.59W. (E.1)

Using (E.1) for a 16 nm × 16 nm SiNW gives around7128 atoms.

Fig. 13 Variation in the number of non-zero (NZ) elements with NA

for the two phonon models (1) Keating VFF and (2) Modified VFF. Forboth the cases NZ varies linearly with NA. MVFF has roughly twicethe number of NZ elements compared to KVFF model

− Non-zero elements in a 〈100〉 SiNW DM: The data fornon-zero elements in the DM for SiNW with W till 6 nmcan be fitted to a quadratic polynomial given by,

NZ(W) = 3156W 2 − 495.5W. (E.2)

Using (E.2) for a 16 nm × 16 nm SiNW yields around800117 non-zero elements in the DM.

− Relation of NZ elements to NA in a 〈100〉 SiNW: Thenumber of non-zero (NZ) elements vary linearly with thenumber of atoms (Fig. 13). Fitting the NZ elements withNA for each wire under study following relations are ob-tained,

NZKVFF ≈ 109.3 × NA, (E.3)

NZMVFF ≈ 205.6 × NA. (E.4)

This shows that the number of NZ elements in MVFFmethod is roughly twice the NZ elements in KVFFmodel.

− Percentage fill factor for a 〈100〉 SiNW DM: The percent-age fill-factor can be derived using (E.1) and (E.2) whichleads to the following expression,

%fill-factor(W) ≈ 0.81

W 2∝ NA−1. (E.5)

Equation (E.5) estimates a 16 nm × 16 nm SiNW DM isfilled only 0.32%. This shows that the DM matrix is verysparsely filled for larger wires.

− Mean assembly time for 〈100〉 SiNW DM: The data formean timeasm of the DM for SiNW with W till 6 nm canbe fitted to a quadratic polynomial given by,

timeasm(W) = 0.9079W 2 − 0.3789W s. (E.6)

Using (E.6) a 16 nm × 16 nm SiNW DM is estimated tobe assembled on single CPU in 226.35 s.

172 J Comput Electron (2010) 9: 160–172

− Eigen solution time per ‘k’ point for 〈100〉 SiNW DM:The eigen values are obtained using the ‘eig’ solver inMATLAB [26]. The time needed for the solution of allthe eigen values (with the eigen vectors) for each mo-mentum vector ‘k’ can be fitted to the following expres-sion,

timeeigen(W) = 5.4 × 10−3W 6.1 s. (E.7)

Thus, the solution time goes with the sixth power of W .Also this expression gives an estimate time for the solu-tion of one k point using the MATLAB eig solver (on aPC) as 1.19e5 seconds (1.4 days). Thus, for very largesystems parallel eigen solvers (like SCALAPACK [36])as well as finding few eigen solutions (using eigs or othersparse eigen solvers, like ARPACK, etc.) is a feasiblemethod.

References

1. Buin, A., Verma, A., Anantram, M.: Carrier-phonon interactionin small cross-sectional silicon nanowires. J. Appl. Phys. 104,053716 (2008)

2. Buin, A.K., Verma, A., Svizhenko, A., Anantram, M.P.:Significant enhancement of hole mobility in [110] siliconnanowires compared to electrons and Bulk silicon. Nano Lett.8(2), 760–765 (2008) pMID: 18205425 [online]. Available:http://pubs.acs.org/doi/abs/10.1021/nl0727314

3. Mingo, N., Yang, L.: Phonon transport in nanowires coated withan amorphous material: an atomistic Green’s function approach.Phys. Rev. B 68(24), 245406 (2003)

4. Mingo, N., Yang, L., Li, D., Majumdar, A.: Predicting the thermalconductivity of Si and Ge nanowires. Nano Lett. 3(12), 1713–1716(2003)

5. Wang, J., Wang, J.-S.: Dimensional crossover of thermal conduc-tance in nanowires. Appl. Phys. Lett. 90(24), 241908 (2007)

6. Peelaers, H., Partoens, B., Peeters, F.M.: Phonon band structureof Si nanowires: a stability analysis. Nano Lett. 9(1), 107–111(2009)

7. Keating, P.N.: Effect of invariance requirements on the elasticstrain energy of crystals with application to the diamond structure.Phys. Rev. 145(2), 637–645 (1966)

8. Sui, Z., Herman, I.P.: Effect of strain on phonons in Si, Ge, andSi/Ge heterostructures. Phys. Rev. B 48(24), 17938–17953 (1993)

9. Fu, H., Ozolins, V., Alex, Z.: Phonons in GaP quantum dots. Phys.Rev. B 59(4), 2881–2887 (1999)

10. McMurry, H., Solbrig, A. Jr., Boyter, J.: The use of valence forcepotentials in calculating crystal vibrations. J. Phys. Chem. Solids28(12), 2359–2368 (1967)

11. Weber, W.: Adiabatic bond charge model for the phonons in dia-mond, Si, Ge, and α-Sn. Phys. Rev. B 15(10), 4789–4803 (1977)

12. Rustagi, K., Weber, W.: Adiabatic bond charge model for thephonons in A(III)B(V) semiconductors. Solid State Commun. 18,673–675 (1976)

13. Markussen, T., Jauho, A.-P., Brandbyge, M.: Heat conductanceis strongly anisotropic for pristine silicon nanowires. Nano Lett.8(11), 3771–3775 (2008)

14. McMurry, H.L., Solbrig, A.W., Boyter, J.K., Noble, C.: The use ofvalence force potentials in calculating crystal vibrations. J. Phys.Chem. Solids 28, 2359–2368 (1967)

15. Zou, J., Balandin, A.: Phonon heat conduction in a semiconductornanowire. J. Appl. Phys. 89(5), 2932–2938 (2001)

16. Zhang, Y., Cao, J.X., Xiao, Y., Yan, X.H.: Phonon spectrum andspecific heat of silicon nanowires. J. Appl. Phys. 102(10), 104303(2007)

17. Li, X., Maute, K., Dunn, M.L., Yang, R.: Strain effects on the ther-mal conductivity of nanostructures. Phys. Rev. B 81(24), 245318(2010)

18. Thonhauser, T., Mahan, G.D.: Phonon modes in Si [111]nanowires. Phys. Rev. B 69(7), 075213 (2004)

19. Zhao, H., Tang, Z., Li, G., Aluru, N.R.: Quasiharmonic models forthe calculation of thermodynamic properties of crystalline siliconunder strain. J. Appl. Phys. 99(6), 064314 (2006)

20. Lazarenkova, O.L., von Allmen, P., Oyafuso, F., Lee, S., Klimeck,G.: Effect of anharmonicity of the strain energy on band offsetsin semiconductor nanostructures. Appl. Phys. Lett. 85(18), 4193–4195 (2004)

21. Hendrikse, Z.W., Elout, M.O., Maaskant, W.J.A.: Computation ofthe independent elements of the dynamical matrix. Comput. Phys.Commun. 86(3), 297–311 (1995)

22. Landauer, R.: Spatial variation of currents and fields due to lo-calized scatterers in metallic conduction. IBM J. Res. Dev. 1(3),223–231 (1957)

23. Wallace, D.C.: Thermodynamics of Crystals. Dover, New York(1998)

24. Weinstein, B.A., Piermarini, G.J.: Raman scattering and phonondispersion in Si and GaP at very high pressure. Phys. Rev. B 12(4),1172–1186 (1975)

25. Dongarra, J.: Survey of sparse matrix storage formats, (1995)[online]. Available: http://www.netlib.org/linalg/html_templates/node90.html

26. Dongarra, J.: Mathworks, Matlab eig reference (2010) [online].Available: http://www.mathworks.com/help/techdoc/ref/eig.html

27. Dongarra, J.: Matlab eig reference (2010) [online]. Available:http://www.mathworks.com/help/techdoc/ref/eigs.html

28. Klimeck, G., Oyafuso, F., Boykin, T.B., Bowen, R.C., von Allmen,P.: Development of a nanoelectronic 3-D (NEMO 3-D) simulatorfor multimillion atom simulations and its application to alloyedquantum dots. Comput. Model. Eng. Sci. (CMES) 3(5), 601–642(2002)

29. Paul A., Luisier, M., Neophytou, N., Kim, R., Geng, J., McLen-nan, M., Lundstrom, M., Klimeck, G.: Band Structure Lab, May2006 [online]. Available: http://nanohub.org/resources/1308

30. Nilsson, G., Nelin, G.: Study of the homology between silicon andgermanium by thermal neutron spectrometry. Phys. Rev. B 6(10),3777–3786 (1972)

31. Electronic archive, new semiconductor materials—characteristicsand properties, Ioffe Physico-Technical Institute Website, 2001,http://www.ioffe.ru/SVA/NSM/Semicond/

32. Madelung, O.: Semiconductors—HandBook, 3rd edn. Springer,Berlin (2004)

33. de Gironcoli, S.: Phonons in Si-Ge systems: An ab initiointeratomic-force-constant approach. Phys. Rev. B 46(4), 2412–2419 (1992)

34. Eryigit, R., Herman, I.P.: Lattice properties of strained GaAs, Si,and Ge using a modified bond-charge model. Phys. Rev. B 53(12),7775–7784 (1996)

35. Lazarenkova, O.L., von Allmen, P., Oyafuso, F., Lee, S., Klimeck,G.: An atomistic model for the simulation of acoustic phonons,strain distribution, and Grüneisen coefficients in zinc-blende semi-conductors. Superlattices Microst. 34(3–6), 553–556 (2003)

36. Blackford, L.S., Choi, J., Cleary, A., D’Azevedo, E., Demmel,J., Dhillon, I., Dongarra, J., Hammarling, S., Henry, G., Petitet,A., Stanley, K., Walker, D., Whaley, R.C.: ScaLAPACK Users’Guide. Society for Industrial and Applied Mathematics, Philadel-phia (1997)