Multiscale electrostatic analysis of silicon nanoelectromechanical systems (NEMS)via heterogeneous quantum models

Yang Xu and N. R. Aluru*Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

Received 4 September 2007; published 11 February 2008

A multiscale method, seamlessly combining semiclassical, effective-mass Schrödinger EMS, and tight-binding TB theories, is proposed for electrostatic analysis of silicon nanoelectromechanical systems NEMS.By using appropriate criteria, we identify the physical models that are accurate in each local region. If the localphysical model is semiclassical, the charge density is directly computed by the semiclassical theory. If the localphysical model is quantum mechanical the EMS or TB model, the charge density is calculated by using thetheory of local density of states LDOS. The LDOS is efficiently calculated from the Green’s function byusing Haydock’s recursion method where the Green’s function is expressed as a continued fraction based onthe local Hamiltonian. Once the charge density is determined, a Poisson equation is solved self-consistently todetermine the electronic properties. The accuracy and efficiency of the multiscale method are demonstrated byconsidering two NEMS examples, namely, a silicon fixed-fixed beam with hydrogen termination surfaces andanother silicon beam switch with 90° single period partial dislocations. The accuracy and efficiency of themultiscale method are demonstrated.

DOI: 10.1103/PhysRevB.77.075313 PACS numbers: 85.85.j, 47.11.St, 41.20.Cv

I. INTRODUCTION

As significant progress is being made in various fields ofnanotechnology, predicting electronic and mechanical prop-erties of semiconductor devices is a compelling challengefrom both the scientific and engineering viewpoints. In com-plex physical systems, the length scales can vary from a fewnanometers to several hundreds of microns and predictingthe physical properties at all length scales accurately andefficiently is a significant computational challenge. Multi-scale computational methods aim to couple different lengthscales into one single framework.1–7 Several multiscalemethods have been proposed in the literature to solve com-plex problems in solid and fluid mechanics. For example, theconcurrent handshaking multiscale method has been shownto successfully simulate crack propagation in siliconnanostructures;1 heterogeneous multiscale methods, whichuse molecular dynamics at the microscale and continuummechanics at the macroscale, have been used to compute themechanical properties of various silicon structures.2,7 In con-trast, there has been comparatively little effort devoted tomultiple scale analysis of electrostatics, which is an impor-tant energy domain for the analysis of nanoelectromechani-cal systems NEMS. In this paper, we propose a multiscalemethod for the electrostatic analysis of silicon nanoelectro-mechanical systems.

Realistic silicon NEMS structures contain bulklike subdo-mains, quantum confinement regions, and defect areas. Typi-cally, bulklike subdomains can be accurately modeled by asemiclassical theory, quantum confinement regions can beaccurately modeled by an effective-mass Schrödinger EMStheory, and defect areas can be modeled by tight-bindingTB methods. The use of any one of these physical theoriesto model the entire NEM structure can either be inaccurate orcomputationally infeasible. As a result, development of amultiscale method that can seamlessly combine variousphysical theories is critical for the accurate and efficient elec-

trostatic analysis of NEMS. Our proposed multiscale schemeis conceptually illustrated in Fig. 1. We deal with two-dimensional electrostatic problems in this paper, but the ap-proach can be easily extended to three-dimensional prob-lems. The semiconductor device is first discretized intopoints—these points are referred to as the Poisson points. Ateach point, by using appropriate criteria, which will be dis-cussed in detail later, we determine if the local region sur-rounding the point can be treated by the semiclassical, EMS,or TB model. When the quantum-mechanical models the

V

Rs

Rs

V

Bottom conductor

Semiconductorstructure

grid pointsPoisson

Schrodingergrid points

Crystal atoms

FIG. 1. Color online Illustration of the multiscale approach fora typical NEMS example consisting a semiconductor beam struc-ture and a bottom conductor. The semiconductor beam region isdiscretized into Poisson grid points. The filled circles represent thePoisson grid points. In the region where the quantum effects areimportant, a sampling region typically circular, but other shapescan also be easily considered is required to compute LDOS at thePoisson point. The radius of the sampling region is denoted by RS.The open circles in the EMS sampling regions represent theSchrödinger grid points. The atomic structure in the TB samplingregions represents the TB atom sites. The LDOS in the EMS andTB regions are calculated from the EMS and the TB Hamiltonian,respectively.

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EMS or TB model are required, one cannot solve the EMSor TB equation in a local region by direct domain decompo-sition to determine the charge distribution, as the boundaryconditions on the wave functions are not known on theboundaries of the local region. Instead, in our multiscalemodel, the charge density is obtained from the local densityof states LDOS.8 To compute the LDOS, only a finite-sizesampling region shown as a circular disk of radius RS in Fig.1 centered at the Poisson grid point is required by applyingKohn’s “nearsightedness” principle.13,14 As shown in Fig. 1,if the sampling region intersects the physical boundaries ofthe semiconductor, then the sampling region is appropriatelymodified. The EMS sampling region is composed ofSchrödinger grid points and the TB sampling region is com-posed of the real atomistic structure. The LDOS is computedfrom the Green’s function of the Hamiltonian matrix by us-ing a recursion method.15 As the number of recursion levelscomputed is usually far less than the dimension of theHamiltonian matrix, the recursion method is an efficient ap-proach to compute the LDOS and other electronic properties.

The multiscale method described above seamlessly com-bines the semiclassical and heterogeneous quantum-mechanical models. Quantum-mechanical effects are in-cluded only in the regions where they are necessary. As aresult, semiconductor nanostructures which can be describedentirely by semiclassical or quantum models or by combinedsemiclassical and quantum models are accurately modeled.The rest of the paper is organized as follows. Section IIdescribes the theory and implementation of the proposedmultiscale model. Section III presents results for several ex-amples demonstrating the accuracy and efficiency of themultiscale method and conclusions are given in Sec. IV.

II. THEORY

A. Physical models

To illustrate the multiscale approach for electrostaticanalysis of semiconductor nanostructures, we consider a gen-eral semiconductor system shown in Fig. 2. We deal with

two-dimensional 2D electrostatic problems in this paper,but the approach can be easily extended to three-dimensionalproblems. Consider a system of objects including semicon-ductors 1 ,2 , . . . ,N0

, where N0 is the number of ob-

jects, embedded in a uniform dielectric medium , as shownin Fig. 2. Potential x and its normal derivative qx aregiven by gx and hx on portions of the boundary of eachsemiconductor, g and h, =1,2 , . . . ,N0, respectively,where x= x ,yT is the position vector of any point. The gov-erning equations for electrostatic analysis along with theboundary conditions are given by8

· s x = − x = − epx − nx + ND+ x

− NA−x in , = 1, . . . ,N0, 1

2x = 0 in , 2

x = gx on g, = 1, . . . ,N0, 3

qx =x

n= hx on h, = 1, . . . ,N0, 4

where x is the charge density, px and nx are hole andelectron densities, respectively, ND

+ x and NA−x are the ion-

ized donor and acceptor concentrations, respectively, s is thepermittivity of the semiconductor material, and e is the el-ementary charge.

In the exterior domain , the Laplace equation is satis-

fied, as shown in Eq. 2. Note that the exterior domain isan open domain. A boundary integral equation BIE of the2D Laplace equation is used to treat the exterior electrostaticproblem,9 i.e.,

xx = =1

N0

xGx,x

ndx

− =1

N0

xn

Gx,xdx + , 5

=1

N0

xn

dx = 0, 6

where is the corner tensor =1 /2 for smooth boundaries,see Ref. 10 for details and Gx ,x= 1

2 ln x−x is theGreen’s function. In 2D problems, the potential at infinity isthe reference potential, . The boundary integral equationsEqs. 5 and 6 are coupled with the Poisson equation Eq.1 through the interface conditions given by11

xBIE = xPoisson on g, = 1,2, . . . ,N0, 7

s xn

Poisson

+ d xn

BIE

= int on h,

Ω

φ2 =0

g2

Γg3

Γg4

Ω

Ω

Γg5

5

h4Γ

2

Γh2

Γ Γ

g1Γ

h1

1Ω Ω

Γh3

Ω3

4

FIG. 2. Color online The electrostatic semiconductorsystem.

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= 1,2, . . . ,N0, 8

where xBIE and xPoisson are the potentials from theboundary integral equation and Poisson equation, respec-

tively, x

n BIE and x

n Poisson are the normal derivativesof the potential from the boundary integral equation andPoisson equation, respectively, d is the permittivity of thedielectric medium, and int is the charge density on the ex-posed surface of the semiconductor. In this paper, for thepurpose of illustrating the multiscale approach, the chargedensity on the surface is assumed to be zero. However, non-zero interface charge density int can also be implementedeasily. The potential x and the charge density x areobtained by solving Eqs. 1–8 self-consistently. In this pa-per, the interior Poisson’s equation Eq. 1 is solved byusing the finite difference method12 and the boundary inte-gral equations Eqs. 5 and 6 are solved by the boundaryelement method.10

In the multiscale approach, the choice of a physical modelfor a particular region is closely related to the accuracy andefficiency of the physical model for that particular region.There are a number of physical models for electrostaticanalysis such as the classical Laplace conductor model,semiclassical Poisson model, effective potential model, EMSmodel, TB model, density functional theory model, etc.8,16

Here, we only discuss three of these physical models, one foreach scale, namely, the semiclassical Poisson model for mi-croscale, the EMS model for nanoscale, and the TB modelfor atomistic scale.

1. Semiclassical model: Semiclassical Poisson equation

When the geometrical characteristic length of the deviceis comparable to the Debye screening length,17 the semiclas-sical model is necessary to compute the charge density dis-tribution. In the semiclassical model, the electron and holedensity in the Poisson equation can be obtained as8

nx = NC2

F1/2EF − ECx

kBT , 9

px = NV2

F1/2EVx − EF

kBT , 10

where NC and NV are the effective density of states of con-duction and valence bands, respectively, F1/2 is the completeFermi-Dirac integral of order 1 /2, ECx is the conductionband energy given by ECx=−ex+

Eg

2 , where Eg is theenergy gap, e is the elementary charge, EVx is the valenceband energy given by EVx=−ex−

Eg

2 , EF is the Fermienergy, kB is the Boltzmann constant, and T is the tempera-ture, which is set to be the room temperature in this work.The main advantage of the semiclassical model is its simplic-ity, but the model can suffer from inaccuracies when quan-tum effects are important.

2. Quantum-mechanical model: Effective-massSchrödinger equation

When the critical size of the device approaches the na-nometer scale, quantum effects such as the carrier quantumconfinement in the semiconductor structure becomesignificant.17 The quantum effects in the device can be ac-counted for by solving the EMS equation in the entire semi-conductor domain. The EMS equation is given by

Hnr = − 2

2

x 1

mx*

x +

2

2

y 1

my*

y nr

+ ernr

= Ennr , 11

where H is the system Hamiltonian based on the effective-mass approximation, is the reduced Planck’s constant, mx

*

and my* are the effective masses along the x and y axes,

respectively, r= x ,yT is the position vector of anySchrödinger point, nr is the nth eigen-wave-function, andEn is the nth eigenvalue of the Hamiltonian H. If mx

* and my*

are electron masses, then the Hamiltonian, nr, and En arecomputed for electrons. If mx

* and my* are hole masses, then

the Hamiltonian, nr, and En are for holes. If we discretizethe domain into NS Schrödinger grid points and the gridspacings along the x axis and the y axis are denoted by xSand yS, respectively, then using the finite differencemethod, the EMS Hamiltonian elements in Eq. 11 can berewritten as

Hi,j = 2

mx*

1

xS2 +

2

my*

1

yS2 + eri when i = j

− 2

2mx*

1

xS2 when points i and j are neighbors along the x axis,

− 2

2my*

1

yS2 when points i and j are neighbors along the y axis

0 otherwise,

12

where i, j=1, . . . ,NS.

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Once the Hamiltonian elements are obtained through Eq.12, we can solve the EMS equation Eq. 11 numerically.After solving the eigensystem, all the eigenenergies andeigen wavefunctions are known. Then, the electron densitynr and hole density pr can be computed by

nr = Nnn

nr2F−1/2EF − En

kBT , 13

pr = Npn

nr2F−1/2En − EF

kBT , 14

where the coefficients Nn and Np are

Nn = gn1

2mn

*kBT

2 1/2

, 15

Np = gp1

2mp

*kBT

2 1/2

, 16

for the conduction band and valence band, respectively, F−1/2is the complete Fermi-Dirac integral of order −1 /2, constantsgn and gp are the semiconductor conduction band degeneracyand valence band degeneracy, respectively, mn

* and mp* are

the density of state masses of electrons and holes, respec-tively, and the summations in Eqs. 13 and 14 are over allthe energy levels. Note that the solution of the EMS equationEq. 11 requires an eigensolution in the entire semiconduc-tor domain. This can be expensive and inefficient whensimulating large devices.

3. Quantum-mechanical model: Tight-binding approach

The EMS approach can be inaccurate when the criticalsize of nanostructures is within a few nanometers or whendefects and other material inhomogeneities in nanostructuresbecome important. To overcome the limitations of the EMSapproach, more accurate quantum-mechanical models are re-quired. Ab initio methods are typically accurate to predictelectronic properties of nanostructures. However, the highcomputational cost prevents their widespread use. An alter-native is to use TB methods, which are somewhat less accu-rate but computationally more efficient compared to ab initiomethods. In this paper, we choose a nearest neighbor sp3d5s*

orthogonal TB approach18 as the atomistic quantum-mechanical model for defects and surface states.

In the TB scheme, the electronic wave functions nr areexpressed as a linear combination of atomic orbitals, nr=ici

nir, where ir is the th atomic orbital wavefunction at site i and ci

n is the nth coefficient. The TBHamiltonian is defined in terms of parametrized matrixelements.19–21 These parametrized matrix elements are usu-ally constructed by fitting to experimental and/or first-principles database of properties of both bulk systems andclusters. The diagonal elements of the Hamiltonian matrixi represent the on-site energies of atomic orbitals. Theatomic orbital , for example, for the sp3d5s* TB model usedin this paper, can represent any of s, px, py, pz, dxy, dyz, dzx,dx2−y2, d3z2−r2, or s* orbitals.18,19 The intersite elements Hi,j

0

describe the bond energies between the th orbital of atom iand the th orbital of atom j, and these are determined byusing the tables of Slater and Koster.19 For example, His,jpx

0 isgiven by

His,jpx

0 = lxVsprij , 17

where rij = R j −Ri is the distance between atoms i and j,Ri= xiyiziT and R j = xjyjzjT are the position vectors of at-oms i and j, respectively, lx denotes the x direction cosine ofthe vector R j −Ri, and Vsprij is the element correspond-ing to the hopping bond energy of forming bond betweenthe s orbital of atom i and the p orbital of atom j.19–21 In thispaper, these elements are taken to follow Harrison’s rule,21–24

i.e.,

Vsprij = sp

2

m0rij2 , 18

where m0 is the electron mass and sp is one of the TBhopping parameters. The TB parameters we use have beenoptimized by Boykin et al.18 and Zheng et al.25 to accuratelyreproduce the band gap and effective masses of bulk Si. Allthe silicon hopping parameters and on-site-energy param-eters of the sp3d5s* TB model are listed in Table I.18 Whensimulating the nanostructure surfaces, the silicon surface at-oms with dangling bonds are passivated with hydrogen at-oms in this paper. The TB parameters for hydrogen atoms,adopted from Zheng et al.,25 are also listed in Table I. TheTB Hamiltonian elements can be written in a general formgiven by

TABLE I. TB parameters for silicon and hydrogen Refs. 18 and25.

Parameter Si-Si interaction Si-H interaction

ss eV −1.95933 −3.99972

ss* eV −1.52230 −1.69770

sp eV 3.02562 4.25175

sd eV −2.28485 −2.10552

s*s* eV −4.24135

s*p eV 3.15565

s*d eV −0.80993

pp eV 4.10364

pp eV −1.51801

pd eV −1.35554

pd eV 2.38479

dd eV −1.68136

dd eV 2.58880

dd eV −1.81400

d0 Å 2.352 1.478

Si H

s eV −2.15168 0.99984

p eV 4.22925

s* eV 19.11650

d eV 13.78950

Zce 4.0 1.0

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Hi,j = i + eri when i = j and =

H0 rij when atoms i and j are nearest neighbors

0 otherwise, 19

where H0 rij are directly obtained from the Slater and Ko-

ster’s table.19

Once the Hamiltonian is obtained, the electronic wavefunctions and eigenenergies of the tight-binding system arecomputed by solving the characteristic equation26,27

HCn = EnCn, 20

where En is the nth eigenenergy of the system and Cn is thecolumn vector comprising the ci coefficients. After the wavefunctions are known, the atomic electron charges can becomputed by

ner = 2n,i

cinir2

eEn−Ef/KT + 1. 21

In a self-consistent approach, the electrostatic potentialcontribution to the on-site terms of the TB Hamiltonian riis updated by solving the Poisson equation. An initial guessof the potential is required to compute the total HamiltonianH. In this paper, we use the semiclassical potential distribu-tion as the initial guess. After the total Hamiltonian is com-puted, Eq. 20 is solved to obtain the eigenenergies and thewave functions of the system. The wave functions are thenused to compute the charge density by using Eq. 21. Next,the Poisson equation is solved to update the potential distri-bution. This process is repeated until the self-consistent so-lution is obtained. Note that the direct solution of the TBequation Eq. 20 requires an eigensolution in the entiresemiconductor domain, which can be costly and inefficientwhen simulating realistic nanoscale systems.

B. Multiscale approach

Typical semiconductor nanostructures contain regionswhere the semiclassical models are valid, but they also con-tain regions where quantum effects are important. If a semi-classical model is used for the entire device, the electronicproperties may not be predicted accurately in the regionwhere quantum effects are important. On the other hand, ifquantum-mechanical models are used for the entire device,the extremely high computational cost limits the size of thedevice that can be simulated. Multiscale models, whichseamlessly combine the semiclassical and quantum models,can be accurate and efficient to predict the electronic prop-erties of semiconductor nanostructures. The use of direct do-main decomposition, where the TB, EMS, and semiclassicalmodels are employed in different regions and combinedthrough interface boundary conditions, can be hard to imple-ment as the wave functions are nonlocal and imposition of awave function as a boundary condition between the semi-classical and quantum regions can be difficult.

To overcome this difficulty, we develop a multiscalemodel which seamlessly combines quantum-mechanical andsemiclassical models. In the region or at the Poisson gridpoints where the semiclassical model is not valid, the Pois-son grid point is represented by heterogeneous fine-scalemodels or the quantum models see Fig. 1. A Hamiltonian isfirst constructed by using the TB atoms or the Schrödingergrid points. Using the Hamiltonian, the elements of theGreen’s function GF matrix are then computed by28–30

lim→0+

Gj,jE + i = lim→0+

E + iI − H−1 j,j

= lim→0+

n

nr jn†r j

E − En + i,

j, j = 1,2, . . . ,Nh, 22

where i=−1, I is the identity matrix, Gj,j denotes the jthrow and jth column entry of the Green’s function matrix G,Nh is the size of the Hamiltonian, E is the energy, and n

†r jis the conjugate value of nr j. The LDOS at position r j,denoted by Nr j ,E, can be expressed as the imaginary partof the diagonal elements of the GF matrix see, e.g., Ref. 28for details,

Nr j,E = − −1 lim→0+

Im Gj,jE + i . 23

The LDOS can be efficiently calculated from the Green’sfunction by using Haydock’s recursion method,15 where theGreen’s function is expressed as a continued fraction basedon the local EMS39 or TB Hamiltonian.

To get the charge density in the EMS region, the EMSHamiltonian elements for electrons and holes need to be con-structed first. If the Hamiltonian of electrons is used in Eq.22, then the LDOS Nr ,E computed in Eq. 23 is thatof the electrons denoted by Ner ,E and if the Hamiltonianof holes is used in Eq. 22, then the hole LDOS denoted byNhr ,E is computed. After Ner ,E and Nhr ,E areknown, the electron density nr and the hole density prcan be obtained by

nr = ECr

Ner,EfeEdE , 24

pr = −

EVr

Nhr,EfhEdE , 25

where feE is the Fermi-Dirac distribution for electronsgiven by

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feE =1

1 + eE−EF/kBT , 26

and fhE is the Fermi-Dirac distribution for holes given by

fhE =1

1 + eEF−E/kBT . 27

Similarly, to get the charge density in the TB region, theLDOS should be first computed using the TB Green’s func-tion. The LDOS of the th orbital at the TB atom site r j,denoted by Nr j , ,E, can also be expressed as the imagi-nary part of the diagonal elements of the GF matrix,

Ner j,,E = − −1 lim→0+

Im Gj,jE + i . 28

The corresponding atomic charge on atom j, qr j, canthen be computed by

qr j = −

−

Ner j,,EfeEdE − Zc , 29

where if atom j is a silicon atom, then ionic core charge Zc=4.0, and if atom j is a hydrogen atom, then Zc=1.0.27 Thus,the charge density r can be obtained by

r = eqrVSi

+ ND+ r − NA

−r , 30

where VSi is the effective volume occupied by a silicon crys-tal atom.

Comparing Eqs. 24, 25, and 30 with Eqs. 13 and14, we note that the wave functions are not required explic-itly in the LDOS approach to compute the electron and holedensity. In addition, for each sampling region, we only needto compute the LDOS at the Poisson point or the corre-sponding Schrödinger point or the corresponding TB atomsite, where the sampling region is centered.

In summary, in the multiscale model described above, in-stead of directly coupling the semiclassical and quantum-mechanical regions, in the regions where semiclassicaltheory is not valid, the LDOS method is applied. This ap-proach enables seamless coupling of the quantum-mechanical theory into the semiclassical theory.

1. Size of the sampling region

As shown in the last section, the size of the samplingregions directly determines the computational cost of themultiscale method. The nearsightedness principle is a usefulconcept that can guide the size of the sampling region. Ac-cording to the nearsightedness principle,13,14 the local elec-tronic properties such as the local charge density only de-pend significantly on the effective external potential in thenearby region. The changes of that potential, no matter howlarge, beyond a distance RS, have a limited effect on the localelectronic properties, which rapidly decay to zero as a func-tion of RS.14 Prodan and Kohn14 gave an approximate expres-sion for the nearsightedness radius Rr0 ,n centered at r0

for an electronic system as

Rr0,n 1

2qeffln

n

n, 31

where qeff is the decay constant of the density matrix givenby qeff=

12Egm* / 2, m* is the effective mass of electrons

holes, n is the local density which can be approximated bythe equilibrium charge density at r0, r0, and n is thefinite maximum asymptotic charge density due to any pertur-bation outside the circle of radius R. Rr0 ,n defines theradius where the change in density at r0 as a result of anyperturbation outside of R does not exceed n. For example,from Eq. 31, if we require the relative density error tosatisfy the condition n

n 510−3, we obtain the nearsight-edness radius for silicon to be approximately 6 nm. Thenearsightedness principle can guide the determination of thesize of sampling regions taking the accuracy level into ac-count. In this paper, we choose the radius of the samplingregion RS to be equal to the nearsightedness radius for sim-plicity.

2. Criterion for choosing tight-binding or effective-massSchrödinger models

If the silicon crystal structure within the sampling regionis not perfect, then the EMS theory is typically not valid, i.e.,if there are defects, material inhomogeneities, or surfacestates non-bulk-like states within the sampling region, thenthe TB method is more appropriate at the sampling point.Another consideration is, for sampling regions with no inho-mogeneities, as long as the TB method gives approximatelythe same local effective masses and band gap as the EMSmethod, then the EMS approach is accurate enough to com-pute the electronic properties in the local region. When thecritical size of the silicon nanostructure is smaller than a fewnanometers, the EMS approach can deviate from the TBmethod. Several papers have addressed this issue.31,32 Whenthe radius of a silicon nanocluster is larger than 2.172 nmequivalent to 32 silicon layers within the sampling region,the deviation of the local effective mass and the band gapobtained by the TB and EMS methods is smaller than a fewpercent at the center region of the cluster.31,32 These resultsare also consistent with the prediction from the nearsighted-ness principle: if we choose sampling regions with radii of1.629 nm 24 silicon layers, 2.172 nm 32 silicon layers,and 2.716 nm 40 silicon layers, the relative density errorsbetween the TB and EMS methods are within 15%, 7%, and3% by using Eq. 31. These conclusions then lead to ourcriterion for determining an appropriate quantum-mechanicalmodel by considering a sampling region of radius 2.716 nm.If the silicon atomic structure within the sampling region is aperfect crystal structure that is, no dangling bonds, defects,impurities, dislocations, surfaces, etc., within the samplingregion, then we can use the EMS method to compute thelocal electronic properties without much loss of accuracy;otherwise, the TB model is needed.

3. Criterion for choosing semiclassical or quantum-mechanicalphysical models

Quantum-mechanical effects diminish when the quantumconfinement is negligible. In this case, semiclassical models

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can be employed to increase the efficiency and avoid thecomplexity of quantum-mechanical methods.16 To determinewhether the local region is quantum-mechanical or semiclas-sical, we use the quantum potential criterion.33–39 When theelectrons are the major carriers in the device, the quantumpotential qx is given by

qx = − 2

8

x 1

mx*

ln nxx

+

y 1

my*

ln nxy

.

32

When the holes are the major carriers in the device, nx isreplaced by px, and the electron effective masses are re-placed by the hole effective masses. To determine whetherthe sampling Poisson point is semiclassical or quantum me-chanical, we first calculate the quantum potential by Eq.32. By comparing the quantum potential qx with thecoulomb potential x, the semiclassical model is used atthe sampling Poisson point if qx /maxxtol, wheretol is the given tolerance; otherwise, the quantum-mechanical model is used at the sampling Poisson point.

C. Dislocation theory

To demonstrate the accuracy and efficiency of the multi-scale method for materials with inhomogeneities, we simu-late a silicon nanostructure with 90° single period partialdislocations. The 90° partial dislocation is one of the mostcommon types of dislocations found in silicon.40,41 It liesalong 110 directions in 111 slip planes and it has a Bur-gers vector of the type 1

6 112.40,42,43 The 90° partial dislo-cation is assumed to have a reconstructed core,45 and severalpossible reconstructions have been suggested. The two whichhave been shown by simulation to be stable46–48 are thesingle period SP and double period structures. In this work,we only focus on the 90° SP partial dislocation. The 90° SPpartial dislocation is believed to undergo reconstruction oftheir cores to eliminate unsaturated bonds and restore four-fold coordination to all atoms. In this 90° SP partial recon-struction, the displacement breaks the mirror symmetry nor-mal to the dislocation line, enabling threefold coordinatedatoms in the unreconstructed core Fig. 3a to come to-gether and bond. This core reconstruction is shown in Fig.3b.

The partial dislocation is associated with shallow statesabove the valence band, i.e., the energy gap is reduced due tothe dislocation reconstruction.49 The reconstruction breaksthe symmetry of the crystal structure. As a result, the appli-cation of the EMS method can be inaccurate, thereby neces-sitating the use of the TB method. For the dislocation geom-etry and positions of the atoms, we use the set of relaxedatomic coordinates for the 90° single period partial disloca-tions from Ref. 44 in our calculation.

D. Algorithm for the self-consistent multiscale model

Algorithm 1 Table II summarizes the procedure for theself-consistent electrostatic analysis of semiconductor nano-structures by using the combined semiclassical/EMS/TBmultiscale approach.

III. RESULTS

A. Two-dimensional fixed-fixed beam case

To illustrate the multiscale model, we first choose a 2Dfixed-fixed silicon beam, as shown in Fig. 4. The x axis ofthe beam is chosen along the 100 direction and the beamcomprises 100 faces on all sides. The p-doped 1020 cm−3silicon beam is clamped above a ground plane. A voltage,Vapp, is applied between the ground plane and the beam. Thesilicon surface dangling bonds are terminated with hydrogenatoms. The Schottky contact effect and tunneling effect areignored in this case for simplicity. The width W and thelength L of the beam are equal to 10.864 nm. The gap Gbetween the beam and the ground plane is set to 2.716 nm.We assume that the beam and the bottom conductor are in-finitely long along the z axis the z axis is pointing out of thepaper. To compute the charge density distribution, we firstdiscretize the domain to Poisson grid points. The Poissongrid spacings along the x axis and the y axis are denoted byxP and yP, respectively.

To understand how the various parameters in the multi-scale model affect the accuracy, efficiency, and convergence

(a)

(b)[110]

[112]_

_

FIG. 3. Color online Core structure of the 90° partial disloca-tion looking down on the 111 slip plane. The dislocation along

the broken line is in the 110 direction. a Unreconstructed core.b SP reconstruction. Note that the bonding breaks the mirror sym-

metry normal to 110 that existed in the unreconstructed core.

appVappV x

y

abce

d

G

Ground Plane

W

L

FIG. 4. Color online A typical nanoswitch consisting of afixed-fixed semiconductor and a bottom conductor, where L=W=10.864 nm and G=2.716 nm with an applied voltage Vapp

=−10.0 V.

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rate of the method, we first study the impact of the samplingradius RS on the convergence rate. The effect of the EMSsampling radius has been investigated in Ref. 39, which gavean optimized EMS sampling radius of 6 nm. Here, we onlyinvestigate the effect of the TB sampling radius. To check theconvergence rate of the multiscale method, we first modelthe entire domain by directly solving the TB eigenproblem asa reference and denote it as the full TB method. Figure 5ashows the error between the multiscale method and the full

TB method as a function of the TB sampling radius RS,where RS is varied from 0.5432 nm equivalent to 8 layers ofsilicon atoms within the sampling region to 4.3456 nm 64layers of atoms. From Fig. 5a, we observe that the errordecays as RS increases. When RS is equal to 2.716 nmequivalent to 24 layers of atoms, the error between themultiscale model and the full TB model is below 4%.

Next, we study how the recursion levels NL influence theconvergence rate for a given RS=2.716 nm. Figure 5bshows the error as a function of NL, when NL is varied from10 to 40 levels. From Fig. 5b, we notice that the errordecays quickly as NL increases. When NL is 30, the errorbetween the multiscale model and the full TB model is lessthan 5%. Because a larger sampling radius and a higher num-ber of recursion levels do not significantly improve the ac-curacy but demand significantly higher computational cost,we choose RS=2.716 nm and NL=30 in this work.

By using the quantum criteria, three kinds of regions areidentified in the silicon beam, as shown in Fig. 6. The TB,EMS, and semiclassical regions are displayed by circles, symbols, and dots, respectively. The big and small circlesrepresent the silicon and hydrogen atoms, respectively. Theoutermost atoms of the silicon beam influenced by the sur-face states are identified as the TB region by using the TBcriterion. In this case, the y component of the electric field inthe middle region of the beam is so strong that the quantumconfinement region is pushed deep into the beam, where thecharacteristic length of the quantum confinement regionis larger than the characteristic length of the surface states.

0 1 2 3 4

0

0.2

0.4

The radius of sampling region Rs

(nm)

Err

or

(a)

0 10 20 30 40

0

0.2

0.4

The recursion levels NL

Err

or

(b)

FIG. 5. Color online Comparison of error between the multi-scale approach and the full TB model as a function of RS and NL.Error is defined as error=1 / max

ref 1 /NPi=1NP refxi−axi2,

where ref and a are the reference full TB solution and the multi-scale solution, respectively, obtained by using xP=yP

=0.2716 nm. When computing the error by varying NL in subplotb, the sampling radius RS is set equal to 2.716 nm.

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

x (nm)

y(n

m)

−0.6 −0.4 −0.2 0 0.2 0.4−6

−4

−2

0

2

4

6

Quantum potential (V)

TB

EMS

Semiclassical

EMS

TB

(a) (b)

FIG. 6. Color online Illustration of the multiscale domain and quantum potential for the fixed-fixed silicon beam with dimensions ofL=W=10.864 nm and G=2.716 nm. The semiclassical model is applicable at the sampling Poisson point if qx /maxtol, where tol

is the given tolerance a value of 510−2 is used here. a The circles, the , and the dots indicate the TB, EMS, and semiclassical regions,respectively. The big and small circles represent silicon and hydrogen atoms, respectively. The TB, EMS, and semiclassical regions occupyabout 72%, 12%, and 16% of the beam, respectively. Plot b shows the quantum potential along the y axis at x=0. Because the quantumpotential is almost negligible in the central region, the semiclassical model is used. In the other regions, where the quantum potential issignificant, the quantum mechanical models are required. Near the surfaces, where the surface states are important, the TB method is applied;otherwise, quantum-mechanical region is identified as the EMS region.

YANG XU AND N. R. ALURU PHYSICAL REVIEW B 77, 075313 2008

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Furthermore, the y component of the electric field in themiddle region is much stronger compared to that at edges. Asa result, two semielliptic EMS regions are identified by thequantum potential criterion, as shown in Fig. 6.

Figure 7 shows the tight-binding LDOS of five atoms de-noted as a to e in Fig. 4 at or near the surface of the siliconbeam. Because the LDOS near the edges of the conductionbands and the valence bands contributes most to the chargedensity due to the Fermi distribution, we focus on the LDOScomparison there. Although the surface states generated bySi-H bonds are mainly located outside the energy gap,50 theycan still significantly change the LDOS near the edges of thevalence and conduction bands when compared to the LDOSof silicon atoms in the interior region. The difference in theLDOS between the surface and interior states induces thenonzero partial charges. The surface states only exist for thefirst few surface layers and decrease quickly as we move intothe interior region.

A comparison of the detailed solutions for the electrondensity and potential obtained by the EMS, TB, and multi-scale methods is presented in Fig. 8. The potential andcharge distributions along the y axis for x=0 cross sectionand along the x axis for y=0 cross section obtained with theEMS, TB, and multiscale models are compared in Figs.8a–8d. From these results, we find that the charge densitywith the EMS method close to the boundary is much differ-ent from that of the multiscale method see Figs. 8c and8d because the EMS method simply ignores all the surfacestates. Although the nonphysical surface states are removed

−15 −10 −5 0 5 100

0.5

1

0

0.5

1

0

0.5

0

0.5

0

0.5

1

Energy (eV)

LDO

S

(a)

(b)

(c)

(d)

(e)

FIG. 7. Color online The tight-binding LDOS of various atomsnear the edge of a silicon beam. The thin lines represent the LDOSof a central silicon atom in the beam. The thick lines are the LDOSof the various surface atoms shown in Fig. 4. a LDOS of thesurface hydrogen atom a shown in Figs. 4. b–e LDOS of sili-con atoms b–e shown in Fig. 4. The surface states decrease quicklyas we move into the inner region from the surface.

−5 0 5−0.6

−0.4

−0.2

0

y (nm)

Pote

ntia

l(V

)

(a)

EMS

TBA

Multiscale

−5 0 5−0.04

−0.02

0

0.02

0.04

(b)

x (nm)

−5 0 5−2

−1

0

1

2

3

y (nm)

Charg

edensi

ty(1

021cm

−3)

(c)

−5 0 5−2

−1

0

1

2

3

x (nm)

(d)

−5 −4 −30

0.5

1

4 5−0.1

−0.05

0

0.05

−5 −4 −3−0.04

−0.02

0

0.02

0.04

FIG. 8. Color online Comparison of the potential obtained from the EMS, TB, and multiscale models. Plots a and b compare thepotential computed by the multiscale, EMS, and TB models along the y axis for x=0 cross section and along the x axis for y=0 cross section,respectively. Plots c and d compare the charge density computed by the multiscale, EMS, and TB models along the y axis for x=0 crosssection and along the x axis for y=0 cross section, respectively.

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by terminating dangling silicon bonds with hydrogen atoms,the nonzero Mulliken partial charges at silicon surface atomsand hydrogen atoms still significantly influence the surfaceelectronic properties. The smoothly varying EMS electrondensity near the boundary results in a different potential dis-tribution compared to the multiscale method see Figs. 8cand 8d insets, in which the electron density oscillates dueto the surface partial charges. Due to the different densitydistribution, the potential distribution obtained from theEMS and multiscale methods is also different see Figs. 8aand 8b. From these results, we observe that the EMSmodel does not correctly predict the electronic propertiesnear surface regions, whereas the multiscale model can re-produce the results of the full TB model. For this example,although the TB region occupies about 72% of the beam, themultiscale method is still about 40 times faster compared todirectly solving the TB model.

B. Two-dimensional fixed-fixed beam with dislocations

To further explore the effectiveness of the multiscalemethod, we simulate a silicon beam with two 90° SP partialdislocations and study the influence of dislocations on elec-tronic properties. The silicon structure is shown in Fig. 9.The width of the beam is denoted by W, the length is L, andthe gap is G. The doping density is the same as in the firstcase and the Schottky contact effects and tunneling effectsare also ignored in this example. The two dislocations lienear the center of the beam along 110 separated by 1.27 nmin the 111 slip plane with opposite Burgers vectors of thetype 1 /6112.

By using the criteria discussed in Sec. II B 3, three kindsof quantum regions are identified, as shown in Fig. 10. Thefirst quantum region is formed by the surface states whichsurround the entire nanostructure. The second kind the twoEMS regions denoted by in Fig. 10 is the quantum con-finement region formed by the applied electric field where

the electrons are confined and the energy levels are quan-tized. The third quantum region is formed by the dislocationswhere the stacking fault is located and the EMS approach isnot accurate. The TB quantum region is about 41% of theentire domain in this case.

The LDOSs of several silicon atoms near the dislocationcore see Fig. 9 for the location of the atoms are plotted inFig. 11. The thin lines represent the LDOS of a referencesilicon atom far from the dislocation and the surface. Thethick lines represent the LDOS of the representative atomsdenoted by 1–10 in Fig. 9 near the dislocation core. Fromthe figure, the dislocation induces a shallow state above the

appVappV

[111]

[112]_

G

Ground Plane

W

L

7

9

2

18

6

54

3

10

x

y

FIG. 9. Color online A silicon beam with two opposite 90° SP partial dislocations is clamped above a ground plane, where L=39.9024 nm, W=19.7505 nm, and G=2.1945 nm with an applied voltage Vapp=−10.0 V. The two partial dislocations located near the

center of the beam have opposite Burgers vectors with a separation of 1.27 nm. The plane of the page is 110. The stacking fault is locatedbetween “” and “.” The dislocation lines are perpendicular to the paper plane. The atomic coordinates of the reconstructed dislocationcores are taken from Ref. 44. Each 90° SP partial dislocation is characterized by the intersection of a fivefold and a sevenfold ring. Forexample, for the dislocation on the right side, the fivefold ring is composed of atoms 2-3-4-5-6-2 and the sevenfold ring contains atoms1-2-6-7-8-9-10-1. The key feature of the relaxed geometry is that all the atoms are now truly fourfold coordinated, i.e., all four neighboringatoms are within a few percent of the ideal bond length and there is no dangling bond in the dislocation core Ref. 44.

−20 −15 −10 −5 0 5 10 15 20−10

−5

0

5

10

x (nm)

y(n

m)

FIG. 10. Color online Illustration of the multiscale domain forthe fixed-fixed silicon beam with dimensions of L=39.9024 nm,W=19.7505 nm, and G=2.1945 nm. The Poisson grid spacings arexP=0.6650 nm and yP=0.3135 nm. The semiclassical model isapplicable at the sampling Poisson point if qx /maxtol,where tol is the given tolerance a value of 510−2 is used here.The circles, the , and the dots indicate the TB, EMS, and semi-classical regions, respectively. The big and small circles representsilicon and hydrogen atoms, respectively. The TB, EMS, and semi-classical regions occupy about 41%, 16%, and 43% of the beam,respectively.

YANG XU AND N. R. ALURU PHYSICAL REVIEW B 77, 075313 2008

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valence band and reduces the energy gap. The dislocationcore results in nonzero partial charges along the dislocationline. The dislocation effects decay quickly when the atoms

are far away from the core. The dislocation reconstructiondue to stacking fault breaks the symmetry of the crystalstructure, thus making it necessary to use the TB method.

TABLE II. A complete algorithm for self-consistent multiscale electrostatic analysis of semiconductor nanostructures.

1: Discretize the domain of the semiconductor beam into NP Poisson grid points for solving the Pois-

son equation and discretize the boundary of the domain (if any) into grid points for solving boundary

integral equation (BIE).39

2: Set k=0, solve the semiclassical Poisson model and obtain the initial potential distribution φ(0), electron

density n(0), and hole density p(0).

3: repeat

4: for each Poisson grid point, i=1 to NP do

5: use the criteria to choose the appropriate physical model at the Poisson grid point

6: if the physical model is TB then

7: define the local sampling region by Eq. (31) according to the desired accuracy level and construct the

entries of the local TB Hamiltonian matrix using φ(k). Compute the LDOS and ρ(k+1)i at the TB atom

site (rj=xi) using Eqs. (28-30)

8: else if the physical model is EMS then

9: define the local sampling region by Eq. (31), discretize the sampling region into Schrodinger grid points,

construct the entries of the local EMS Hamiltonian matrix by using φ(k), and compute the LDOS and

n(k+1)i and p

(k+1)i at the Schrodinger point (rj=xi) using Eqs. (24-25)

10: else if the physical model is semiclassical then

11: compute the charge density n(k+1)i and p

(k+1)i using Eq. (9) and Eq. (10)

12: end if

13: end for

14: solve the coupled BIE/Poisson equations to update the potential distribution φ(k+1)

15: k = k + 1

16: until a self-consistent solution is obtained when max|n(k)i − n

(k−1)i | < θρ

tol, max|p(k)i − p

(k−1)i | < θρ

tol

and max|φ(k)i − φ

(k−1)i | < θφ

tol, where θρtol and θφ

tol are the error tolerances for the charge density and

potential, respectively.

17: Output the results.

Algorithm 1

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Since the conduction band energy and electron densitynear the dislocation core are the interesting features, we fo-cus on the distribution along the positive x axis for y=0 crosssection obtained by the EMS, TB, and multiscale modelssee Fig. 12. The charge density deviation between the mul-tiscale and EMS models indicates that the EMS approach isnot accurate near the dislocation and surface regions and theTB model is needed. The multiscale method is about 8 timesslower compared to the full EMS method but over 600 timesfaster than the full TB method. The additional cost of themultiscale method compared to the EMS method is due tothe fact that the TB quantum region occupies 41% of thebeam. From these results, we can conclude that the multi-scale method is an accurate and efficient tool to compute theelectronic properties of silicon nanostructures.

IV. CONCLUSIONS

A heterogeneous multiscale model combining semiclassi-cal, effective-mass Schrödinger, and tight-binding ap-

proaches has been presented in this paper for the electrostaticanalysis of silicon nanoelectromechanical systems. Quantumcriteria are used to identify semiclassical, EMS, and TB re-gions. In the regions where quantum effects are significant,the charge densities are computed by considering samplingregions and the LDOS. The LDOS is efficiently calculatedby using Haydock’s recursion method. The multiscalemethod has been shown to be accurate by comparing withthe full TB method. When the TB regions are significant, themultiscale method has been shown to be several orders ofmagnitude faster compared to the direct solution of the fullTB method. When the quantum regions are small, the multi-scale method approaches the efficiency of the semiclassicalmethod. By considering examples of silicon NEMS with hy-drogen surface termination and 90° SP partial dislocations,we have also shown that the multiscale method efficientlyand seamlessly coupled the different physical models intoone framework without loss of accuracy.

ACKNOWLEDGMENTS

This work is supported by the National Science Founda-tion under Grant Nos. 0228390 and 0601479, by DARPA/MTO, and by SRC.

*Corresponding author: aluru@uiuc.edu; http://www.uiuc.edu/aluru

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−5−4−3−2−1 0 1 2 3 4 50

0.3

0

0.3

0

0.3

0

0.3

0

0.3

Energy (eV)

LDO

S

(1)

(2)

(3)

(4)

(5)

−5−4−3−2−1 0 1 2 3 4 5Energy (eV)

(6)

(7)

(8)

(9)

(10)

FIG. 11. Color online The LDOS of ten silicon atoms denotedin Fig. 9 near the dislocation core. The thick and thin lines repre-sent the LDOS of the atoms near the dislocation core and for aninterior silicon atom far from the dislocation in the beam, respec-tively. 1–10 LDOS of atoms 1–10 see Fig. 9 which surroundthe dislocation core. The influence of the dislocation states de-creases quickly when the atoms are far away from the dislocationcore.

0 5 10 15 20−0.04−0.02

00.020.040.06

x (nm)

Pot

entia

l(V

) (a) EMS

TBA

Multiscale

0 5 10 15 20−4

−2

0

2

4

x (nm)Cha

rge

dens

ity(1

021cm

−3 )

(b)

FIG. 12. Color online Comparison of the potential and chargedensity obtained from the multiscale, EMS, and TB models. Plotsa and b compare the potential and charge distribution computedby the multiscale, EMS, and TB models along the positive x axisx0 for y=0 cross section.

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