Universita degli Studi di MilanoFacolta di Scienze Matematiche, Fisiche e Naturali

Laurea Triennale in Fisica

Tight-Binding Calculation ofSilicon Bands

RELATORE: Dott. Nicola Manini

CORRELATRICE: Dott.ssa Katalin Gaal-Nagy

Eugenio Cinquanta

Matricola n◦ 626351A.A. 2005/2006

Tight-Binding Calculation

of Silicon Bands

Eugenio Cinquanta

Dipartimento di Fisica, Universita di Milano,

Via Celoria 16, 20133 Milano, Italia

Abstract

In this work we implement a tight-binding calculation of the energy bands of silicon.This traditional method is still employed as a useful approximation for the electronicmotion in solids. The tight-binding wavefunctions are taken as linear combinationsof atomic orbitals located at each atom in the crystal, based on phase factors e(ik·R)

(R are the position of such atoms) for coefficients. To address silicon, we implementa fcc lattice structure with 2 atoms per cell and four orbitals per atom, representingthe atomic 3s and 3p. The method involves the diagonalization of an 8 × 8 matrixat each k-point in the Brillouin zone. We reconstruct the bands along a standardpath through special k-points and we integrate the bands over the Brillouin zone toreconstruct the density of states of the system.

Advisor: Dr. Nicola Manini

Co-Advisor: Dr. Katalin Gaal-Nagy

Contents

1 Theory 2

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Electrons in solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Equation of motion for Bloch functions . . . . . . . . . . . . . 3

1.3 The Tight-binding method . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Implementation and Results 7

2.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 The bands of silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1

Chapter 1

Theory

1.1 Introduction

In this work we apply the tight-binding method to the calculation of the bandsof silicon. This method is a simple way to obtain useful information about thebehavior of electron in solids based on few assumptions. We apply this method toSi because of the great quantity of refence data available in the literature, allowingus to compare the results of our parametric calculation with experimental data andab-initio calculations. Clearly the purpose of this simple excercise is a demonstrationof the simplicity of the implementation of a TB code and an superficial analysis ofits points of strength and of its weakness.

1.2 Electrons in solids

We introduce the present work with a brief reminder of the general framework describ-ing the dynamic of electrons in solids. Indicated with Te the kinetic energy of electronand with Vne and Vee the interaction potential of nuclei-electron and electron-electronrespectively, the motion of electrons moving in a solid is governed by:

Hψ(r) = [Te + Vne + Vee]ψe(r) = Eeψe. (1.1)

This equation describes a many-body problem whose solution is immensely difficult.A mean-field approach simplifies the system mapping the N -electron equation to aset of single-electron equations for the motion of an electron in the mean field of thenuclei and the other N − 1 electrons. We represent the electron-electron and theelectron-nuclei interactions with an effective one-electron potential V(r) which hasthe same periodicity of the lattice:

V (r) = V (r + R) (1.2)

2

for all Bravais lattice vectors R. We assume therefore perfect lattice periodicity. Inthis way, we have to find solutions of a Schrodinger equation of the form:

Hψ =

[

−h2

2m∇2 + V (r)

]

ψ = Eψ. (1.3)

According to Bloch’s theorem it is possible to write the eigenstates of Eq.(1.3) asBloch functions:

ψjk(r) = eik·rϕ

jk(r) (1.4)

ϕjk(r + R) = ϕ

jk(r) (1.5)

where ϕnk has the same periodicity of the Bravais lattice. Clearly, we have:

ψjk(r + R) = eik·Rψ

jk(r) (1.6)

which indicates that the eigenstates of Eq. (1.3) are a irreducible representations ofthe lattice symmetry group. Furthermore, according to Eq. (1.4) and (1.6), all elec-tronic eigenfunctions have a nontrivial spatial dependence only within one primitiveunit cell. In any other cell displaced by TR, where TR is a translational operator, the

wavefunction is equal to that in the original cell multiplied by a phase factor eik·R.

1.2.1 Equation of motion for Bloch functions

In a crystalline solid a single-electron state is labeled by an irreducible representationof the Bravais lattice group. The vector k can be chosen within a primitive zone of thereciprocal lattice, e.g. the first Brillouin zone. This wavevector contains informationon how the wavefunction changes under the action of the operator of translation froma lattice cell to the next one.

Starting from the kinetic energy:

∇2eik·rϕk(r) = (1.7)

eik·r∇2ϕk(r) + 2ikeik·r∇ϕk(r) − |k|2eik·rϕk(r) = eik·r(∇ + ik)2ϕk(r)

by substituting this result into Eq. (1.3), and dividing by the phase factor eik·r, weobtain the equation for ϕk(r):

[

−h2

2m(∇ + ik)2 + V (r)

]

ϕkj(r) = Ekjϕkj(r) (1.8)

This equation governs the dynamics of an electron characterized by wave vector k.Indeed, following Bloch’s theorem, it is not necessary to solve Eq. (1.8) in the wholecrystal but it is sufficient to solve it in a single cell of the direct lattice. However, onceϕkj is determined in a cell, then the wavefunction ψj(r) is completely determined inthe whole crystal through Eq. (1.4).

We observe that Eq. (1.8) depends analytically on k, thus, for a given quantumnumber j, a solution depends on k as a continuous function. These functions areprecisely the energy bands Ekj of the solid represented by the periodic potentialV (r).

3

1.3 The Tight-binding method

The tight-binding (TB) method consists in expanding the crystal single-electron statein linear combinations of atomic orbitals substantially localized at the various atomicpositions of the crystal. Thus we can decompose the Hamiltonian (1.3) in two terms

H = Hat + ∆V (r) (1.9)

where Hat includes the kinetic energy and the effective potential of the ion located(atomic Hamiltonian), e.g at R = 0, and ∆V (r) contains the potential generated byall others ions in the lattice except the one already counted.Assuming that the atomic problem Hatϕ = Eatϕ is solved, we can build a Bloch’ssum as in Section 1.2, by combining atomic states localized around each atom in thelattice:

ψnk(r) = Z∑

R

eik·Rϕn(r − R) (1.10)

where Z is a normalization constant and ϕn are a basis of atomic eigenfunctions. Onethen expands a generic wavefunction as

ψk(r) =∑

n

cn(k)ψnk((r). (1.11)

In order for ψk to be a crystal eigenstate, it must solve the equation[

Hat + ∆V (r)]

ψk(r) = Ekψk. (1.12)

We can now multiply this expression on the left by ψ∗mk

and integrate it over all r

space. This integration maps eq. (1.12) to a matrix equation:∑

m

Amn(k)cn(k) = Ek

∑

m

Bmncn(k), (1.13)

whereAmn(k) = N

∑

R

eik·R ×∫

ϕ∗m(r)Hϕn(r −R)d3r, (1.14)

and the overlap term on the right-hand side is

Bmn(k) = N∑

R

eik·R ×∫

ϕ∗m(r)ϕn(r− R)d3r. (1.15)

Using the decomposed Hamiltonian (1.9), for the energy term (1.14) we have:

Amn(k) = N∑

R

eik·R × [∫

ϕ∗m(r)∆V (r)ϕn(r −R)d3r (1.16)

+Eatm

∫

ϕ∗m(r)ϕn(r − R)d3r] =

= N∑

R

eik·R × [∫

ϕ∗m(r)∆V (r)ϕn(r −R)d3r + Eat

mBmn(k),

where we used the fact that ϕ∗m(r) are left eigenstates of Hat. Thus, the TB matrix

eigenvalue equation is written in the form∑

n

Mmn(k)cn(k) = (E − Eatm)

∑

n

Bmn(k)cn(k) (1.17)

4

withMmn(k) =

∑

R

eik·R ×∫

ϕ∗m(r)∆V (r)ϕn(r − R)d3r (1.18)

The resulting TB energy matrix is:

Eat1 +M11(k) M12(k) M13(k) . . .M21(k) Eat

2 +M22(k) M13(k) . . .M31(k) M32(k) Eat

3 +M33(k) . . ....

......

. . .

whose generalized eigenvalues in the sense of Eq. (1.17) provide the energy bands weare looking for.

The TB method is in principle exact and displays many attractive qualitativefeatures, since it gives solutions showing all the correct symmetry of the energybands at an arbitrary point in the Brillouin zone. It is rather easy to get solutionsfor energy bands where other methods become too difficult to carry out. On theother hand, we pay this great generality with difficult in the need of many integralsthat become rather complicate possibly expensive. The first difficult, is due to theBloch’s sum (1.10). Indeed, the atomic orbitals ϕn(r) are orthogonal to each otherbut clearly Bloch’s sum generally are not. So, the matrix Bmn(k), which contains theoverlap elements between states located at different atoms, has nontrivial off-diagonalelements which make the generalized eigenvalue problem (1.17) slightly involved. Asecond, more serious, difficulty arises because the TB method is exact only as long aswe include all the infinite set of atomic levels. Clearly this will bring us to a infinitecomputational problem which is not possible to solve. A truncation is thereforeneeded in practice.

In our work we make a few simplifications in order to overcome these problems.We notice that, in case of extremely localized atomic orbitals, all contributions to theoverlap matrix of integrals like:

∫

ϕ∗m(r)ϕn(r − R)d3r (1.19)

become negligible when we are integrating over two different atoms (R 6= 0). Thisjustifies the choice of an orthogonal basis, i.e. Bmn = δmn. Another method whichproduces an unit overlap matrix consist in using Wannier states Ref. [4] which areorthogonal by construction, but contain contributions of neighbors atoms. Further-more we note that hopping terms like those of Eq. (1.18) decrease as distance betweenatoms increases, but never vanish completely. In an approximate TB calculation wedecide to truncate the sum of Eq. (1.18) only to first or second neighbors neglectingthe contributions of higher-order neighbors. Finally, it is not possible to include inour calculation the contributions of all unoccupied atomic levels sitting above the lastoccupied state. In order to solve this problem we have considered three possibilities:the first one is including a finite number, say two or three, of unoccupied state inthe TB matrix and carry out a very precise highest band calculation. A simpler

5

phenomenological approximation consists in considering a non-physical state, say s∗,which contains effective contributions from some unoccupied levels. Alternativelyone can neglect completely all unoccupied levels and consider only the occupied ones:clearly this method is the least accurate for the top (conduction) bands. We will tochoose this third possibility in order to simplify our implementation.

Summarizing all our approximations we obtain a new simplified TB matrix eigen-value problem:

∑

n

Amn(k)cn(k) = Ecm(k) (1.20)

where:Amn(k) = Eat

mδmn +∑

R

eik·R ×∫

ϕ∗m(r)∆V (r)ϕn(r −R)d3r (1.21)

where the sum over R is extended only over nearest neighbors or, possibly, first andsecond neighbors. This new eigenvalue problem consists in a diagonalization of aN ×N A(k) matrix, where N is the product of the number of atoms and the numberof orbitals involved, which is finite, usually small number.

Now we focus our attention on the matrix elements (1.18) and on how to writethem in a simple way using some parameters in agreement with the previous assump-tions. In order to have further simplifications we consider the two-center approxima-

tion, which consists in neglecting integrals involving three or more different centers.The resulting hopping matrix contains a potential which depends on the positions as|r−R|. For example, if we consider s and p atomic levels we have four wavefunctionswhich describe orbital s and orbitals px, py, pz respectively. The resulting matrixelements are indicated by the following labels Ref. [4]:

∫

ϕ∗s(r)V (r −R)ϕs(r− R)dr V (ssσ)

∫

ϕ∗s(r)V (r −R)ϕx(r −R)dr lxV (spσ)

∫

ϕ∗x(r)V (r − R)ϕx(r − R)dr l2xV (ppσ + (1 − l2x)V (ppπ))

∫

ϕ∗x(r)V (r − R)ϕy(r −R)dr lxly[V (ppσ) − V (ppπ)]

∫

ϕ∗x(r)V (r − R)ϕz(r − R)dr lxlz[V (ppσ) − V (ppπ)]

We note that the independent integrals are four and all the other integral areobtained by cyclic permutations of indices x, y, z. It follows that hopping matrixelements can be expressed by few independent parameters which can be numericallyevaluated. These parameters have been labelled V (ssσ), V (spσ), V (ppπ) and theprecise value of each integral is obtained by multiplying these terms by suitabledirector cosines li = Ri

|R| (i = z, y, z) of the two-center vector R.

6

Chapter 2

Implementation and Results

In the present chapter we sketch our implementation of the TB method appliedon a finite silicon crystal, followed by an application to the calculation of its bandstructure. In the end we present our computed density of band states.

2.1 Implementation

In our implementation of the TB method for the silicon crystal we include only partlyoccupied atomic levels 3s, 3p and we restrict to first and second nearest neighbors.We implement a calculation of the simplified eigenvalue problem (1.20). Si realizes adiamond lattice illustrated in Fig. 2.1, this structure can be described as two inter-penetrating fcc lattices displaced by (0, 0, 0) 1

4(1, 1, 1) along the body diagonal. This

leads to a N = 8 secular problem. Therefore we must modify the expression for theTB matrix Mmn(k) adding a sum over the atoms in the fcc cell

Mmn,d(k) =∑

R,d

eik·R ×∫

ϕ∗m(r)∆V (r− R − d)ϕn(r − R − d)d3r. (2.1)

Figure 2.2 illustrates the primitive vectors of the fcc lattice, which form a basisof generators of the lattice. Each point in the lattice is given by the relation:

R = n1 · a1 + n2 · a2 + n3 · a3 (2.2)

where ni are the integer coefficients of the linear combination and

a1 = a(0,1

2,1

2)

a2 = a(1

2, 0,

1

2)

a3 = a(1

2,1

2, 0)

7

Figure 2.1: Diamond lattice structure. We fix an atom, labeled d1 and indicatenearest neighbors (pv) and some second neighbors (sv).

Figure 2.2: Primitive vectors of the fcc lattice

8

Figure 2.3: The reciprocal lattice of the fcc lattice defined by a1, a2, a3 is a bcc latticegenerated by b1 = 2π

a3 a2 ×a3, b2 = 2πa3 a3 ×a1, b3 = 2π

a3 a1 ×a2. The drawn polyhedronrepresent the first Brillouin zone with the conventional notation for high symmetrypoints.

with the lattice parameter a of the cubic cell. All points of the type (2.2) form aninfinite array named fcc lattice.

The primitive cell is define as the minimal volume which fills up the wholespace without overlapping by applying lattice translations R. This elementary par-allelepiped of volume (a1 × a2) · a3 contains all the relevant informations about thewhole lattice. A useful primitive cell is the Wigner-Seitz cell which is defined as theset of all points closer to a given lattice point R than to any other lattice point R

′

. Inour TB simulation we chose the k points inside the Wigner-Seitz cell of the reciprocallattice, called the first Brillouin zone (fig. 2.3).

The two basis atoms of the fcc lattice for the diamond structure are located atthe position d1 = (0, 0, 0) and d2 = a(1

4, 1

4, 1

4). As is illustrated in fig. 2.1, the distance

between first neighbors atoms is√

34a and between the second neighbors is 1√

2a. Thus,

as we truncate as cut-off distance our calculation at first or second neighbors, we mustchose the two previous values. This means that any atom at distances greater than√

34a or 1√

2a, respectively, is excluded from the sum 2.1 for the matrix elements.

The core of the calculation involves a double loop over the atoms in the cell,and over neighboring cells to the reference cell taken at R = 0. A check of thedistances distinguishes between diagonal terms (where this distance vanishes), off-

9

hopping matrix parameters

α(Ry) β(Ry/Bohr) γ(Ry/Bohr2) δ(Bohr−1

2 )Hssσ 219.560813651 -16.2132459618 -15.5048968097 1.26439940008Hspσ 10.127687621 -4.4036811240 0.2266767834 0.99267109054Hppσ -22.959028107 1.7207707741 1.4191307713 1.03136916513Hppπ 10.265449263 4.6718241428 -2.2161562721 1.11134828469

Table 2.1: parameters for the calculation of hopping matrix elements through Eq.(2.3)

hopping integral (eV)1st neighbors 2nd neighbors

Hssσ -1.77416 -0.0891011Hspσ -0.84730 -0.105639Hppσ 1.53114 0.388664Hppπ -0.716431 -0.126518

diagonal matrix elements (eV)Es 1Ep -3.40277

Table 2.2: Diagonal and hopping elements of the TB matrix eq. (2.1)

diagonal contributions to the sums in (2.2), and neglected terms because the distanceexceeds the imposed cutoff value. A double loop over the four orbitals completes thestructure of the core and contains a cumulation of the contributes to the sum definingthe matrix element Mmn(k). After this matrix is complete by filled, its eigenvaluesare computed by means of a routine taken from Ref. [9]. The core of the code isthen surrounded by an external loop over the k points. This is implementeted intwo alternative fashions. To compute the bands along high symmetry directions, apath composed by five straight segments through the special points L = (1

2, 1

2, 1

2),

Γ = (0, 0, 0), X = (1, 0, 0), W = (1, 12, 0), K = (3

4, 3

4, 0) and back to Γ (as shown

in fig. 2.3) is generated. Alternatively, when we address the density of state, weimplement a fine grid over one quarter of the Brillouin zone (taking advantage ofsymmetry), and cumulate a histogram of occurencies of the band energies.

To determine the Hamiltonian values, namely hopping matrix elements betweenorbitals of different atoms and diagonal energies, we choose a numerical parametriza-tion based on fits of ab-initio calculations. The relations which give us hoppingintegrals are:

Vll′n = (αll

′n + βll

′np+ γll

′np

2)e−δ2

ll′n

p(2.3)

where p is the distance of the other atom. In table 2.1 we report all the parameterswe use in the parametrization (2.3).

In table 2.2 we report the resulting off-diagonal (hopping) matrix elements, andthe diagonal energy we employed.

10

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

ener

gy (

eV)

L Γ X W K Γ

Figure 2.4: Silicon bands including 1st neighbors only in the sum (2.1)

2.2 The bands of silicon

In this section we present our results for bands of silicon.

We explore the Brillouin zone along L, Γ, X, W, K and back to Γ. The resultingelectronic bandstructure is illustrated in Fig. 2.4 and 2.5. In Fig. 2.6 and 2.7 weshow the silicon bands reported in Ref. [6] and Ref. [10]. for comparison. InFig. 2.4 is represented a TB calculation including only 1st neighbors, whereas inFig. 2.5 are illustreted bands calculated including in the implementation also secondneighbors. We can see how code containing only nearest neighbors gives poor bands,symmetries are not respected, the shape of the bands is substantially different frombands founded with an empirical nonlocal pseudo-potential calculation Fig. 2.6. InFig. 2.7 are plotted bands calculated with an ab-initio method. If we include alsosecond neighbors our calculation becomes more precise and the resulting bands arepretty good.

By confronting Fig. 2.5 with results ilustrated in Fig. 2.6 and Fig. 2.7 we can saythat agreement is good for lower (valence) bands but quite poor for high (conduction)bands. Indeed, it is evident that for the first three bands the shape, symmetries andenergy values are correct and thus our implementation seems to be acceptable. Onthe contrary, for higher bands we note that dependecies are profondly different. Thiswas expected because in our code we have not inserted any unoccupied levels above3s and 3p. This produces deformations in top band’s shapes. Another importantphysical feature of a bandstructure is the gap value. Silicon is a semiconductor,namely an isolant with a small energy gap (energy distance between the highest

11

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

ener

gy (

eV)

L Γ X W K Γ

Figure 2.5: silicon bands cutting of on 2nd neighbors

Figure 2.6: Silicon bands from [6]. Dashed line and solid line illustrates bands calcu-late with local pseudo- potential and nonlocal pseudo-ptential respectively

12

Figure 2.7: silicon bands calculated with an ab-initio method

occupied and lowest unoccupied band). With our calculation we obtain an indirectgap semiconductor, namely a semiconductor in which minimum of conduction bandand maximum of valence band occur at different values for k (see Fig. 2.5). Thisresult is in agreement with the well known silicon structure. However, with theparameters used here, the resulting gap is about 0.22eV and thus it is substantiallysmaller than the experimentally measured 1.14eV. This difference is to be attributedto the neglect of overlap corrections Bmn. In fact, by tuning the atomic s and penergies we could tune the gap value. Better agreement to more sofisticated ab-initio

calculations, could probably be obtained with a more enlarged TB basis and includingthe orthogonality corrections.

13

-12 -10 -8 -6 -4 -2 0 2 4 6 8energy (eV)

0

0.1

0.2

0.3

dens

ity o

f st

ates

(ar

bitr

ary

units

)

Figure 2.8: Normalized density of states of the bands computed including first andsecond neighbors with 106 k points in the Brillouin zone.

2.3 Density of states

Figure 2.8 represents histogram of a 106 calculations of the bands for different k

points, equally spread in a volume of 18

of the Brillouin zone. Figure 2.9 representsan analogous calculation of the density of states based on the method of Ref. [6],compared to a measured density of states (restricted to the valence band). ComparingFig. 2.8 with Fig. 2.9 and with Fig. 2.10, we find a fair qualitative agreement withresult found with an empiricall nonlocal pseudo-potential method and with resultsobtained with ab-initio calculation Ref. [10]. The band gap of our calculation is avisible feature.

The 3-peaks stimature of valence density of states is reported qualitatively, eventhough with several difference to experiment, especially, the major dip near −4eV .

14

Figure 2.9: Calculated electronic densities of states compared to experimental for Si[7]. Theoretical results are from [6], experimental are from [7]. Local and nonlocalpseudo-potential methods are illustrated with dashed and solid lines respectively

-12 -10 -8 -6 -4 -2 0 2 4 6 8energy (eV)

0

0.2

0.4

0.6

0.8

1

dens

ity o

f st

ates

(ar

bitr

ary

units

)

Figure 2.10: Density of states calculated with an ab-initio method, Ref. [10].

15

2.4 Conclusion

In summary, we have verified the simplicity of implementation of the TB method,thanks to several drastic simplifying approximations. With a simple orthogonal TBand few unoptimized phenomenological parameters we obtain decent bands with sym-metry properties. In particular, we reconstruct the semiconducting character of Sil-icon by observing an indirect band gap between a top of the valence band at Γ anda bottom of the conduction band near X. Poor results for the gap energy are re-conducible to unoptimized parameters and to the orthogonal approximation for Bmatrix. On the other hand, top conduction bands are inaccurate due to exclusion of4s, 3d and higher orbitals. The density of states confirms the outcome of the bandcalculation which only considers special directions. TB can be a quick and powerfulmethod to address simple properties of complex system with minimal effort. It canbe, and indeed is, used as a first approach to gather basic insight in new systembefore more quantitative investigations ca be carried out.

16

Bibliography

[1] N.W Aschcroft and N. D. Mermin, Solid State Physics, (Cornell University, 1976).

[2] Nicola Manini, Lecture Notes, Dipartimento diFisica, (Universita degli Studi di Milano, 2004),http://www.mi.infm.it/manini/dida/note_struttura1.v1.07

[3] G. Grosso and G. Pastori Parravicini, Solid State Physics, (Academic Press, Lon-don 2000).

[4] J.C. Slater and G.F. Koster, Phys. Rev. B 94, 6 (1954).

[5] N. Bernstein, M. J. Mehl, D. A. Papaconstantopoulos, N. I. Papanicolaou, MartinZ. Bazant and Efthimios Kaxiras, Phys. Rev. B 62, 4477 (2000).

[6] J.R Chelikowsky, M.L. Cohen, Phys. Rev. B 14, 556 (1976).

[7] R. Pollack, L. Ley, S. Kowalcyzk, D.A. Shirley, J. Joannopoulos, D.J. Chadi andM.L. Cohen, Phys. Rev. Lett. 29, 1103 (1973).

[8] W.D. Grobman and D.E. Eastman, Phys. Rev. Lett. 29, 1508 (1972).

[9] W.H Press, S.A. Teukolsky, and W.T. Wetterling and B.P. Flannery, Numerical

Recipies in c++, (Cambridge University Press, 2002).

[10] Katalin Gaal-Nagy, private communication (2005).

17