Tight-binding formulation of electronic band structure of 2D hexagonal materials

CHAPTER THREE

Tight-binding formulation of electronic band structure of 2D hexagonal materials Clifford M. Krowne Quantum and Classical Field Theories Electromagnetics Technology Branch, Electronics Science and Technology Division, Naval Research Laboratory, Washington, DC, United States e-mail address: [email protected]

Contents 3.1. π and σ orbitals in graphene 3.2. Relationship between atomic orbitals and crystalline wavefunction 3.3. Assessment of overlap between atomic orbitals 3.4. Reduction of the spatially varying Schrödinger equation into a solvable system References Further reading

23 24 30 37 43 43

3.1. π and σ orbitals in graphene Our interest is in the low energy electronic states, not too far from the Fermi energy. This is consistent with placing attention on the π bonds in a material covalently arranged with stronger σ bonds, as in graphene. Use of the tight-binding method is aligned with the desire to obtain analytical solutions for the electronic band structure which can be applied to both transport problems in electronic materials and devices and emission devices. Original work for graphene’s honeycomb lattice was done by Wallace (1947), long ago, but the sophisticated uses for this material and other 2D materials has had to await the nanoscience era of today. Graphene σ band structure is a result of the hybridization of the 2s and 2p orbitals, and so the hybridization is denoted sp2 , because two p orbitals are involved. Because graphene is planar, one can view this involving the 2px and 2py orbitals hybridizing with the 2s orbital, or in quantum  mechanical   terms,  the |2s orbital wavefunction hybridizing with the 2px and 2py orbital wavefunctions. Hybridization forms three resultant orbital wavefunctions, Advances in Imaging and Electron Physics, Volume 210 ISSN 1076-5670 https://doi.org/10.1016/bs.aiep.2019.01.003

Copyright © 2019 Elsevier Inc. All rights reserved.

23

24

C.M. Krowne

given as  2 sp = √1 |2s − 1



2   2py 3

3   √  2   1  1 2 3 sp = √ |2s + 2px + 2py 2 3 2 2 3  2 sp = − √1 |2s + 3

3

  √

2 3   1   − 2px + 2py 3 2 2 

(3.1a) (3.1b) 

(3.1c)



The last 2p orbital 2pz , with orbital 2pz wavefunction, remains unhybridized, providing the extra electron per C atom for π bonding and electronic motion. This last orbital is perpendicular to the xy-plane containing the C atom graphene crystal structure, and is associated with the π bonds, our focus here. The main axis of each of the three hybridized   orbital wavefunctions are rotated by 120° as one progresses through sp2i , i = 1, 2, 3. Tight-binding treatment can be found elsewhere (Saito, Dresselhaus, & Dresselhaus, 1998), especially for handling the σ orbitals. And a good discussion is also found in Blatt (1968), as to the basic ideas behind it, specifically in Chapter 4: Electrons in a Periodic Lattice, Section 4.6 – Tight-Binding Approximation. This method is also referred to as the linear combination of atomic orbitals (LCAO), is most appropriate whenever the electrons are well localized about their attractive ion cores, and the overlap of the electron wavefunction centered at one lattice point into the adjacent unit cells is relatively small. Also the method is briefly mentioned by Kane (1982) in his examination of band structure. So one should be aware that using this method, although very attractive from the analytical standpoint of view, has its limitations, since the σ bonding electrons will by necessity have much more limited wavefunction extents into other adjacent unit cells, than the more mobile π electrons. So with this understanding in mind, we continue the investigation.

3.2. Relationship between atomic orbitals and crystalline wavefunction Electronic orbitals φa describe each of the atoms in the 2D lattice. They must be superpositioned to form the total electronic wavefunction, utilizing the orbitals specific to the type of tight-binding approach in-

Tight-binding formulation of electronic band structure

25

tended. Here the tight-binding approach will utilize the 2pz orbital   atomic wave function, for the π bonds. Since the 2pz wavefunction 2pz must be identical for all similar A atoms, and identical for all B atoms, what must differentiate the actual atomic wavefunction in the lattice for each atom must be a positional factor. This positional factor arises from the Bloch Theorem, which states that the most general solution of the one-electron Schrödinger equation for an electron in a crystal is of the form ψk (r) = eik·r uk (r)

(3.2)

with uk (r) a function of r having the same spatial periodicity as the crystal lattice, uk (r + Ri ) = uk (r)

(3.3)

with Ri being the translation vector of the lattice, constructed from a combination of the real space Bravais unit cell lattice vectors Ri = na1 + ma2 ;

n, m = integers

(3.4)

The translation vectors, if we are in the A sublattice, will take one from any particular A reference origin, to any other atomic orbital A atom site. Likewise, the translation vectors, if we are in the B sublattice, will take one from any particular B reference origin, to any other atomic orbital B atom site. However, because the Bravais unit cell lattice vectors a1 and a2 span a unit cell with two atoms per cell, say an A atom at its origin, and an internal B atom, there is no translational symmetry from an A atom to a B atom. This is because the closest distance between an A atom and a B atom is less than either Bravais unit cell lattice vector magnitude, being a fractional value and precluding use of Ri . Proof of the Bloch Theorem follows below. For a discussion of this also refer to Blatt (1968), Chapter 4: Electrons in a Periodic Lattice, Section 4.3 – Bloch Theorem. ψk (r) is a solution of the Schrödinger equation, H (r)ψk (r) = εk ψk (r)

(3.5)

with Hamiltonian H (r), H (r) = −

h¯ 2 2 ∇ + V (r) 2me

(3.6)

26

C.M. Krowne

V (r) is the total scalar potential energy at a point r that the electron experiences in the periodic lattice consisting of the atoms, each atom made up of its electrons and its positive atomic core. Therefore, V (r) must also be periodic. Since the mobile electron from each atom is considered completely free, our 2pz electron, the V (r) consists therefore of effective positive ion cores ZCeff = ZC − 11 = 12 − 11 = +1 charge. Consequently, V (r + Ri ) = V (r)

(3.7)

Define the translation operator Ti such that when it acts on the scalar function F (r), Ti F (r) = F (r + Ri )

(3.8)

Now let Ti act on the eigenvalue equation (3.5), Ti {H ψk (r)} = Ti {εk ψk (r)}

(3.9)

H (r + Ri )ψk (r + Ri ) = εk Ti {ψk (r)}

(3.10)

gives

However, translated Hamiltonian will be, using (3.7) and ∇r+Ri = ∇r , H (r + Ri ) = −

2 h¯ 2  h¯ 2 2 ∇r+Ri V (r + Ri ) = − ∇ + V (r) = H (r) (3.11) 2me 2me

meaning the Hamiltonian also has the periodicity of the lattice. Substituting (3.11) into (3.10) yields H (r)Ti {ψk (r)} = εk Ti {ψk (r)}

(3.12)

Therefore, εk is both an eigenfunction of the Hamiltonian and of the translation operator! Eq. (3.12) resulted from the commutation of the H and Ti operators, going from the left hand sides of (3.9) to (3.12): Ti {H ψk (r)} = H (r)Ti {ψk (r)}

⇒

Ti H ψk (r) = H (r)Ti ψk (r)

(3.13)

Operator commutation implies simultaneous eigenfunctions of both operators. Specify the eigenfunction of the Ti operator by itself with the statement Ti ψk (r) = μi ψk (r)

(3.14)

27

Tight-binding formulation of electronic band structure

One can choose μi to be a complex scalar number of magnitude unity, since a non-unity magnitude can be reabsorbed into the wavefunction. Therefore, set μi = eik·Ri

(3.15)

where k is in general a complex crystal momentum vector such that momentum p is hk. ¯ For cases when the electron experiences no losses in its movement, k will be a real vector, otherwise it could have imaginary components corresponding to wavefunction decay or growth. Apply two successive translation operators to the wavefunction, first Tj , then Ti , Ti Tj ψk (r) = Ti μj ψk (r) = μj Ti ψk (r) = μj μi ψk (r)

(3.16)

Now reversing the order of the translation operators, first Ti , then Tj , Tj Ti ψk (r) = Tj μi ψk (r) = μi Tj ψk (r) = μi μj ψk (r)

(3.17)

Since scalars commute, left-hand-sides of (3.16) and (3.17) are equal, Ti Tj ψk (r) = μj μi ψk (r) = μi μj ψk (r) = Tj Ti ψk (r)

(3.18)

making the translation operators commute: T i Tj = Tj Ti

(3.19)

Utilizing (3.15) in (3.18), 



Ti Tj ψk (r) = Tj Ti ψk (r) = eik· Ri +Rj ψk (r)

(3.20)

So it is observed that successive T operators result in a translation by the sum of the translation vectors. Now define a reduced wavefunction uk (r) which has the exponential factor knocked out, uk (r) = e−ik·r ψk (r)

(3.21)

Hit both sides of this uk (r) equation with the translation operator Tj ,



Ti uk (r) = Ti e−ik·r ψk (r) = e−ik·(r+Ri ) ψk (r + Ri ) = e−ik·(r+Ri ) Ti ψk (r)

(3.22)

28

C.M. Krowne

and using (3.15) and (3.14), Ti uk (r) = e−ik·(r+Ri ) Ti ψk (r) = e−ik·(r+Ri ) eik·Ri ψk (r) = e−ik·r ψk (r) = uk (r)

(3.23)

or Ti uk (r) = uk (r)

(3.24)

By (3.8), this equation also reads uk (r + Ri ) = uk (r)

(3.25)

showing that the function uk (r) is periodic in the lattice. By (3.21), multiplying both sides by eik·r , we finally obtain ψk (r) = eik·r uk (r)

(3.26)

which with (3.25), constitutes the Bloch Theorem given above in (3.2) and (3.3). Because uk (r) has the periodicity of the lattice, its value at any lattice point is the same at any other lattice point. Therefore, it must represent the atomic orbital wavefunction φ a (r) of the electron in the particular orbital of interest, namely our 2pz π type orbital. r is the distance of the electron from the central reference point, rc = 0. Thus we set φa (r) = φa (r; rc = 0) = uk (r)

(3.27)

Shifting the orbital to any other reference lattice point, say Ri , requires looking at φa (r − Ri ), since the electron will be orbiting at ri = r − Ri about the new location reference point Ri and from the central reference point will be at r = ric = Ri + ri . So by the translation operator in (3.14) and (3.15), the orbital wavefunction at any location Ri must be Ti ψk (r) = μi ψk (r);

μi = eik·Ri ;

ψk (r) → φa (ri )

(3.28)

where attention is applied to evaluating at the relative vector location desired, namely ri . From (3.28) is found (see Fig. 3.1 for orbital shifting in 2D direct space), Ti φa (ri ) = eik·Ri φa (r − Ri )

(3.29a)

Tight-binding formulation of electronic band structure

29

Figure 3.1 Orbital shifting effects in 2D direct space. Shown are the reference location, lattice location, and atomic orbital wavefunctions in the region about the reference l l atom φasub,c (rc ) or about the lattice atom φasub (ri ). Shifting is accomplished by the lattice vector R i .

Superposition of all the orbital wavefunctions at all unit cell Bravais locations for a particular atom type, be it type A or type B, by (3.29), yields φka (r) =



eik·Ri φa (r − Ri )

(3.29b)

Ri

Formula (3.29b) is a generic relationship for a particular type of atom. It must be made specific for the two types of atoms in our hexagonal honeycomb lattice, types of atoms A and B. For atoms of type A,   ψkA (r) = φka (r)A = eik·Ri φaA (r − Ri )

(3.30a)

Ri

and for type B,   ψkB (r) = φka (r)B = eik·Ri φaB (r − Ri )

(3.30b)

Ri

Because there are these two atom types, corresponding to the two atom types within the unit cell Bravais lattice, the total electronic wavefunction must be the superposition of wavefunctions ψkA (r) and ψkB (r). That is, ψk (r) =



alksub ψklsub (r) = ak ψkA (r) + bk ψkB (r)

(3.31)

sub

Index lsub is the sublattice atom index type, and can equal generally any number of atom types contained within the Bravais unit cell lattice. If we consider the first atom type, 1 or A, acting as the central reference point,

30

C.M. Krowne

rlsub =1 = rlsub =A = 0, then the other atom type locations within the Bravais unit cell lattice will be rlsub =1 . For the case of just one other atom type B, r = ric = Ri + rlsub =B + ri , making (3.30) become   

 ψkB (r) = φka (r)B = eik·Ri φaB r − Ri + rlsub =B

(3.32)

Ri

Generally (3.32) becomes l

ψksub (r) =







eik·Ri φalsub r − Ri + rlsub



(3.33)

Ri

making (3.31) is generalized to ψk (r) =



l

ψksub (r) =

sub

 lsub





eik·Ri φalsub r − Ri + rlsub



(3.34)

Ri

3.3. Assessment of overlap between atomic orbitals Carbon has the electronic configuration [He] 2s2 2p2 = 1s2 2s2 2p2 . To complete the p orbital, 4 more electrons are required. Three are involved in the formation of strong covalent σ bonds, and the remaining electron for the π bonds. From Cohen-Tannoudji, Diu, and Laloe (1977), see Chapter VII: Particle in a Central Potential; The Hydrogen Atom. Solution of its energy eigenvalue equation, 



h¯ 2 e2 − ∇2 − ϕ(r) = Eϕ(r) 2μ r

(3.35)

gives ϕ(r) as 1 r

ϕklm (r) = ukl (r )Ylm (θ, φ)

(3.36)

Radial wavefunction ukl (r) is determined by its radial equation, 



h¯ 2 d2 l(l + 1)h¯ 2 e2 − + − ukl (r ) = Ekl ukl (r ) 2μ dr 2 2μr 2 r

(3.37)

with boundary condition, ukl (r = 0) = 0

(3.38)

31

Tight-binding formulation of electronic band structure

Figure 3.2 Radial variation Rn,l (r) of the atomic wavefunction, scaled by a power of the Bohr radius a0 , versus the normalized radius r/a0 . Results are shown for two hydrogen shells (n = 1, 2) and two angular momentum cases (l = 0, 1) corresponding to the s orbital and the p orbital.

All radial wavefunctions go to zero as r → 0. However, the total radial wavefunction Rnl (r ), does not go to zero for l = 0 (see Fig. 3.2 for a plot of a few cases), and is given by 1 Rnl (r ) = ukl (r ); r

k=n−l

(3.39)

Our case would use Z = 6 for carbon, and the spectroscopic p notation corresponds to l = 1 for the 2pz orbital, of the n = 2 shell. Rnl (r )|n=2,l=1 = 0 as r → 0. Angular variation is given by the spherical harmonics Ylm (θ, φ) 

Ylm (θ, φ) = (−1)m

(2l + 1)(l − m)! m Pl (cos θ )eimφ 4π(l + m)!

(3.40)

where Plm (y) are the associated Legendre polynomials. Selecting m = 0 for the azimuthal quantum number, for the moment, with l = 1, requires for the Legendre polynomial Plm==10 (cos θ ) = cos(θ ), making Ylm (θ, φ) reduce to 

Y10 (θ, φ) = Ylm==1 0 (θ, φ) =

1 3 cos θ 2 π

(3.41)

To make simple plots, it is convenient to look at the squared magnitude of (3.40),   m

Y (θ, φ)2 = (2l + 1) (l − m)! P m (cos θ ) 2 l l 4π (l + m)!

(3.42)

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C.M. Krowne

Figure 3.3 Atomic orbital angular Ylm (θ, φ) effects in direct space. Plotted is its magnitude squared |Ylm (θ, φ)|2 which relates to probability of finding the electron spatially, but only in terms of its angular behavior. The orbitals are rotationally symmetric about the z-axis for those particular ones shown, which are the angular momentum quantum number l = 0, 1, 2 (the s, p, and d orbitals) with azimuthal quantum number set to m = 0.

as shown for a few m = 0 (l = 0, 1, 2) cases in Fig. 3.3. For our simple case m = 0, l = 1,   0 Y (θ, φ)2 = 3 cos2 θ 1 4π

(3.43)

This yields two distorted ellipsoidal lobes of revolution about the z-axis, pointing above and below the xy-plane of the graphene (see Fig. 3.3). Total radial wavefunction Rnl (r ) for our simple case m = 0, l = 1, can be found noting that Rnl (r ) = Rkl (r )|k=n−l

(3.44)

making (plotted in Fig. 3.2) 1 r −r /(2a0 ) Rn=2,l=1 (r ) = Rkl (r )|k=2−l,l=1 = Rk=1,l=1 (r ) = (2a0 )−3/2 √ e (3.45) 3 a0 Here a0 is the Bohr radius, 0.52 Å. Combining (3.36) and (3.39), ϕnlm (r) = ϕklm (r) = Rkl (r )Ylm (θ, φ)

(3.46)

which for our simple case m = 0, l = 1, ϕn=k+l=2,l=1,m=0 (r) = ϕk=1,l=1,m=0 (r) = Rk=1,l=1 (r )Ylm==1 0 (θ, φ)

(3.47)

33

Tight-binding formulation of electronic band structure

Figure 3.4 2p orbitals Rn,l (r)Ylm (θ, φ) in direct space, the 2px , 2py , and 2pz orbitals. They are formed from quantum numbers l = 1, m = ±1 (the 2px and 2py ), and l = 1, m = 0 (the 2pz ). Actual orbitals are solid cylinders created by rotations about, respectively, the axes x, y, z for orbitals 2px , 2py , 2pz . Orbital lobe polarity is shown by the “+” or “−” signs with the lobes of the orbitals.

and inserting solutions (3.41) and (3.45), ϕn=k+l=2,l=1,m=0 (r) =

1 r −r /(2a0 ) 1  e cos θ 4 2π (a0 )3 a0

(3.48)

This wavefunction has the two lobed appearance found in (3.43), but now we must distinguish its sign by noting that for z < 0, or π < θ < 2π , below the xy-plane, it takes on negative values. The n = 2 (k = 1) three 2p orbitals are shown in Fig. 3.4. It is the exponential in (3.48) which controls the decay of one atom location 2pz orbital wavefunction to spread and overlap with another atom 2pz orbital wavefunction. Atom-to-atom distance gives the minimum distance, and so forms the test condition for determination to see if indeed the tight-binding assumption of low orbital overlap actually occurs. Look back at Section 2.5, to find in (2.58), (2.63b), and (2.68), those distances given between, respectively A–B atoms, A–A atoms, and A–B atoms. Knowing aCC = 1.42 Å for carbon atoms in graphene, those expressions are evaluated here, dnn = aCC = 1.42 Å;

√

dnnn = 3aCC = 2.46 Å;

dnnnn = 2aCC = 2.84 Å

(3.49)

For nn, nnn, and nnnn, the exponential evaluates to 

e−r /(2a0 ) nn = e−dnn /(2a0 ) = e−aCC /(2a0 ) = e−1.42 Å/(2×0.52 Å) = 0.255

(3.50a)

34

C.M. Krowne



√

e−r /(2a0 ) nnn = e−dnnn /(2a0 ) = e−  e−r /(2a0 ) 

nnnn

3aCC /(2a0 )

= e−2.46 Å/(2×0.52 Å) = 0.094 (3.50b)

= e−dnnnn /(2a0 ) = e−2aCC /(2a0 ) = e−2.84 Å/(2×0.52 Å) = 0.065 (3.50c)

Now it is known that the actual probability to overlap is the magnitude squared of the wavefunction. But before this is done, because the 2pz orbital has a counteracting r /a0 linear growth factor to offset the exponential decay, actual wavefunction drop must be recalculated, giving (given are the  wavefunctions ϕ¯210 normalized to the constant leading factor in (3.48), 1/4 2π (a0 )2 ) 

r −r /(2a0 )  dnn −dnn /(2a0 ) aCC −aCC /(2a0 ) e = e = e  a0 a0 a0 nn (3.51a) 1.42 −1.42 Å/(2×0.52 Å) = = 0.696 e 0.52 √  r −r /(2a0 )  dnnn −dnnn /(2a0 ) 3aCC −√3aCC /(2a0 ) ϕ¯ 210 (dnnn ) = e = e = e  a0 a0 a0 nnn (3.51b) 2.46 −2.46 Å/(2×0.52 Å) = = 0.445 e 0.52   r dnnnn −dnnnn /(2a0 ) 2aCC −2aCC /(2a0 ) ϕ¯ 210 (dnnnn ) = e−r /(2a0 )  = e = e a0 a0 a0 nnnn (3.51c) 2.84 −2.84 Å/(2×0.52 Å) = = 0.355 e 0.52 ϕ¯ 210 (dnn ) =

Clearly, the original decays are severely offset by the linear growth factor. With ϕ¯210 provided, squared magnitudes are   ϕ¯ 210 (dnn )2 =



  ϕ¯ 210 (dnnn )2 =

r a0



  ϕ¯ 210 (dnnnn )2 =

2

r a0



e

    dnn −dnn /(2a0 ) 2 e = (0.696)2 = 0.484  =  a0

−r /a0 

nn

2

r a0

  −r /a0  e  

 =

nnn

2

  −r /a0  e  

dnnn a0 

= nnnn

2

(3.52a) e−dnnn /a0 = (0.445)2 = 0.198

dnnnn −dnnnn /(2a0 ) e a0

2

(3.52b) = (0.355)2 = 0.126

(3.52c) Eq. (3.52a) shows why graphene is such a good conductor, as one A atom nearly shares its electron with the closest neighboring atom, a B atom,

35

Tight-binding formulation of electronic band structure

the nearest neighbor, at just below the 50% level. Next nearest neighboring atom to A, another A atom, has overlap below the 20% level seen in (3.52b). Next next nearest neighboring atom to A, a B atom, is down to roughly the 10% level in (3.52c). Last two equations seem consistent with a tight-binding approach, whereas the first one shows why the tight-binding approach is suggestive, an adjunct to more numerically, and accurate approaches, such as density functional theory first principles calculations (Martinez, Calle-Vallejo, Krowne, & Alonso, 2012; Martinez, Abad, Calle-Vallejo, Krowne, & Alonso, 2013). To confirm that the above estimate calculations of the orbital–orbital overlaps are appropriate, look at an overlap integral, to be discussed at much greater detail later, which comes out of the analysis: ¨

tnAIp i =



∗

d2 r φ A (r) V φ Ii (r + RAIi )

(3.53)

Here p is the order of the nearest neighbor location; p = 2, 3, 4 for nn, nnn, nnnn. Atom A is the reference atom, from which all other distances measured. RAIi is the vector between reference atom A and orbital Ii , being considered for the overlap integral, with index i being the particular atom in the set of I atoms, there being, respectively, 3, 6, and 3 of them for the nn, nnn, and nnnn. Recognizing that r = rA , r + RAIi = rIi , use of orbital 2pz expression (3.48) for orbital wavefunctions φA and φIi in (3.53), considering

V constant, it is the orbital product which appears in the integrand:

∗ φ A (r) φ Ii (rIi ) = φ A (r)φ Ii (rIi ) 





1 1 1 rA −rA /(2a0 ) 1 rIi −rI /(2a0 )   e cos θA e i cos θIi = 4 2π (a0 )3 a0 4 2π (a0 )3 a0 1 1 rA rIi −rA +rI /(2a0 ) i = 5 e cos θA cos θIi (3.54) 2 π (a0 )3 (a0 )2

When the point P being considered, at the end of the vector r = rA , lies in the zx -plane (call it the zx -plane with x in the RAIi direction), with h the distance this point is above the line joining atoms A and Ii , the orbital product can be reduced using the following relations (see Fig. 3.5): h = rIi cos θIi = rA cos θA

(3.55a)

rIi x = rIi sin θIi ;

(3.55b)

RAIi

rAx = rA sin θA   = RAI  = rI x + rAx

i

i

(3.55c)

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C.M. Krowne

Figure 3.5 Overlap between 2pz orbitals in direct space. Shown is a cut through the x z-plane. The two atoms are labeled A and Ii and are separated by a lattice spacing R AIi . Geometry necessary to determine the orbital wavefunction overlap is shown.

Thus, in the φ A (r)φ Ii (rIi ) product in (3.54), the linear-angular product part simplifies drastically using h to 

rA cos θA rIi cos θIi h h h r A rI i cos θA cos θIi = = = 2 a0 a0 a0 a0 a0 (a0 )

2

(3.56)

For the x -directed separation RAIi given by (3.55c), inserting Eqs. (3.55b), 



RAIi = rIi sin θIi + rA sin θA = rIi 1 − cos2 θIi + rA 1 − cos2 θA   2 2   = rI i 1 − h / rI i + rA 1 − h / rA

(3.57)

Considering the separation RAIi between atom types is known, and one of the local distances to its atom type, say rA also given, then the other rIi may be solved for by isolating it, squaring both sides and extracting it by a square root,  − h2 = RAIi − (rA )2 − h2 ⇒  2   2 rIi = h2 + RAIi − (rA )2 − h2

  2

rI i

(3.58)

37

Tight-binding formulation of electronic band structure

Taking the positive square root, 

rI i =



h2

2



+ RAIi −

(rA )

− h2

2

(3.59)

Placing both (3.56) and (3.59) into (3.54),



∗ 1 φ (r) φ Ii (rIi ) = 5



  1 rA rI i − rA +rIi /(2a0 ) cos θ cos θ e A I i 2 π (a0 )3 (a0 )2

A

1

1 = 5 2 π (a0 )3



h a0

⎛

2





−⎝rA + h2 +

⎞ 2  2 2 RAIi − (rA ) −h ⎠/(2a0 )

(3.60)

e

3.4. Reduction of the spatially varying Schrödinger equation into a solvable system Only way to convert the spatially varying Schrödinger equation (3.5) into a form which can be numerically evaluated, is to eliminate the vector spatial variable r by integrating it out. By (3.31), total wavefunction consisted as the sum over the sublattice atoms. For carbon, that was two types labeled A and B. Eq. (2.9) in vector form appears as ψk (r) =



ψkA (r)



ψkB (r)



ak bk



(3.61)

Its Hermitian conjugate is  

† ψk (r) = ψk (r) = ψkA (r)

†

 =

ak bk

†



ψkA (r)

ψkB (r)

ψkB (r)





ak bk



†

†

(3.62a)

or

†  ψk (r) = ψk (r) = (ak )∗

†

=



a∗k

b∗k





∗

 ∗ 

ψkA (r) ∗ ψkB (r)

bk 

 ! "∗  ψkA (r) ! B "∗ ψk (r)

(3.62b)

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C.M. Krowne

Multiplying the Schrödinger equation from the left by ψk† (r), (3.5) becomes ψk† (r)H (r)ψk (r) = εk ψk† (r)ψk (r)

(3.63)

Integrating over two dimensional space for our 2D crystalline system, ¨

¨ †

d2 rψk† (r)ψk (r)

d rψk (r)H (r)ψk (r) = εk 2

(3.64)

Substitute in ψk† (r) and ψk (r) from (3.61) and (3.62b) into (3.64), ¨

d2 r



a∗k

b∗k

¨ = εk

d2 r





   ∗  a  ψkA (r) k H ψkA (r) ψkB (r) ∗ ψkB (r) bk  (3.65)     a  ψ A∗ (r)  k ∗ A B k bk ψk (r) ψk (r) ∗ ψkB (r) bk



a∗k

Knowing that quantum mechanically the Hamiltonian H operator is scalar here, performing outer products, allows identification of two matrix operators,  Hk (r) =

 =  Sk (r) =

 =

∗

ψkA (r) ∗ ψkB (r)





H

ψkA (r)

∗

∗

ψkA (r)H (r)ψkA (r) ∗ ψkB (r)H (r)ψkA (r) ∗

ψkA (r) ∗ ψkB (r) ∗







ψkA (r)H (r)ψkB (r) ∗ ψkB (r)H (r)ψkB (r)

ψkA (r)

ψkA (r)ψkA (r) ∗ ψkB (r)ψkA (r)

ψkB (r)

ψkB (r) ∗



(3.66a)



ψkA (r)ψkB (r) ∗ ψkB (r)ψkB (r)



(3.66b)

Eqs. (3.66) represent the Hamiltonian tested by the A and B atom total wavefunctions Hk (r), and the same and mixed products of the total A and B atom wavefunctions Sk (r) (self-matrix of the sublattice wavefunctions). Notice that the second equation of (3.66) can be obtained from the first by formally letting H → 1. Now integral equation (3.65) can be written

39

Tight-binding formulation of electronic band structure

compactly as ¨

d2 r



a∗k

b∗k

¨ = εk





2

d r

 Hk (r)

∗

ak

∗



bk

ak bk

 

Sk (r)

ak bk



(3.67)

Matrix operations and integration order can be switched in (3.67), so the double integral can be pulled past the coefficient row or column vectors. 

a∗k

b∗k

= εk

  a k d2 rHk (r)

 ¨

bk



∗

∗

ak

  ak d rSk (r)

 ¨

(3.68)

2

bk

bk

The integrated out Hamiltonian and self-matrices, are identified as H¯ k =

¨

d2 rHk (r);

S¯k =

¨

d2 rSk (r)

(3.69)

Inserting (3.66) into (3.69) gives 

   A∗ (r) ψ k H¯ k = d2 rHk (r) = d2 r H ψkA (r) ψkB (r) ∗ B ψk (r)   ˜ ˜ 2 A∗ 2 rψ A∗ (r)H (r)ψ A (r) d d r ψk (r)H (r)ψkB (r) k k ˜ 2 B∗ = ˜ 2 B∗ d rψk (r)H (r)ψkA (r) d rψk (r)H (r)ψkB (r) ¨

¨



 ∗  ψkA (r)  A B (r) d rSk (r) = d r ( r ) ψ ψ ∗ k k ψkB (r)  ˜  ˜ ∗ ∗ d2 rψ A (r)ψkA (r) ˜ d2 rψkA (r)ψkB (r) = ˜ 2 kB∗ ∗ d rψk (r)ψkA (r) d2 rψkB (r)ψkB (r)

S¯k =

¨

(3.70a)

¨

2

2

(3.70b)

so that (3.68) can be rewritten as 

a∗k

b∗k



 H¯ k

ak bk

 = εk



a∗k

b∗k



 S¯k

ak bk



(3.71)

40

C.M. Krowne

Collecting terms on the left-hand-side, 

∗

∗

ak

bk

!

H¯ k − εk S¯k

"



ak bk

 =0

(3.72)

Stripping off the left row matrix hitting the remaining double product, !

H¯ k − εk S¯k

"





ak bk

=0

(3.73)

what is present is recognizable from linear matrix theory, as our implicit determinantal equation. That is, in order for (3.73) to have a solution for any arbitrary column matrix Ssub of the sublattice coefficients weighting the two types of atoms available, A and B, 

Ssub =

ak bk



(3.74)

with (3.73) streamlined to ! " H¯ k − εk S¯k Ssub = 0

(3.75)

the determinant of the matrix hitting Ssub must be zero, giving the secular equation !

"

det H¯ k − εk S¯k = 0

(3.76)

This is the solution for a non-trivial Ssub when it is not null, when at least one of its elements are not zero. For a given k value, (3.76) will give two energy eigenvalue solutions εkλ , for λ = 1, 2 bands, associated with the fact that two sublattices of atoms exist, the A and B sublattices for graphene or some other two atom Bravais unit cell in real space. In general, the bands are labeled λ = 1, 2, 3, . . . , Nac

(3.77)

where Nac is the number of atoms inside the Bravais unit cell. The problem may be generalized for any number of atoms in the Bravais unit cell. Clearly, secular equation (3.76) is already general. However, (3.31)

41

Tight-binding formulation of electronic band structure

must be generalized to ψk (r) =

Nac 

alksub ψklsub (r)

sub =1

=



⎤

⎡

ψk1 (r)

ψk2 (r)

l

···

ψkNac (r)

a1k ⎢  ⎢ a2k ⎥ ⎥ ⎢ ⎢ ⎣

.. .

l

(3.78)

⎥ ⎥ ⎦

akNac which also requires the total sublattice row wavefunction vector be defined as ψksub (r) =



ψk1 (r)

···

ψk2 (r)

l

ψkNac (r)



(3.79)

while the sublattice coefficient column matrix of (3.74) generalize to ⎡

⎤

a1k ⎢ a2 ⎥ ⎢ k ⎥

Ssub = ⎢ ⎢

⎥ ⎥ ⎦

.. .

⎣

l

(3.80)

akNac Matrix equivalent of (3.78) is ψk (r) = ψksub (r)Ssub

(3.81)

Continuing the generalization, the Hamiltonian tested by any number of atoms in the unit cell, by atom total wavefunctions, Hk (r), and the same and mixed products of any number of atoms in the unit cell, by atom total wavefunctions, Sk (r) (self-matrix of the sublattice wavefunctions), are now ⎡ ⎢ ⎢ Hk (r) = ⎢ ⎢ ⎣ ⎡ ⎢ =⎢ ⎣

∗

ψk1 (r) ∗ ψk2 (r) .. . ∗

ψkNac (r)

⎤ ⎥  ⎥ ⎥ H ψ 1 (r) k ⎥ ⎦

∗ ψk1 (r)H (r)ψk1 (r)

ψkNac (r)

···

ψk2 (r)



.. .

··· .. .

∗ ψk1 (r)H (r)ψkNac (r)

ψkNac (r)H (r)ψk1 (r)

···

ψkNac (r)H (r)ψkNac (r)

∗

l

∗

.. .

⎤ ⎥ ⎥ ⎦

(3.82a)

42

C.M. Krowne

⎡ ⎢ ⎢ Sk (r) = ⎢ ⎢ ⎣ ⎡ ⎢ =⎢ ⎣

⎤

∗

ψk1 (r) ∗ ψk2 (r) .. .

⎥ ⎥ ⎥ ψ 1 (r) k ⎥ ⎦

∗

ψkNac (r)

∗ ψk1 (r)ψk1 (r)

ψkNac (r)

···

ψk2 (r)

.. .

··· .. .

ψkNac (r)ψk1 (r)

···

ψkNac (r)ψkNac (r)

.. .

∗

l

(3.82b)

⎤

∗ ψk1 (r)ψkNac (r)

∗



⎥ ⎥ ⎦

Inserting (3.82) into (3.69) yields the integrated matrix versions, H¯ k =

¨

d2 rHk (r) ⎡ ⎢ ⎢ d r⎢ ⎢ ⎣

¨ =

2

⎡ ˜ ⎢ =⎢ ⎣

S¯k =

˜

⎤

∗

ψk1 (r) ∗ ψk2 (r) .. .

⎥  ⎥ ⎥ H (r) ψ 1 (r) k ⎥ ⎦

∗

ψkNac (r) ∗

˜

··· .. .

d2 rψk1 (r)H (r)ψk1 (r) .. .

∗

d2 rψkNac (r)H (r)ψk1 (r)

˜

···

ψkNac (r)

···

ψk2 (r)

∗



d2 rψk1 (r)H (r)ψkNac (r) l

∗

.. .

⎤ ⎥ ⎥ ⎦

d2 rψkNac (r)H (r)ψkNac (r) (3.83a)

¨

d2 rSk (r) ⎡ ¨

=

⎢ ⎢ d r⎢ ⎢ ⎣ 2

⎡ ˜ ⎢ =⎢ ⎣

˜

⎤

∗

ψk1 (r) ∗ ψk2 (r) .. . ∗

ψkNac (r)

⎥ ⎥ ⎥ ψ 1 (r) k ⎥ ⎦

∗

d2 rψk1 (r)ψk1 (r) .. .

∗

d2 rψkNac (r)ψk1 (r)

··· .. . ···

···

ψk2 (r)

ψkNac (r)



(3.83b) ˜ ˜

∗

d2 rψk1 (r)ψkNac (r) .. .

l

∗

d2 rψkNac (r)ψkNac (r)

⎤ ⎥ ⎥ ⎦

The ij matrix element of Hk (r), Hkij (r), is obtained from (3.82a) by inspection, ij

∗

j

Hk (r) = ψki (r)H (r)ψk (r)

(3.84a)

43

Tight-binding formulation of electronic band structure

Similarly by inspection, the ij matrix element of Sk (r), Skij (r), is obtained from (3.82b), ij

∗

j

Sk (r) = ψki (r)ψk (r)

(3.84b)

From (3.69), taking the ijth matrix element under the double spatial integration ij H¯ k =

¨

d2 rHkij (r);

S¯k =

¨

d2 rSkij (r)

(3.85)

References Blatt, F. J. (1968). Physics of electronic conduction in solids. New York: McGraw-Hill. Cohen-Tannoudji, C., Diu, B., & Laloe, F. (1977). Quantum mechanics. Wiley. Kane, E. O. (1982). Energy band theory. In T. S. Moss, & W. Paul (Eds.), Handbook on semiconductors: Vol. 1. Band theory and transport properties (pp. 193–217). Amsterdam: North-Holland. Martinez, J. I., Abad, E., Calle-Vallejo, F., Krowne, C. M., & Alonso, J. A. (2013). Tailoring structural and electronic properties of RuO2 nanotubes: Many-body approach and electronic transport. Physical Chemistry Chemical Physics, 15(35), 14715–14722. Martinez, J. I., Calle-Vallejo, F., Krowne, C. M., & Alonso, J. A. (2012). First-principles structural & electronic characterization of ordered SiO2 nanowires. Journal of Physical Chemistry C, 116, 18973–18982. Saito, R., Dresselhaus, G., & Dresselhaus, M. S. (1998). Physical properties of carbon nanotubes. Imperial College Press. Wallace, P. R. (1947). The band theory of graphite. Physical Review, 71, 622. Erratum: Physical Review, 72, 258.

Further reading Ahn, J.-H., Lee, M.-J., Heo, H., Sung, J. H., Kim, K., Hwang, H., & Jo, M.-H. (2015). Deterministic two-dimensional polymorphism growth of hexagonal n-type SnS2 and orthorhombic p-type SnS crystals. Nano Letters, 15, 3703–3708. Bjorken, J. D., & Drell, S. D. (1964). Relativistic quantum mechanics. New York: McGraw-Hill. Butcher, P. N., & Fawcett, W. (1966). Calculation of the velocity-field characteristics for gallium-arsenide. Physics Letters, 21, 489. Dwight, H. B. (1961). Tables of integrals and other mathematical data (4th ed.). New York: The Macmillan Co. Fuchs, J.-N., & Goerbig, M. O. (2008). Introduction to physical properties of graphene, lecture notes. Geim (2007). The rise of graphene. Nature Materials, 6, 183–191. Harrington, R. F. (1961). Time harmonic electromagnetic fields. New York: McGraw-Hill. See Appendix A, (A-13), 8th equation. Hu, H., Wang, Z., & Liu, F. (2014). Half metal in two-dimensional hexagonal organometallic framework. Nanoscale Research Letters, 9, 960.

44

C.M. Krowne

Julian, M. M. (2008). Foundations of crystallography with computer applications. Boca Raton, FL: CRC Press. Kittel, C. (1968). Introduction to solid state physics. New York: Wiley. Kittel, C., Knight, W. D., & Ruderman, M. A. (1965). Berkeley physics course: Vol. 1. Mechanics. New York: McGraw-Hill. Leggett, A. (2010). Lecture 5. Graphene: Electronic band structure and Dirac fermions. In Physics 769. Selected topics in condensed matter physics. University of Illinois at Urbana– Champaign. Littlejohn, M. A., Hauser, J. R., & Glisson, T. H. (1977). Velocity-field characteristics of GaAs with –L–X conduction band ordering. Journal of Applied Physics, 48, 4587. Malterre, D., Kierren, B., Fagot-Revurat, Y., Didiot, C., García de Abajo, F. J., Schiller, F., . . . Ortega, J. E. (2011). Symmetry breaking and gap opening in two-dimensional hexagonal lattices. New Journal of Physics, 13, 013026. Martin, P. M. (2018a). Graphene: Single and stacked layer basics. Vacuum Technology & Coating, 19(7), 6–13. Martin, P. M. (2018b). Active thin films: Graphene and related materials. Vacuum Technology & Coating, 19(5), 6–10. Martin, P. M. (2018c). Active thin films: Graphene nanoribbons. Vacuum Technology & Coating, 19(9), 6–14. Martin, P. M. (2018d). Active thin films: Graphene related materials. Vacuum Technology & Coating, 19(10), 6–13. McKelvey, J. P. (1966). Solid state and semiconductor physics. New York: Harper & Row. Neto, A. H. C., Guinea, F., Peres, N. M. R., Novoselov, K. S., & Geim, A. K. (2009). The electronic properties of graphene. Reviews of Modern Physics, 81, 109–162. Osofsky, M. S., Hernández-Hangarter, S. C., Nath, A., Wheeler, V. D., Walton, S. G., Krowne, C. M., & Gaskill, D. K. (2016). Functionalized graphene as a model system for the two-dimensional metal–insulator transition. Scientific Reports, 6, 19939. Osofsky, M. S., Krowne, C. M., Charipar, K. M., Bussmann, K., Chervin, C. N., Pala, I. R., & Rolison, D. R. (2016). Disordered RuO2 exhibits two dimensional, low-mobility transport and a metal–insulator transition. Scientific Reports, 6, 21836. Peskin, M. E., & Schroeder, D. V. (1995). An introduction to quantum field theory. Westview Press. Phani, A. S., Woodhouse, J., & Fleck, N. A. (2006). Wave propagation in two-dimensional periodic lattices. Journal of the Acoustical Society of America, 119. Rees, H. D., & Gray, K. W. (1976). Indium phosphide: A semiconductor for microwave devices. IEE Journal on Solid-State and Electron Devices, 1, 1. Reich, S., Thomsen, C., & Maultzsch, J. (2004). Carbon nanotubes – basic concepts and physical properties. Berlin: Wiley–VCH Verlag. Ruch, J. G., & Kino, G. S. (1967). Measurements of the velocity-field characteristics for gallium-arsenide. Applied Physics Letters, 10, 40. Sadurni, E. (2013). Propagators in two-dimensional lattices. arXiv:1306.0261v1 [quant-ph]. Schiff, L. I. (1968). Quantum mechanics. New York: McGraw-Hill. Snoke, D. W., & Keeling, J. (2017). The new era of polaritons. Physics Today, 70, 54. Sun, L., Banhart, F., & Warner, J. (2015). Two-dimensional materials under electron irradiation. MRS Bulletin, 40, 29. Symons, D. D., & Fleck, N. A. (2008). The imperfection sensitivity of isotropic twodimensional elastic lattices. Journal of Applied Mechanics, 75, 051011. Sze, S. M. (1981). Physics of semiconductor devices (2nd ed.). New York: Wiley.

Tight-binding formulation of electronic band structure

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Van Hove, L. (1953). The occurrence of singularities in the elastic frequency distribution of a crystal. Physical Review, 89, 1189–1193. Yamamoto, M., Shimazaki, Y., Borzenets, I. V., & Tarucha, S. (2015). Valley Hall effect in two-dimensional hexagonal lattice. Journal of the Physical Society of Japan, 84, 121006.