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Evaluation of the matrix elements for the tight-binding formulation of 2D hexagonal materials
CHAPTER FOUR
Evaluation of the matrix elements for the tight-binding formulation 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 4.1. Determination of an arbitrary Hamiltonian and self-matrix elements 4.2. Secular equation of the system using the Hamiltonian 4.3. Nearest neighbor hopping and overlap integrals 4.4. Next nearest neighbor hopping and overlap integrals References Further reading
47 54 55 60 64 64
4.1. Determination of an arbitrary Hamiltonian and self-matrix elements From the last Section 3.4 of Chapter 3, solving the tight-binding problem amounts to finding the integrated matrix elements for both the Hamiltonian and self, shown in (3.85). Orbital wavefunctions are required for the lsub = i and lsub = j sublattices within the Bravais unit cell, obtainable from (3.33) as ψki (r) =
Rlat
j
ψk (r) =
eik·Rlat φai r − Rlat + ri eik·Rlat φaj r − Rlat + rj
(4.1a) (4.1b)
Rlat
Note that in order to avoid index confusion, summation over lattice positional vectors Rlat has been changed from i to lat, making Ri → Rlat . For Advances in Imaging and Electron Physics, Volume 210 ISSN 1076-5670 https://doi.org/10.1016/bs.aiep.2019.01.004
Copyright © 2019 Elsevier Inc. All rights reserved.
47
48
C.M. Krowne
the Hamiltonian matrix element, utilizing (3.83a) in (3.84a), and (4.1), ij H¯ k =
¨
d2 rHkij (r)
¨
∗
d2 rψki (r)H (r)ψkj (r)
=
⎡
¨
d2 r ⎣
=
⎤∗ eik·Rlat 1 φa r − Rlat 1 + ri ⎦ i
R
× H (r)
lat 1
=
d2 r
∗
e−ik·Rlat 1 φai r − Rlat 1 + ri
Rlat 1
× H (r)
eik·Rlat 2 φaj r − Rlat 2 + rj
Rlat 2
¨
eik·Rlat 2 φaj r − Rlat 2 + rj
(4.2)
Rlat 2
Pulling the lattice summations outside of the double integral, ij H¯ k =
¨
∗
d2 re−ik·Rlat 1 φai r − Rlat 1 + ri
Rlat 1 Rlat 2
× H (r)eik·Rlat 2 φaj r − Rlat 2 + rj ∗ ¨ = d2 re−ik· Rlat 1 −Rlat 2 φai r − Rlat 1 + ri Rlat 1 Rlat 2
× H (r)φaj r − Rlat 2 + rj ¨ ∗ = d2 reik·Rlat 3 φai r − Rlat 1 + ri H (r)φaj r − Rlat 2 + rj Rlat 1 Rlat 2
=
¨
∗
∗
Rlat 1 Rlat 2
= =
¨
1
⎭ Rlat 3 ¨
∗
Rlat 3
eik·Rlat 3
¨
∗
∗
d2 r eik·Rlat 3 φai r H (r + ri )φaj r − Rlat 3 − rij
Rlat 3
= Nsys
d2 r eik·Rlat 3 φai r H (r + ri )φaj r − Rlat 3 − rij
Rlat 1
= Nsys
d2 r eik·Rlat 3 φai r H (r + ri )φaj r − Rlat 3 − rij
Rlat 1 Rlat 2 ⎧ ⎫ ⎨ ⎬ ¨
⎩
d2 r eik·Rlat 3 φai r H (r + Rlat 1 + ri )φaj r − Rlat 3 − rij
d2 r φai r H (r + ri )φaj r − Rlat 3 − rij
(4.3)
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Evaluation of the matrix elements for the tight-binding formulation
Here a number of procedures were employed to arrive at (3.47). First the exponential terms were collected together into one factor. Next a third lattice position was defined, integration variable shifted, and the inner summation changed: Rlat 3 = − (Rlat 1 − Rlat 2 )
(4.4a)
rij = ri + rj
(4.4b)
r = r − Rlat 1 + ri
(4.4c)
dr = dr − d Rlat 1 + ri = dr
Rlat 1 Rlat 2
→
Rlat 1 Rlat 1 +Rlat 3
→
(4.4d)
(4.4e)
Rlat 1 Rlat 3
Inner summation variable can drop Rlat 1 since the same range is covered, namely all of the lattice points available by using the Bravais unit cell vectors. Last three lines in (4.3) arose because the Hamiltonian (Kane, 1982) must be periodic in Bravais lattice positional vectors (Julian, 2008), and the integral has no Rlat 1 dependence, leading to the factor of Nsys atoms in the system being considered. Finally, the primed spatial integral variable r , can be relabeled r as there no other integral variables. ij H¯ k = Nsys
eik·Rlat 3
¨
∗
d2 rφai (r) H (r + ri )φaj r − Rlat 3 − rij
(4.5)
Rlat 3
Hamiltonian must be examined. By (3.6), for an electron associated with lattice position Rlat , its atomic orbital description is fairly accurate, calling its vector in reference to that local origin r¯ lat . a (¯rlat ) = − Hlat
h¯ 2 2 ∇ + V a (¯rlat ) 2me r¯lat
(4.6)
In order to handle electrons localized about any lattice site Rlat , r¯ lat must be generalized to r¯ lat = rlat − Rlat . a a Hlat (rlat ) = Hlat (¯rlat ) = −
h¯ 2 2 ∇ + V a (rlat − Rlat ) 2me rlat
(4.7)
In assessing the overlap between orbitals in Chapter 3, Section 3.3, atomic hydrogenic solutions were studied. Here V a (¯rlat ) is the atomic solution for the electron attached to the lat lattice location. There are as we just mentioned, Nsys such locations, so all the other orbital locations will perturb or
50
C.M. Krowne
modify the fairly accurate atomic solution. a Hlat 1 (rlat 1 ) = Hlat 1 (rlat 1 ) +
Nsys
V a (rlat 1 − Rlat 2 )
(4.8)
lat 2=1 lat 2=lat 1
Total Hamiltonian for the entire system must consist of a sum over all the system electrons, and this would then be for Zv valence mobile electrons per atom for a single type of site atom like graphene, where Zv < Z, where Z is the positive core charge, or the atomic number of the element. Graphene has Zv = 1. For multiple atom types, causing several sublattice atom types, or several atoms per Bravais unit cell, the relation is more complicated requiring a summation over sublattices types:
Nel =
Zv Nsys ; NT ,sub = 1 NT ,sub i=1 Zv,subi Nsys,subi ; NT ,sub = 1
(4.9)
Total number of sublattices is NT ,sub = Nac
(4.10)
where from earlier this number must be the atom count in the Bravais unit cell Nac . Total number of atoms in the system must be those in each sublattice group within the system, summed up, Nsys =
Nac
Nsys,subi
(4.11)
i=1
Although the kinetic part of each atomic Hamiltonian is the same as in (4.7), potential energy part may be different, and this is allowed for by a indexing V a with subi , giving Vsub . Lattice location Rlat;subi also now has i subi associated with it: a Hlat ;subi (rlat;subi ) = −
h¯ 2 2 a ∇ + Vsub (rlat;subi − Rlat;subi ) i 2me rlat ;subi
(4.12)
Upgraded atomic Hamiltonian at lattice location lat 1 in sublattice subi , lat 1; subi , Rlat 1:subi , taking again into account all the other orbital locations which perturb or modify the fairly accurate atomic solution, whether in
Evaluation of the matrix elements for the tight-binding formulation
51
the same sublattice or in another sublattice, is Hlat 1;subi (rlat 1;subi ) a = Hsub (rlat 1;subi ) + i
Nac
Nsys;subj
j=1
lat 2;subj =1 lat 2;subj =lat 1;subi
a Vsub (rlat 1;subi − Rlat 2:subj ) i
(4.13)
Here rlat 1;subi is the actual position in space being considered, r, referenced to the particular sublattice atom location ri , and the particular lattice location shifted by a Bravais unit cell lattice vector rlat 1;subi = r − ri − Rlat 1
(4.14)
Substituting (4.14) into (4.13), one obtains ca ca a Hlat 1;subi (r) = Hsubi (r − ri − Rlat 1 ) = Hsubi (r − ri − Rlat 1 )
+
Nac
Nsys;subj
j=1
lat 2;subj =1 lat 2;subj =lat 1;subi
a Vsub (r − ri − Rlat 1 − Rlat 2:subj ) i
(4.15)
Total Hamiltonian for the entire system is the sum of the individual corrected or perturbed Hamiltonians Hlat 1;subi (r) from (4.15) over all the lattice locations (lat 1) for all of the sublattices (subi ), Rlat 1:subi , H (r) =
Nac
Nsys,subi
i=1 lat 1;subi =1
ca Hlat 1;subi (r)
(4.16)
Considering second term as a perturbation in (4.15), small enough to isolate and label as a Vsub (r − ri i
− Rlat 1 ) =
Nac
Nsys;subj
j=1
lat 2;subj =1 lat 2;subj =lat 1;subi
a Vsub (r − ri − Rlat 1 − Rlat 2:subj ) i
(4.17) the expression in (4.15) becomes ca a a Hlat 1;subi (r) = Hsubi (r − ri − Rlat 1 ) + Vsubi (r − ri − Rlat 1 )
(4.18)
52
C.M. Krowne
Placing (4.18) in (4.16), to obtain the system Hamiltonian, H (r) =
Nsys,subi
Nac
i=1 lat 1;subi =1
a a Hsub (r − ri − Rlat 1 ) + Vsub (r − ri − Rlat 1 ) i i
(4.19)
Once the system Hamiltonian is available, as it is from (4.19), the ijth matrix element of it can be found from (4.5), by inserting (4.19), being careful to change the indexing i to n to avoid confusion with the orbital indexes i and j. ij H¯ k = Nsys
eik·Rlat 3
Rlat 3
= Nsys
eik·Rlat 3
¨
∗
d2 rφai (r) H (r + ri )φaj r − Rlat 3 − rij ¨
∗
d2 rφai (r)
Rlat 3
×
Nac
Nsys,subn
i=n lat 1;subn =1
a a Hsub (r − Rlat 1 ) + Vsub (r − Rlat 1 ) n n
× φaj r − Rlat 3 − rij ¨ ∗ ik·Rlat 3 e d2 rφai (r) = Nsys
(4.20)
Rlat 3
×
sys,subn Nac N
n=1 lat 1;subn =1
+ Nsys
e
a Hsub (r − Rlat 1 )φaj r − Rlat 3 − rij n
ik·Rlat 3
¨
∗
d2 rφai (r)
Rlat 3
×
sys,subn Nac N
a Vsub (r − Rlat 1 )φaj r − Rlat 3 − rij n
n=1 lat 1;subn =1
If it is argued that the projection of the total atomic Hamiltonian onto the jth sublattice orbital wavefunction selects out that orbital and hits it with an energy eigenvalue characteristic of it, in the first integral of (4.20), then ⎧ Nac ⎨ ⎩
Nsys,subn
n=1 lat 1;subn =1
⎫ ⎬ a Hsub ( r − R ) φaj r − Rlat 3 − rij ≈ ε j φaj r − Rlat 3 − rij lat 1 n ⎭
(4.21)
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Evaluation of the matrix elements for the tight-binding formulation
Here we see that Bravais lattice vector Rlat 1 goes through its range until it finds the optimum one equaling Rlat 3 . This is when the local atomic Hamiltonian coincides with its local orbital. Eq. (4.21) forms a reasonable approximation. In doing this, the small misalignment between the Hamil tonian argument r − Rlat 1 and the orbital argument r − Rlat 3 − rij , when Rlat 1 = Rlat 3 , rij is ignored. For the potential energy term, second integral term of (4.21) ⎫ ⎬ a Vsub ( r − R ) φaj r − Rlat 3 − rij lat 1 n ⎭ ⎩ n=1 lat 1;subn =1 a φ j r − Rlat 3 − rij = Vsub j a ⎧ Nac ⎨
Nsys,subn
(4.22)
Once again, it is seen that the Bravais lattice vector Rlat 1 goes through its range until it finds the optimum one equaling Rlat 3 . This is when the local atomic potential energy coincides with its local orbital. Eq. (4.22) forms a reasonable approximation. In doing this, the small misalignment between the potential energy argument r − Rlat 1 and the orbital argument r − Rlat 3 − rij , when Rlat 1 = Rlat 3 , rij is ignored. Take the results of (4.21) and (4.22) and set them into (4.20), to obtain the H¯ kij matrix element ij H¯ k = Nsys
eik·Rlat 3
¨
Rlat 3
+ Nsys
eik·Rlat 3
Rlat 3
= Nsys ε
j
e
ik·Rlat 3
Rlat 3 a + Nsys Vsub j
∗
d2 rφai (r) εj φaj r − Rlat 3 − rij ¨
∗
a d2 rφai (r) Vsub φ j r − Rlat 3 − rij j a
¨
∗
d2 rφai (r) φaj r − Rlat 3 − rij e
ik·Rlat 3
¨
∗
(4.23)
d2 rφai (r) φaj r − Rlat 3 − rij
Rlat 3
ij ij = Nsys ε j sk + tk
Here sijk and tkij are defined by sijk =
eik·Rlat 3
Rlat 3 a tkij = Vsub j
Rlat 3
¨
∗
d2 rφai (r) φaj r − Rlat 3 − rij eik·Rlat 3
¨
∗
d2 rφai (r) φaj r − Rlat 3 − rij
(4.24)
(4.25)
54
C.M. Krowne
4.2. Secular equation of the system using the Hamiltonian Recall from (3.76) that we have a descriptive equation determining the system behavior, and stating it again here, we have det H¯ k − εk S¯k = 0
(3.76)
Recognize the self-matrix term for the system is S¯k = Nsys sk
(4.26)
where elements of sk are given by (4.24). System Hamiltonian matrix elements (4.23) may be expressed in matrix form as H¯ k = Nsys (sk ε + tk )
(4.27)
with ε being considered the diagonal matrix, ⎡ ⎢ ⎢ ε=⎢ ⎢ ⎣
ε1
0
0
ε2
.. .
0
⎤
···
0 0 ⎥ ⎥
0 ..
0 0
⎥ ⎥
0 ⎦
.
0
(4.28)
ε Nac
With this information, the secular equation of the system (3.76) may be written in matrix form, utilizing the Hamiltonian matrix form of (4.27), det Nsys (sk ε + tk ) − εk Nsys sk = 0
(4.29)
Factoring out Nsys , and rearranging terms
Nsys det tk − sk εkλ − ε
=0
(4.30)
Dividing out the total system number of electrons, leaves the final secular equation form, det tk − sk εkλ − ε = 0
(4.31)
where the explicitly branch index λ on the eigenenergy solution of the system εkλ , is inserted. This eigenenergy value is a number. Here tk is the
55
Evaluation of the matrix elements for the tight-binding formulation
hopping matrix, sk the overlap matrix. Postfactor of sk in (4.31) in general looks like ⎡
1 ⎢ ⎢ 0 λ
0 1
0
0 0
ε k − ε = εk ⎢ ⎢ .. ⎣ . λ
⎡ ⎢ ⎢ =⎢ ⎢ ⎣
εkλ − ε 1
0 .. .
···
0 ..
.
0
⎡
⎤
0 0 ⎥ ⎥
⎢ ⎢ ⎥−⎢ ⎥ ⎢ 0 ⎦ ⎣
1 ···
εkλ − ε 2
0
0
0
0
ε2
.. .
..
0 ..
.
0
⎤
0 0 ⎥ ⎥ ⎥ ⎥
0 ⎦ ε Nac
(4.32)
⎤
0 0
.
0
···
0 0
0
0 0 0
ε1
⎥ ⎥ ⎥ ⎥ ⎦
0 εk − ε Nac λ
For the case where the onsite energies εj are all the same for the various sublattice sites j, the sk postfactor matrix εkλ − ε reduces to a number, or in matrix form, εkλ − ε = εkλ − ε0 I
(4.33)
with I being the identity matrix. Clearly, (4.33) applies to the case of graphene where sublattice sites A and B have carbon atoms. Secular equation then reduces to det tk − εkλ − ε0 sk = 0
(4.34)
4.3. Nearest neighbor hopping and overlap integrals Integral to be evaluated for hopping integral is (4.25), and in relation to an A atom site, nearest neighboring atoms are three B atoms (see Fig. 4.1), located by the direct space lattice vectors v1nn , v2nn , and v3nn , given respectively by Eqs. (2.54), (2.56), and (2.57). That equation is a tkij = Vsub j
Rlat 3
eik·Rlat 3
¨
∗
d2 rφai (r) φaj r − Rlat 3 − rij
(4.25)
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C.M. Krowne
Figure 4.1 Nearest neighbor (nn) atom locations in 2D direct space lattice. a i , i = 1, 2, 3 are the Bravais unit cell direct space lattice vectors; a 3 = a 2 − a 1 . Nearest neighboring atoms Bi are shown. Distance between the reference A atom and the B3 atom is rB3 A . Other Bj atom locations are found by a i shifts.
a , the integral beSetting i = A, j = B, and dropping the j index on Vsub j comes, noting the summation is over B nearest neighbors
a tkAB = Vsub
eik·Rlat
¨
∗
B
d2 rφaA (r) φa j r − Rlat − rAB3
Rlat =0(j=3),−a2 (j=1), −a3 (j=2) (nn atoms B3 ,B1 , B2 ; counterclockwise)
(4.35) where the choice of lattice vector with respect to the B3 atom at position vertically below the reference A by a carbon–carbon spacing aCC , −aCC yˆ , (2.57) makes Rlat = 0, −a2 , −a3
(4.36)
This shifts the B atom orbital at atom B3 (Rlat = 0) to that at B1 by (see Fig. 4.1) − Rlat − rAB3 − = −Rlat + rAB3 = a2 + rAB3
(4.37)
For shifting to the B2 atom, from the B3 (Rlat = 0) atom, an amount of − Rlat − rAB3 − = −Rlat + rAB3 = a3 + rAB3
is required.
(4.38)
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Evaluation of the matrix elements for the tight-binding formulation
Figure 4.2 Orbital shifting effects in 2D direct space with reference to an origin. Top diagram for any two adjacent A and B atoms. Bottom diagram for a central reference A atom and a B atom located directly below it (a B3 atom). The hydrogenic atom orbitals l
=A
l
=Bj
l
=A
are for the top diagram are φasub (rm(ref ) ) at atom A, and φasub (riB ) = φasub ref (r − R lat − rBi Aref ) at atom B. For bottom diagram, setting i, m(ref ) → Bi=3 Aref the orbital l =Bj=3 about the particular B3 atom becomes φasub (riB ) = φalsub =Aref (r − R lat − rB A ). i=3 ref
Here a3 is a sublattice vector obtained by taking the difference of the two Bravais sublattice vectors a2 and a1 , a 3 = a2 − a1
(4.39)
Eqs. (2.31) provide a2 and a1 , so that a3 is easily determined, √
a3 =
√ √ √ √ 3aCC 3aCC xˆ + 3yˆ − 3aCC xˆ = −xˆ + 3yˆ 2 2
(4.40)
Reason why a1 does not appear in (4.36) is that B3 atom acts as the reference and the other two atoms are located with both vertical and horizontal offsets, precluding use of a sublattice vector with a pure horizontal offset. Expanding tkAB by writing out the summation, using the shifting information contained in (4.36)–(4.38), with the shifting process shown in Fig. 4.2,
58
C.M. Krowne
tkAB
a = Vsub
e
ik·Rlat
¨
∗
B
d2 rφaA (r) φa j r − Rlat − rAB3
Rlat =0(j=3),−a2 (j=1), −a3 (j=2) (nn atoms B3 ,B1 , B2 ; counterclockwise)
¨ ⎫ ⎧ ik·0 2 A∗ B3 ;ref ⎪ ⎪ ⎪ ⎪ e d r φ r r + r φ ( ) AB3 ⎪ ⎪ a a ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¨ ⎬ ⎨ ∗ a −ik·a2 2 A B1 d rφa (r) φa r + a2 + rAB3 +e = Vsub ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¨ ⎪ ⎪ ⎪ ⎪ ∗ ⎪ 2 A B2 ⎭ ⎩ + e−ik·a3 d rφa (r) φa r + a3 + rAB3 ⎪
(4.41) Now we know that the orbitals in A and B atoms are the same for graphene. Furthermore, expecting the orbital environments of the wavefunctions at B1 , B2 , and B3 to be the same, that is φaB1 r + a2 + rAB3 = φaB3 ;ref φaB2 r + a3 + rAB3 = φaB3 ;ref
r + rAB3 r + rAB3
(4.42)
because shifts of Bravais lattice vectors occur, (4.41) simplifies to ¨ ⎫ ⎧ ik·0 2 A∗ B ⎪ ⎪ ⎪ ⎪ e d r φ r r + r φ ( ) AB3 a a ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¨ ⎬ ⎨ ∗ AB a −ik·a2 2 A B d rφa (r) φa r + rAB3 +e tk = Vsub ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¨ ⎪ ⎪ ⎪ ⎪ ∗ ⎪ − ik · a 2 A B 3 ⎭ ⎩ +e d rφa (r) φa r + rAB3 ⎪ ¨ ∗ a eik·0 + e−ik·a2 + e−ik·a3 d2 rφaA (r) φaB r + rAB3 = Vsub = 1 + e−ik·a2 + e−ik·a3 tkAB3 = 1 + e−ik·a2 + e−ik·a3 t
(4.43)
Here the tkAB3 hopping integral is defined as ¨
t
= tkAB3
a = Vsub
∗
d2 rφaA (r) φaB r + rAB3
(4.44)
where the superscript information was dropped finally, and the k index for electron momentum is dropped because it is absent in the integral. Short point about the orbital assignments in (4.42), and their prior use in (4.41). If we consider the basic orbital, as origin based, and unshifted,
59
Evaluation of the matrix elements for the tight-binding formulation
then it is denoted by φaB (r). Its form would be precisely that given by (3.48) as ϕn=k+l=2,l=1,m=0 (r) which could be assigned as φaB (r) = ϕn=k+l=2,l=1,m=0 (r)
Then the orbital at location of B3 atom must be φaB3 (r) = φaB (r) shifted by r
AB3
= φaB r + rAB3 = φaB3 ;ref r + rAB3
Defining the sum of the phase shifts for the positive Bravais lattice appearing in the factor hitting the basic hopping integral (Fuchs & Goerbig, 2008), as γk = 1 + eik·a2 + eik·a3
(4.45)
total nearest neighbor hopping integral from (4.41) becomes tkAB = γk∗ t
(4.46)
What about the BA hopping integral tkBA ? Consider taking tkAB from (4.41) and reversing the A and B indices, giving the integral a tkBA = Vsub
eik·Rlat
Rlat (nn atoms B3 ,B1 , B2 ; counterclockwise) a = Vsub
e
ik·Rlat
Rlat (nn atoms B3 ,B1 , B2 ; counterclockwise) a = Vsub
eik·Rlat
¨ ¨ ¨
Rlat (nn atoms B3 ,B1 , B2 ; counterclockwise) a = Vsub
Rlat =−Rlat (nn atoms B3 ,B1 ,B2 ; counterclockwise)
e
∗
d2 rφaB (r) φaA r − Rlat − rB3 A
−ik·Rlat
¨
∗
d2 r φaB r + Rlat − rB3 A φaA r
∗
d2 r φaA r φaB r + Rlat − rB3 A
∗
d2 r φaA r φaB r + −Rlat + rAB3
⎧ ⎫∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¨ ⎨ ⎬ ∗ a −ik·Rlat 2 A B = Vsub e d r φa r φa r − Rlat − rAB3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rlat =0,−a2 ,−a3 ⎪ ⎪ ⎪ ⎪ (nn atoms B3 ,B1 ,B2 ; ⎩ ⎭ counterclockwise) ∗ = tkAB (4.47)
60
C.M. Krowne
The result follows from the change of variable,
r = r − Rlat − rB3 A ;
r = r + Rlat − rB3 A ;
d2 r = d2 r
(4.48a)
with rB3 A = −rAB3
(4.48b)
recognizing that the nearest neighbor lattice vectors to arrive at B sites from A should be the same set no matter how labeled, Rlat or Rlat . Nearest neighbor overlap integral matrix elements can be found from (4.24), and related to hopping matrix elements: sijk
=
e
ik·Rlat 3
¨
∗
d2 rφai (r) φaj r − Rlat 3 − rij
(4.49)
Rlat 3
=
tkij a V =1
Thus, setting i = A, j = B in (4.49),
AB sAB k = tk V a =1 =
⎧ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ Rlat =0,−a2 ,−a3 ⎪ ⎪ ⎩ (nn atoms B3 ,B1 ,B2 ; counterclockwise)
eik·Rlat
⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭
t|V a =1 = γk∗ t|V a =1 = γk∗ s
(4.50) What about the BA overlap integral sBA k ? By (4.49),
∗
BA AB sBA k = tk V a =1 = tk
V a =1
∗ ∗ = tkAB V a =1 = sAB k
(4.51)
4.4. Next nearest neighbor hopping and overlap integrals Next nearest neighboring sites to the A atom, are the six A atoms located at 60° increments, and located by the direct space lattice vectors, v1nnn , v2nnn , v3nnn , v4nnn , v5nnn , and v6nnn (Fig. 2.4). They are labeled counterclockwise, and the first three v1nnn , v2nnn , v3nnn are given by, respectively, (2.59), (2.60), and (2.61). Again, (4.25) needs to be evaluated for the hop-
61
Evaluation of the matrix elements for the tight-binding formulation
Figure 4.3 Next nearest neighbor (nnn) atom locations in 2D direct space lattice. Central atom A acts as the reference atom. From it, all the other six next nearest A atoms Ai are found by translations equal to one of the three Bravais unit cell direct space lattice vectors a i , i = 1, 2, 3. A4 = A1¯ , A5 = A2¯ , A6 = A3¯ , a 3 = a 2 − a 1 .
ping integral, but this time for the next nearest neighbors. In relation to the A atom, next nearest A atoms are found for atoms A1 , A2 , and A3 , respectively, by direct space lattice vector shifts a1 , a2 , and a3 . For atoms A4 = A1¯ , A5 = A2¯ , and A6 = A3¯ , direct space lattice vector shifts are −a1 , −a2 , and −a3 (see Fig. 4.3).
a tkAA = Vsub
eik·Rlat
¨
∗
A
d2 rφaA (r) φa j r − Rlat − rAAj
Rlat =a1 (j=1),a2 (j=2),a3 (j=3), −a1 (j=1¯ ),−a2 (j=2¯ ),−a3 (j=3¯ ) (nnn atoms A1 ,A2 ,A3 , A4 ,A5 ,A6 ; counterclockwise) a = Vsub
3¯
e
ik·Rlat
¨
∗
A
∗
A
d2 rφaA (r) φa j r − Rlat − rAAj
j=1 Rlat =aj (nnn atoms A1 , A2 ,A3 ,A4 ,A5 ,A6 ; counterclockwise) a = Vsub
3¯
eik·Rlat
¨
d2 rφaA (r) φa j r − aj − rAAj
j=1 Rlat =aj (nnn atoms A1 , A2 ,A3 ,A4 ,A5 ,A6 ; counterclockwise) a = Vsub
3¯
e
ik·aj
¨
∗
A
∗
A
j=1 a = Vsub
3¯ j=1
eik·aj
d2 rφaA (r) φa j r − aj − aj ¨
d2 rφaA (r) φa j (r)
62
C.M. Krowne
a = Vsub
⎛ =⎝
3¯ j=1
3¯
eik·aj
¨
∗
d2 rφaA (r) φaA r − aj
⎞
AA eik·aj ⎠ tnnn
(4.52)
j=1 AA between the central A atom and the Here the bare hopping integral tnnn outlying six Aj atoms (no phase factors) is defined as
¨
∗
AA a tnnn = Vsub
d2 rφaA (r) φaA r − aj
(4.53)
This integral is actually independent of the index j because all next nearest neighboring A atoms are the same magnitude of distance away from the central A atom. ⎛ ⎛ ⎞ 3¯ AA =⎝ tkAA = ⎝ eik·aj ⎠ tnnn ⎛ =⎝
=⎝
eik·aj +
eik·aj +
=⎝
=⎝
⎞
AA eik·aj ⎠ tnnn
⎞
AA eik·aj ⎠ tnnn
j=−1,−2,−3
eik·aj +
j=1,2,3
⎛
j=1¯ ,2¯ ,3¯
j=1,2,3
⎛
AA eik·aj ⎠ tnnn
j=1,2,3,1¯ ,2¯ ,3¯
j=1
j=1,2,3
⎛
⎞
⎞
AA e−ik·aj ⎠ tnnn
j=1,2,3
"
(4.54)
⎞
#
AA AA eik·aj + e−ik·aj ⎠ tnnn = 2tnnn
j=1,2,3
3
cos(k · aj )
j=1
It is possible to write the relationship (4.54) in another way, noting the previous definition of γk in (4.45). For its magnitude squared, |γk |2 = γk γk∗ ∗ = 1 + eik·a2 + eik·a3 1 + eik·a2 + eik·a3 =3+2
3 j=1
cos(k · aj )
(4.55)
63
Evaluation of the matrix elements for the tight-binding formulation
Using the last result in (4.54), the complete hopping integral (with phase factors) is AA tkAA = 2tnnn
3
cos(k · aj )
j=1
(4.56)
AA 1 |γk |2 − 3 = 2tnnn 2 AA |γk |2 − 3 = tnnn
One could have just as easily worked with the B atom sites, instead of the A atom sites. In this case, tkBB
a = Vsub
e
ik·Rlat
¨
∗
B
d2 rφaB (r) φa j r − Rlat − rBBj
Rlat =a1 (j=1),a2 (j=2),a3 (j=3), −a1 (j=1¯ ),−a2 (j=2¯ ),−a3 (j=3¯ ) (nnn atoms B1 ,B2 ,B3 , B4 ,B5 ,B6 ; counterclockwise) a = Vsub
3¯
eik·Rlat
¨
∗
B
∗
B
d2 rφaB (r) φa j r − Rlat − rBBj
j=1 Rlat =aj (nnn atoms B1 , B 2 ,B 3 ,B 4 ,B 5 ,B 6 ; counterclockwise) a = Vsub
3¯
e
ik·Rlat
¨
d2 rφaB (r) φa j r − aj − rBBj
j=1 Rlat =aj (nnn atoms B1 , B 2 ,B 3 ,B 4 ,B 5 ,B 6 ; counterclockwise) a = Vsub
3¯
eik·aj
¨
∗
d2 rφaA (r) φaA r − aj
j=1
⎛ ⎞ 3¯ AA =⎝ eik·aj ⎠ tnnn j=1
= tkAA
(4.57)
with ¨ BB tnnn
a = Vsub a = Vsub AA = tnnn
¨
∗
∗
d2 rφaB (r) φaB r − aj
d2 rφaA (r) φaA r − aj
(4.58)
64
C.M. Krowne
Eq. (4.58) allows one to define a next nearest neighbor bare hopping integral strength, AA BB = tnnn tnnn = tnnn
(4.59)
Because the overlap integrals arise from the hopping integrals by driving the perturbing potential energy to unity, as seen in (4.49), using (4.57),
BB AA AA sBB k = tk V a =1 = tk V a =1 = sk
(4.60)
References Fuchs, J.-N., & Goerbig, M. O. (2008). Introduction to physical properties of graphene, lecture notes. Julian, M. M. (2008). Foundations of crystallography with computer applications. Boca Raton, FL: CRC Press. 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.
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. Blatt, F. J. (1968). Physics of electronic conduction in solids. New York: McGraw-Hill. Butcher, P. N., & Fawcett, W. (1966). Calculation of the velocity-field characteristics for gallium-arsenide. Physics Letters, 21, 489. Cohen-Tannoudji, C., Diu, B., & Laloe, F. (1977). Quantum mechanics. Wiley. Dwight, H. B. (1961). Tables of integrals and other mathematical data (4th ed.). New York: The Macmillan Co. 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. 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.
Evaluation of the matrix elements for the tight-binding formulation
65
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. 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. 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]. Saito, R., Dresselhaus, G., & Dresselhaus, M. S. (1998). Physical properties of carbon nanotubes. Imperial College Press. 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. Van Hove, L. (1953). The occurrence of singularities in the elastic frequency distribution of a crystal. Physical Review, 89, 1189–1193.
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Wallace, P. R. (1947). The band theory of graphite. Physical Review, 71, 622. Erratum: Physical Review, 72, 258. 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.
Evaluation of the matrix elements for the tight-binding formulation 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 4.1. Determination of an arbitrary Hamiltonian and self-matrix elements 4.2. Secular equation of the system using the Hamiltonian 4.3. Nearest neighbor hopping and overlap integrals 4.4. Next nearest neighbor hopping and overlap integrals References Further reading
47 54 55 60 64 64
4.1. Determination of an arbitrary Hamiltonian and self-matrix elements From the last Section 3.4 of Chapter 3, solving the tight-binding problem amounts to finding the integrated matrix elements for both the Hamiltonian and self, shown in (3.85). Orbital wavefunctions are required for the lsub = i and lsub = j sublattices within the Bravais unit cell, obtainable from (3.33) as ψki (r) =
Rlat
j
ψk (r) =
eik·Rlat φai r − Rlat + ri eik·Rlat φaj r − Rlat + rj
(4.1a) (4.1b)
Rlat
Note that in order to avoid index confusion, summation over lattice positional vectors Rlat has been changed from i to lat, making Ri → Rlat . For Advances in Imaging and Electron Physics, Volume 210 ISSN 1076-5670 https://doi.org/10.1016/bs.aiep.2019.01.004
Copyright © 2019 Elsevier Inc. All rights reserved.
47
48
C.M. Krowne
the Hamiltonian matrix element, utilizing (3.83a) in (3.84a), and (4.1), ij H¯ k =
¨
d2 rHkij (r)
¨
∗
d2 rψki (r)H (r)ψkj (r)
=
⎡
¨
d2 r ⎣
=
⎤∗ eik·Rlat 1 φa r − Rlat 1 + ri ⎦ i
R
× H (r)
lat 1
=
d2 r
∗
e−ik·Rlat 1 φai r − Rlat 1 + ri
Rlat 1
× H (r)
eik·Rlat 2 φaj r − Rlat 2 + rj
Rlat 2
¨
eik·Rlat 2 φaj r − Rlat 2 + rj
(4.2)
Rlat 2
Pulling the lattice summations outside of the double integral, ij H¯ k =
¨
∗
d2 re−ik·Rlat 1 φai r − Rlat 1 + ri
Rlat 1 Rlat 2
× H (r)eik·Rlat 2 φaj r − Rlat 2 + rj ∗ ¨ = d2 re−ik· Rlat 1 −Rlat 2 φai r − Rlat 1 + ri Rlat 1 Rlat 2
× H (r)φaj r − Rlat 2 + rj ¨ ∗ = d2 reik·Rlat 3 φai r − Rlat 1 + ri H (r)φaj r − Rlat 2 + rj Rlat 1 Rlat 2
=
¨
∗
∗
Rlat 1 Rlat 2
= =
¨
1
⎭ Rlat 3 ¨
∗
Rlat 3
eik·Rlat 3
¨
∗
∗
d2 r eik·Rlat 3 φai r H (r + ri )φaj r − Rlat 3 − rij
Rlat 3
= Nsys
d2 r eik·Rlat 3 φai r H (r + ri )φaj r − Rlat 3 − rij
Rlat 1
= Nsys
d2 r eik·Rlat 3 φai r H (r + ri )φaj r − Rlat 3 − rij
Rlat 1 Rlat 2 ⎧ ⎫ ⎨ ⎬ ¨
⎩
d2 r eik·Rlat 3 φai r H (r + Rlat 1 + ri )φaj r − Rlat 3 − rij
d2 r φai r H (r + ri )φaj r − Rlat 3 − rij
(4.3)
49
Evaluation of the matrix elements for the tight-binding formulation
Here a number of procedures were employed to arrive at (3.47). First the exponential terms were collected together into one factor. Next a third lattice position was defined, integration variable shifted, and the inner summation changed: Rlat 3 = − (Rlat 1 − Rlat 2 )
(4.4a)
rij = ri + rj
(4.4b)
r = r − Rlat 1 + ri
(4.4c)
dr = dr − d Rlat 1 + ri = dr
Rlat 1 Rlat 2
→
Rlat 1 Rlat 1 +Rlat 3
→
(4.4d)
(4.4e)
Rlat 1 Rlat 3
Inner summation variable can drop Rlat 1 since the same range is covered, namely all of the lattice points available by using the Bravais unit cell vectors. Last three lines in (4.3) arose because the Hamiltonian (Kane, 1982) must be periodic in Bravais lattice positional vectors (Julian, 2008), and the integral has no Rlat 1 dependence, leading to the factor of Nsys atoms in the system being considered. Finally, the primed spatial integral variable r , can be relabeled r as there no other integral variables. ij H¯ k = Nsys
eik·Rlat 3
¨
∗
d2 rφai (r) H (r + ri )φaj r − Rlat 3 − rij
(4.5)
Rlat 3
Hamiltonian must be examined. By (3.6), for an electron associated with lattice position Rlat , its atomic orbital description is fairly accurate, calling its vector in reference to that local origin r¯ lat . a (¯rlat ) = − Hlat
h¯ 2 2 ∇ + V a (¯rlat ) 2me r¯lat
(4.6)
In order to handle electrons localized about any lattice site Rlat , r¯ lat must be generalized to r¯ lat = rlat − Rlat . a a Hlat (rlat ) = Hlat (¯rlat ) = −
h¯ 2 2 ∇ + V a (rlat − Rlat ) 2me rlat
(4.7)
In assessing the overlap between orbitals in Chapter 3, Section 3.3, atomic hydrogenic solutions were studied. Here V a (¯rlat ) is the atomic solution for the electron attached to the lat lattice location. There are as we just mentioned, Nsys such locations, so all the other orbital locations will perturb or
50
C.M. Krowne
modify the fairly accurate atomic solution. a Hlat 1 (rlat 1 ) = Hlat 1 (rlat 1 ) +
Nsys
V a (rlat 1 − Rlat 2 )
(4.8)
lat 2=1 lat 2=lat 1
Total Hamiltonian for the entire system must consist of a sum over all the system electrons, and this would then be for Zv valence mobile electrons per atom for a single type of site atom like graphene, where Zv < Z, where Z is the positive core charge, or the atomic number of the element. Graphene has Zv = 1. For multiple atom types, causing several sublattice atom types, or several atoms per Bravais unit cell, the relation is more complicated requiring a summation over sublattices types:
Nel =
Zv Nsys ; NT ,sub = 1 NT ,sub i=1 Zv,subi Nsys,subi ; NT ,sub = 1
(4.9)
Total number of sublattices is NT ,sub = Nac
(4.10)
where from earlier this number must be the atom count in the Bravais unit cell Nac . Total number of atoms in the system must be those in each sublattice group within the system, summed up, Nsys =
Nac
Nsys,subi
(4.11)
i=1
Although the kinetic part of each atomic Hamiltonian is the same as in (4.7), potential energy part may be different, and this is allowed for by a indexing V a with subi , giving Vsub . Lattice location Rlat;subi also now has i subi associated with it: a Hlat ;subi (rlat;subi ) = −
h¯ 2 2 a ∇ + Vsub (rlat;subi − Rlat;subi ) i 2me rlat ;subi
(4.12)
Upgraded atomic Hamiltonian at lattice location lat 1 in sublattice subi , lat 1; subi , Rlat 1:subi , taking again into account all the other orbital locations which perturb or modify the fairly accurate atomic solution, whether in
Evaluation of the matrix elements for the tight-binding formulation
51
the same sublattice or in another sublattice, is Hlat 1;subi (rlat 1;subi ) a = Hsub (rlat 1;subi ) + i
Nac
Nsys;subj
j=1
lat 2;subj =1 lat 2;subj =lat 1;subi
a Vsub (rlat 1;subi − Rlat 2:subj ) i
(4.13)
Here rlat 1;subi is the actual position in space being considered, r, referenced to the particular sublattice atom location ri , and the particular lattice location shifted by a Bravais unit cell lattice vector rlat 1;subi = r − ri − Rlat 1
(4.14)
Substituting (4.14) into (4.13), one obtains ca ca a Hlat 1;subi (r) = Hsubi (r − ri − Rlat 1 ) = Hsubi (r − ri − Rlat 1 )
+
Nac
Nsys;subj
j=1
lat 2;subj =1 lat 2;subj =lat 1;subi
a Vsub (r − ri − Rlat 1 − Rlat 2:subj ) i
(4.15)
Total Hamiltonian for the entire system is the sum of the individual corrected or perturbed Hamiltonians Hlat 1;subi (r) from (4.15) over all the lattice locations (lat 1) for all of the sublattices (subi ), Rlat 1:subi , H (r) =
Nac
Nsys,subi
i=1 lat 1;subi =1
ca Hlat 1;subi (r)
(4.16)
Considering second term as a perturbation in (4.15), small enough to isolate and label as a Vsub (r − ri i
− Rlat 1 ) =
Nac
Nsys;subj
j=1
lat 2;subj =1 lat 2;subj =lat 1;subi
a Vsub (r − ri − Rlat 1 − Rlat 2:subj ) i
(4.17) the expression in (4.15) becomes ca a a Hlat 1;subi (r) = Hsubi (r − ri − Rlat 1 ) + Vsubi (r − ri − Rlat 1 )
(4.18)
52
C.M. Krowne
Placing (4.18) in (4.16), to obtain the system Hamiltonian, H (r) =
Nsys,subi
Nac
i=1 lat 1;subi =1
a a Hsub (r − ri − Rlat 1 ) + Vsub (r − ri − Rlat 1 ) i i
(4.19)
Once the system Hamiltonian is available, as it is from (4.19), the ijth matrix element of it can be found from (4.5), by inserting (4.19), being careful to change the indexing i to n to avoid confusion with the orbital indexes i and j. ij H¯ k = Nsys
eik·Rlat 3
Rlat 3
= Nsys
eik·Rlat 3
¨
∗
d2 rφai (r) H (r + ri )φaj r − Rlat 3 − rij ¨
∗
d2 rφai (r)
Rlat 3
×
Nac
Nsys,subn
i=n lat 1;subn =1
a a Hsub (r − Rlat 1 ) + Vsub (r − Rlat 1 ) n n
× φaj r − Rlat 3 − rij ¨ ∗ ik·Rlat 3 e d2 rφai (r) = Nsys
(4.20)
Rlat 3
×
sys,subn Nac N
n=1 lat 1;subn =1
+ Nsys
e
a Hsub (r − Rlat 1 )φaj r − Rlat 3 − rij n
ik·Rlat 3
¨
∗
d2 rφai (r)
Rlat 3
×
sys,subn Nac N
a Vsub (r − Rlat 1 )φaj r − Rlat 3 − rij n
n=1 lat 1;subn =1
If it is argued that the projection of the total atomic Hamiltonian onto the jth sublattice orbital wavefunction selects out that orbital and hits it with an energy eigenvalue characteristic of it, in the first integral of (4.20), then ⎧ Nac ⎨ ⎩
Nsys,subn
n=1 lat 1;subn =1
⎫ ⎬ a Hsub ( r − R ) φaj r − Rlat 3 − rij ≈ ε j φaj r − Rlat 3 − rij lat 1 n ⎭
(4.21)
53
Evaluation of the matrix elements for the tight-binding formulation
Here we see that Bravais lattice vector Rlat 1 goes through its range until it finds the optimum one equaling Rlat 3 . This is when the local atomic Hamiltonian coincides with its local orbital. Eq. (4.21) forms a reasonable approximation. In doing this, the small misalignment between the Hamil tonian argument r − Rlat 1 and the orbital argument r − Rlat 3 − rij , when Rlat 1 = Rlat 3 , rij is ignored. For the potential energy term, second integral term of (4.21) ⎫ ⎬ a Vsub ( r − R ) φaj r − Rlat 3 − rij lat 1 n ⎭ ⎩ n=1 lat 1;subn =1 a φ j r − Rlat 3 − rij = Vsub j a ⎧ Nac ⎨
Nsys,subn
(4.22)
Once again, it is seen that the Bravais lattice vector Rlat 1 goes through its range until it finds the optimum one equaling Rlat 3 . This is when the local atomic potential energy coincides with its local orbital. Eq. (4.22) forms a reasonable approximation. In doing this, the small misalignment between the potential energy argument r − Rlat 1 and the orbital argument r − Rlat 3 − rij , when Rlat 1 = Rlat 3 , rij is ignored. Take the results of (4.21) and (4.22) and set them into (4.20), to obtain the H¯ kij matrix element ij H¯ k = Nsys
eik·Rlat 3
¨
Rlat 3
+ Nsys
eik·Rlat 3
Rlat 3
= Nsys ε
j
e
ik·Rlat 3
Rlat 3 a + Nsys Vsub j
∗
d2 rφai (r) εj φaj r − Rlat 3 − rij ¨
∗
a d2 rφai (r) Vsub φ j r − Rlat 3 − rij j a
¨
∗
d2 rφai (r) φaj r − Rlat 3 − rij e
ik·Rlat 3
¨
∗
(4.23)
d2 rφai (r) φaj r − Rlat 3 − rij
Rlat 3
ij ij = Nsys ε j sk + tk
Here sijk and tkij are defined by sijk =
eik·Rlat 3
Rlat 3 a tkij = Vsub j
Rlat 3
¨
∗
d2 rφai (r) φaj r − Rlat 3 − rij eik·Rlat 3
¨
∗
d2 rφai (r) φaj r − Rlat 3 − rij
(4.24)
(4.25)
54
C.M. Krowne
4.2. Secular equation of the system using the Hamiltonian Recall from (3.76) that we have a descriptive equation determining the system behavior, and stating it again here, we have det H¯ k − εk S¯k = 0
(3.76)
Recognize the self-matrix term for the system is S¯k = Nsys sk
(4.26)
where elements of sk are given by (4.24). System Hamiltonian matrix elements (4.23) may be expressed in matrix form as H¯ k = Nsys (sk ε + tk )
(4.27)
with ε being considered the diagonal matrix, ⎡ ⎢ ⎢ ε=⎢ ⎢ ⎣
ε1
0
0
ε2
.. .
0
⎤
···
0 0 ⎥ ⎥
0 ..
0 0
⎥ ⎥
0 ⎦
.
0
(4.28)
ε Nac
With this information, the secular equation of the system (3.76) may be written in matrix form, utilizing the Hamiltonian matrix form of (4.27), det Nsys (sk ε + tk ) − εk Nsys sk = 0
(4.29)
Factoring out Nsys , and rearranging terms
Nsys det tk − sk εkλ − ε
=0
(4.30)
Dividing out the total system number of electrons, leaves the final secular equation form, det tk − sk εkλ − ε = 0
(4.31)
where the explicitly branch index λ on the eigenenergy solution of the system εkλ , is inserted. This eigenenergy value is a number. Here tk is the
55
Evaluation of the matrix elements for the tight-binding formulation
hopping matrix, sk the overlap matrix. Postfactor of sk in (4.31) in general looks like ⎡
1 ⎢ ⎢ 0 λ
0 1
0
0 0
ε k − ε = εk ⎢ ⎢ .. ⎣ . λ
⎡ ⎢ ⎢ =⎢ ⎢ ⎣
εkλ − ε 1
0 .. .
···
0 ..
.
0
⎡
⎤
0 0 ⎥ ⎥
⎢ ⎢ ⎥−⎢ ⎥ ⎢ 0 ⎦ ⎣
1 ···
εkλ − ε 2
0
0
0
0
ε2
.. .
..
0 ..
.
0
⎤
0 0 ⎥ ⎥ ⎥ ⎥
0 ⎦ ε Nac
(4.32)
⎤
0 0
.
0
···
0 0
0
0 0 0
ε1
⎥ ⎥ ⎥ ⎥ ⎦
0 εk − ε Nac λ
For the case where the onsite energies εj are all the same for the various sublattice sites j, the sk postfactor matrix εkλ − ε reduces to a number, or in matrix form, εkλ − ε = εkλ − ε0 I
(4.33)
with I being the identity matrix. Clearly, (4.33) applies to the case of graphene where sublattice sites A and B have carbon atoms. Secular equation then reduces to det tk − εkλ − ε0 sk = 0
(4.34)
4.3. Nearest neighbor hopping and overlap integrals Integral to be evaluated for hopping integral is (4.25), and in relation to an A atom site, nearest neighboring atoms are three B atoms (see Fig. 4.1), located by the direct space lattice vectors v1nn , v2nn , and v3nn , given respectively by Eqs. (2.54), (2.56), and (2.57). That equation is a tkij = Vsub j
Rlat 3
eik·Rlat 3
¨
∗
d2 rφai (r) φaj r − Rlat 3 − rij
(4.25)
56
C.M. Krowne
Figure 4.1 Nearest neighbor (nn) atom locations in 2D direct space lattice. a i , i = 1, 2, 3 are the Bravais unit cell direct space lattice vectors; a 3 = a 2 − a 1 . Nearest neighboring atoms Bi are shown. Distance between the reference A atom and the B3 atom is rB3 A . Other Bj atom locations are found by a i shifts.
a , the integral beSetting i = A, j = B, and dropping the j index on Vsub j comes, noting the summation is over B nearest neighbors
a tkAB = Vsub
eik·Rlat
¨
∗
B
d2 rφaA (r) φa j r − Rlat − rAB3
Rlat =0(j=3),−a2 (j=1), −a3 (j=2) (nn atoms B3 ,B1 , B2 ; counterclockwise)
(4.35) where the choice of lattice vector with respect to the B3 atom at position vertically below the reference A by a carbon–carbon spacing aCC , −aCC yˆ , (2.57) makes Rlat = 0, −a2 , −a3
(4.36)
This shifts the B atom orbital at atom B3 (Rlat = 0) to that at B1 by (see Fig. 4.1) − Rlat − rAB3 − = −Rlat + rAB3 = a2 + rAB3
(4.37)
For shifting to the B2 atom, from the B3 (Rlat = 0) atom, an amount of − Rlat − rAB3 − = −Rlat + rAB3 = a3 + rAB3
is required.
(4.38)
57
Evaluation of the matrix elements for the tight-binding formulation
Figure 4.2 Orbital shifting effects in 2D direct space with reference to an origin. Top diagram for any two adjacent A and B atoms. Bottom diagram for a central reference A atom and a B atom located directly below it (a B3 atom). The hydrogenic atom orbitals l
=A
l
=Bj
l
=A
are for the top diagram are φasub (rm(ref ) ) at atom A, and φasub (riB ) = φasub ref (r − R lat − rBi Aref ) at atom B. For bottom diagram, setting i, m(ref ) → Bi=3 Aref the orbital l =Bj=3 about the particular B3 atom becomes φasub (riB ) = φalsub =Aref (r − R lat − rB A ). i=3 ref
Here a3 is a sublattice vector obtained by taking the difference of the two Bravais sublattice vectors a2 and a1 , a 3 = a2 − a1
(4.39)
Eqs. (2.31) provide a2 and a1 , so that a3 is easily determined, √
a3 =
√ √ √ √ 3aCC 3aCC xˆ + 3yˆ − 3aCC xˆ = −xˆ + 3yˆ 2 2
(4.40)
Reason why a1 does not appear in (4.36) is that B3 atom acts as the reference and the other two atoms are located with both vertical and horizontal offsets, precluding use of a sublattice vector with a pure horizontal offset. Expanding tkAB by writing out the summation, using the shifting information contained in (4.36)–(4.38), with the shifting process shown in Fig. 4.2,
58
C.M. Krowne
tkAB
a = Vsub
e
ik·Rlat
¨
∗
B
d2 rφaA (r) φa j r − Rlat − rAB3
Rlat =0(j=3),−a2 (j=1), −a3 (j=2) (nn atoms B3 ,B1 , B2 ; counterclockwise)
¨ ⎫ ⎧ ik·0 2 A∗ B3 ;ref ⎪ ⎪ ⎪ ⎪ e d r φ r r + r φ ( ) AB3 ⎪ ⎪ a a ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¨ ⎬ ⎨ ∗ a −ik·a2 2 A B1 d rφa (r) φa r + a2 + rAB3 +e = Vsub ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¨ ⎪ ⎪ ⎪ ⎪ ∗ ⎪ 2 A B2 ⎭ ⎩ + e−ik·a3 d rφa (r) φa r + a3 + rAB3 ⎪
(4.41) Now we know that the orbitals in A and B atoms are the same for graphene. Furthermore, expecting the orbital environments of the wavefunctions at B1 , B2 , and B3 to be the same, that is φaB1 r + a2 + rAB3 = φaB3 ;ref φaB2 r + a3 + rAB3 = φaB3 ;ref
r + rAB3 r + rAB3
(4.42)
because shifts of Bravais lattice vectors occur, (4.41) simplifies to ¨ ⎫ ⎧ ik·0 2 A∗ B ⎪ ⎪ ⎪ ⎪ e d r φ r r + r φ ( ) AB3 a a ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¨ ⎬ ⎨ ∗ AB a −ik·a2 2 A B d rφa (r) φa r + rAB3 +e tk = Vsub ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¨ ⎪ ⎪ ⎪ ⎪ ∗ ⎪ − ik · a 2 A B 3 ⎭ ⎩ +e d rφa (r) φa r + rAB3 ⎪ ¨ ∗ a eik·0 + e−ik·a2 + e−ik·a3 d2 rφaA (r) φaB r + rAB3 = Vsub = 1 + e−ik·a2 + e−ik·a3 tkAB3 = 1 + e−ik·a2 + e−ik·a3 t
(4.43)
Here the tkAB3 hopping integral is defined as ¨
t
= tkAB3
a = Vsub
∗
d2 rφaA (r) φaB r + rAB3
(4.44)
where the superscript information was dropped finally, and the k index for electron momentum is dropped because it is absent in the integral. Short point about the orbital assignments in (4.42), and their prior use in (4.41). If we consider the basic orbital, as origin based, and unshifted,
59
Evaluation of the matrix elements for the tight-binding formulation
then it is denoted by φaB (r). Its form would be precisely that given by (3.48) as ϕn=k+l=2,l=1,m=0 (r) which could be assigned as φaB (r) = ϕn=k+l=2,l=1,m=0 (r)
Then the orbital at location of B3 atom must be φaB3 (r) = φaB (r) shifted by r
AB3
= φaB r + rAB3 = φaB3 ;ref r + rAB3
Defining the sum of the phase shifts for the positive Bravais lattice appearing in the factor hitting the basic hopping integral (Fuchs & Goerbig, 2008), as γk = 1 + eik·a2 + eik·a3
(4.45)
total nearest neighbor hopping integral from (4.41) becomes tkAB = γk∗ t
(4.46)
What about the BA hopping integral tkBA ? Consider taking tkAB from (4.41) and reversing the A and B indices, giving the integral a tkBA = Vsub
eik·Rlat
Rlat (nn atoms B3 ,B1 , B2 ; counterclockwise) a = Vsub
e
ik·Rlat
Rlat (nn atoms B3 ,B1 , B2 ; counterclockwise) a = Vsub
eik·Rlat
¨ ¨ ¨
Rlat (nn atoms B3 ,B1 , B2 ; counterclockwise) a = Vsub
Rlat =−Rlat (nn atoms B3 ,B1 ,B2 ; counterclockwise)
e
∗
d2 rφaB (r) φaA r − Rlat − rB3 A
−ik·Rlat
¨
∗
d2 r φaB r + Rlat − rB3 A φaA r
∗
d2 r φaA r φaB r + Rlat − rB3 A
∗
d2 r φaA r φaB r + −Rlat + rAB3
⎧ ⎫∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¨ ⎨ ⎬ ∗ a −ik·Rlat 2 A B = Vsub e d r φa r φa r − Rlat − rAB3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rlat =0,−a2 ,−a3 ⎪ ⎪ ⎪ ⎪ (nn atoms B3 ,B1 ,B2 ; ⎩ ⎭ counterclockwise) ∗ = tkAB (4.47)
60
C.M. Krowne
The result follows from the change of variable,
r = r − Rlat − rB3 A ;
r = r + Rlat − rB3 A ;
d2 r = d2 r
(4.48a)
with rB3 A = −rAB3
(4.48b)
recognizing that the nearest neighbor lattice vectors to arrive at B sites from A should be the same set no matter how labeled, Rlat or Rlat . Nearest neighbor overlap integral matrix elements can be found from (4.24), and related to hopping matrix elements: sijk
=
e
ik·Rlat 3
¨
∗
d2 rφai (r) φaj r − Rlat 3 − rij
(4.49)
Rlat 3
=
tkij a V =1
Thus, setting i = A, j = B in (4.49),
AB sAB k = tk V a =1 =
⎧ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ Rlat =0,−a2 ,−a3 ⎪ ⎪ ⎩ (nn atoms B3 ,B1 ,B2 ; counterclockwise)
eik·Rlat
⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭
t|V a =1 = γk∗ t|V a =1 = γk∗ s
(4.50) What about the BA overlap integral sBA k ? By (4.49),
∗
BA AB sBA k = tk V a =1 = tk
V a =1
∗ ∗ = tkAB V a =1 = sAB k
(4.51)
4.4. Next nearest neighbor hopping and overlap integrals Next nearest neighboring sites to the A atom, are the six A atoms located at 60° increments, and located by the direct space lattice vectors, v1nnn , v2nnn , v3nnn , v4nnn , v5nnn , and v6nnn (Fig. 2.4). They are labeled counterclockwise, and the first three v1nnn , v2nnn , v3nnn are given by, respectively, (2.59), (2.60), and (2.61). Again, (4.25) needs to be evaluated for the hop-
61
Evaluation of the matrix elements for the tight-binding formulation
Figure 4.3 Next nearest neighbor (nnn) atom locations in 2D direct space lattice. Central atom A acts as the reference atom. From it, all the other six next nearest A atoms Ai are found by translations equal to one of the three Bravais unit cell direct space lattice vectors a i , i = 1, 2, 3. A4 = A1¯ , A5 = A2¯ , A6 = A3¯ , a 3 = a 2 − a 1 .
ping integral, but this time for the next nearest neighbors. In relation to the A atom, next nearest A atoms are found for atoms A1 , A2 , and A3 , respectively, by direct space lattice vector shifts a1 , a2 , and a3 . For atoms A4 = A1¯ , A5 = A2¯ , and A6 = A3¯ , direct space lattice vector shifts are −a1 , −a2 , and −a3 (see Fig. 4.3).
a tkAA = Vsub
eik·Rlat
¨
∗
A
d2 rφaA (r) φa j r − Rlat − rAAj
Rlat =a1 (j=1),a2 (j=2),a3 (j=3), −a1 (j=1¯ ),−a2 (j=2¯ ),−a3 (j=3¯ ) (nnn atoms A1 ,A2 ,A3 , A4 ,A5 ,A6 ; counterclockwise) a = Vsub
3¯
e
ik·Rlat
¨
∗
A
∗
A
d2 rφaA (r) φa j r − Rlat − rAAj
j=1 Rlat =aj (nnn atoms A1 , A2 ,A3 ,A4 ,A5 ,A6 ; counterclockwise) a = Vsub
3¯
eik·Rlat
¨
d2 rφaA (r) φa j r − aj − rAAj
j=1 Rlat =aj (nnn atoms A1 , A2 ,A3 ,A4 ,A5 ,A6 ; counterclockwise) a = Vsub
3¯
e
ik·aj
¨
∗
A
∗
A
j=1 a = Vsub
3¯ j=1
eik·aj
d2 rφaA (r) φa j r − aj − aj ¨
d2 rφaA (r) φa j (r)
62
C.M. Krowne
a = Vsub
⎛ =⎝
3¯ j=1
3¯
eik·aj
¨
∗
d2 rφaA (r) φaA r − aj
⎞
AA eik·aj ⎠ tnnn
(4.52)
j=1 AA between the central A atom and the Here the bare hopping integral tnnn outlying six Aj atoms (no phase factors) is defined as
¨
∗
AA a tnnn = Vsub
d2 rφaA (r) φaA r − aj
(4.53)
This integral is actually independent of the index j because all next nearest neighboring A atoms are the same magnitude of distance away from the central A atom. ⎛ ⎛ ⎞ 3¯ AA =⎝ tkAA = ⎝ eik·aj ⎠ tnnn ⎛ =⎝
=⎝
eik·aj +
eik·aj +
=⎝
=⎝
⎞
AA eik·aj ⎠ tnnn
⎞
AA eik·aj ⎠ tnnn
j=−1,−2,−3
eik·aj +
j=1,2,3
⎛
j=1¯ ,2¯ ,3¯
j=1,2,3
⎛
AA eik·aj ⎠ tnnn
j=1,2,3,1¯ ,2¯ ,3¯
j=1
j=1,2,3
⎛
⎞
⎞
AA e−ik·aj ⎠ tnnn
j=1,2,3
"
(4.54)
⎞
#
AA AA eik·aj + e−ik·aj ⎠ tnnn = 2tnnn
j=1,2,3
3
cos(k · aj )
j=1
It is possible to write the relationship (4.54) in another way, noting the previous definition of γk in (4.45). For its magnitude squared, |γk |2 = γk γk∗ ∗ = 1 + eik·a2 + eik·a3 1 + eik·a2 + eik·a3 =3+2
3 j=1
cos(k · aj )
(4.55)
63
Evaluation of the matrix elements for the tight-binding formulation
Using the last result in (4.54), the complete hopping integral (with phase factors) is AA tkAA = 2tnnn
3
cos(k · aj )
j=1
(4.56)
AA 1 |γk |2 − 3 = 2tnnn 2 AA |γk |2 − 3 = tnnn
One could have just as easily worked with the B atom sites, instead of the A atom sites. In this case, tkBB
a = Vsub
e
ik·Rlat
¨
∗
B
d2 rφaB (r) φa j r − Rlat − rBBj
Rlat =a1 (j=1),a2 (j=2),a3 (j=3), −a1 (j=1¯ ),−a2 (j=2¯ ),−a3 (j=3¯ ) (nnn atoms B1 ,B2 ,B3 , B4 ,B5 ,B6 ; counterclockwise) a = Vsub
3¯
eik·Rlat
¨
∗
B
∗
B
d2 rφaB (r) φa j r − Rlat − rBBj
j=1 Rlat =aj (nnn atoms B1 , B 2 ,B 3 ,B 4 ,B 5 ,B 6 ; counterclockwise) a = Vsub
3¯
e
ik·Rlat
¨
d2 rφaB (r) φa j r − aj − rBBj
j=1 Rlat =aj (nnn atoms B1 , B 2 ,B 3 ,B 4 ,B 5 ,B 6 ; counterclockwise) a = Vsub
3¯
eik·aj
¨
∗
d2 rφaA (r) φaA r − aj
j=1
⎛ ⎞ 3¯ AA =⎝ eik·aj ⎠ tnnn j=1
= tkAA
(4.57)
with ¨ BB tnnn
a = Vsub a = Vsub AA = tnnn
¨
∗
∗
d2 rφaB (r) φaB r − aj
d2 rφaA (r) φaA r − aj
(4.58)
64
C.M. Krowne
Eq. (4.58) allows one to define a next nearest neighbor bare hopping integral strength, AA BB = tnnn tnnn = tnnn
(4.59)
Because the overlap integrals arise from the hopping integrals by driving the perturbing potential energy to unity, as seen in (4.49), using (4.57),
BB AA AA sBB k = tk V a =1 = tk V a =1 = sk
(4.60)
References Fuchs, J.-N., & Goerbig, M. O. (2008). Introduction to physical properties of graphene, lecture notes. Julian, M. M. (2008). Foundations of crystallography with computer applications. Boca Raton, FL: CRC Press. 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.
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. Blatt, F. J. (1968). Physics of electronic conduction in solids. New York: McGraw-Hill. Butcher, P. N., & Fawcett, W. (1966). Calculation of the velocity-field characteristics for gallium-arsenide. Physics Letters, 21, 489. Cohen-Tannoudji, C., Diu, B., & Laloe, F. (1977). Quantum mechanics. Wiley. Dwight, H. B. (1961). Tables of integrals and other mathematical data (4th ed.). New York: The Macmillan Co. 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. 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.
Evaluation of the matrix elements for the tight-binding formulation
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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. 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. 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]. Saito, R., Dresselhaus, G., & Dresselhaus, M. S. (1998). Physical properties of carbon nanotubes. Imperial College Press. 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. Van Hove, L. (1953). The occurrence of singularities in the elastic frequency distribution of a crystal. Physical Review, 89, 1189–1193.
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Wallace, P. R. (1947). The band theory of graphite. Physical Review, 71, 622. Erratum: Physical Review, 72, 258. 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.