First principles study of graphene related materials 1

Part I: Introductory material This part consists of three introductory chapters to give the reader sufficient background material to understand the Ph.D work that is reported in the later chapters In Chap 1, a review of the physical and chemical properties of graphene and hexagonal boron nitride is given in order to provide motivation for studying these materials A review of the fabrication methods of these materials is also presented Chap is devoted to the discussion of density-functional theory, which is the workhorse for the computational simulations done in this thesis In Chap 3, the Landauer approach for electronic and thermal transport is discussed, and we show the derivation for the equations that we use to calculate the thermoelectric properties of graphene-related materials in the later chapters Chapter Introduction to graphene-related materials Abstract: In this chapter, we briefly introduce the physical and chemical properties of graphene, graphene nanoribbons and quantum dots, and hexagonal boron nitride 1.1 Physical and chemical properties of graphene Unit cell of graphene – Graphene is composed of a two-dimensional hexagonal network of C atoms, as shown in Fig 1.1(a) The primitive unit cell (PUC) of graphene is made up of two C basis atoms which are labeled A and B Graphene is a bipartite lattice because the atoms A and B can be grouped into two different sublattices where atoms A are only bonded to atoms B, and vice versa 10 The lattice vectors of the primitive unit cell (PUC) are √ a1 = 3acc √ ˆ ˆ x+ y 2 √ and a2 = 3acc √ ˆ ˆ x− y , 2 (1.1) where acc = 1.42 Å is the C–C bond length of graphene The entire structure of graphene can be constructed by translating the PUC in any general direction R = n1 a1 + n2 a2 , where ni ∈ Z The corresponding lattice vectors for the reciprocal-space PUC are 2π b1 = √ 3acc √ x+y ˆ ˆ 2π and b2 = √ 3acc √ x−y , ˆ ˆ (1.2) CHAPTER INTRODUCTION TO GRAPHENE-RELATED MATERIALS b1 (b) a2 a2 Κ' a1 A a1 b2 (a) Μ B Y Γ Κ X Figure 1.1: (a) The atomistic model of the hexagonal network of graphene; the grey spheres represent C atoms The primitive unit cell (PUC) of graphene is denoted by lattice vectors a1 and a2 and is composed of two C basis atoms A and B Assuming a C–C bond length of 1.42 Å, |a1 | = |a2 | = 2.46 Å (b) The corresponding reciprocal lattice vectors b1 and b2 of the real-space unit cell denoted in (a) The corners of the light blue hexagon represent all the reciprocal lattice points closest to origin, and the 1st Brillouin zone in dark blue is constructed from the lines that bisect the vectors connecting the origin to the nearest reciprocal lattice points The letters Γ, M, K, and K’ mark the high-symmetry points of the 1st Brillouin zone which obey the relation · b j = 2πδi j , where δi j is the Kronecker delta Similarly, the entire reciprocal lattice is obtained by translating the reciprocal PUC in any general direction G = m1 b1 + m2 b2 , mi ∈ Z Bonding in graphene – The 2s, 2px , and 2py orbitals of each C atom in graphene hybridize to form sp2 orbitals that lie in the x − y plane and are separated from each other by 120◦ Within the x − y plane of graphene, the C atoms are connected via σ bonds due to the overlap of these sp2 orbitals The remaining 2pz orbitals of the C atoms are oriented perpendicular to the x − y plane, and overlap to form π bonds When one attempts to connect the C atoms in graphene by double bonds to fulfill the octet rule, three equivalent Clar structures 11 can be constructed, as shown in Fig 1.2 Since a benzenoid ring appears in each hexagon ring of graphene when all three Clar structures are superimposed on each other, the π electrons from the 2pz orbitals are fully delocalized over the entire graphene structure The highly delocalized nature of the π electrons lead to high electronic conductance in graphene, as will be discussed later The partial double bond character of the C–C bonds makes graphene mechanically strong: theoretical calculations showed that the Young’s modulus of graphene is 1.05 TPa for uniaxial tensile strain applied along the plane of graphene; 12 experimentally, a break- CHAPTER INTRODUCTION TO GRAPHENE-RELATED MATERIALS = Figure 1.2: The three equivalent Clar structures of graphene ing strength of 42 Nm-1 and Young’s modulus of 1.0 TPa was measured for graphene in the out-of-plane direction 13 This great mechanical strength makes graphene capable of withstanding high electron current densities without structural failure 14 In comparison, 15 Cu has a Young’s modulus of 130 GPa The geometric size of the gap in the middle of each hexagon ring of graphene is estimated to be ∼ 0.064 nm, when the C–C bond length and van der Waals’ radius of the C atoms are assumed to be 0.142 nm and 0.11 nm, respectively Not only is the gap size much smaller than small molecules like He and H2 , but the delocalized π electron cloud also works to repel molecules: 16 this makes graphene ideal to be used as an impermeable gas membrane 17 Electronic band structure and conductance of graphene – The many interesting properties of graphene are due to the unique characteristics of its electronic band structure, which will be derived here using the tight-binding approach 7,18 Since the interesting electronic properties of graphene are due to the electrons in the π-band, and the 2pz orbital does not overlap significantly with the 2s, 2px , and 2py orbitals, we shall use pz atomic orbital functions centered on the A and B basis atoms in the nth PUC of a graphene supercell as the minimal basis set; i.e., pz (r − rA − Rn ) and pz (r − rB − Rn ), (1.3) where Rn is the position vector of the nth PUC in the graphene supercell consisting of N PUCs, and rX is the position vector of basis atom X from the origin of the nth PUC CHAPTER INTRODUCTION TO GRAPHENE-RELATED MATERIALS The electronic wavefunction of the π electrons in graphene can be expressed as Ψ(k, r) = cA (k) pA (k, r) + cB (k) pB (k, r), ˜z ˜z (1.4) pX (k, r) = √ ∑ eik·Rn pz (r − rX − Rn ), ˜z N n (1.5) with the Bloch sums and cX are the coefficients representing the contribution of each Bloch sum to the electronic wavefunction We need to solve the Schrödinger equation ˆ HΨ(k, r) = E(k)Ψ(k, r) (1.6) Substituting Eq 1.4 into Eq 1.6, we get      HAB   cA     HAA − E   =  ;  HBA HBB − E cB (1.7) if we assume that the overlap matrices S = pA (k, r) pB (k, r) = The matrix ele˜z ˜z ments are HAA = ˆ R ,A ∑ eik·(Rn −Rn) pRn,A H pz n , z N n,n (1.8) HAB = ˆ R ,B ∑ eik·(Rn −Rn) pRn,A H pz n z N n,n (1.9) and ∗ Note that HAB = HBA If only nearest-neighbor interactions are taken into account, then ˆ z HAA = p0,A H p0,A , z (1.10) CHAPTER INTRODUCTION TO GRAPHENE-RELATED MATERIALS which we shall assign a value of for convenience Also, HAB = ˆ z ˆ z p0,A (k, r) H p0,B (k, r) + e−ik·a1 p0,A (k, r) H p−a1 ,B (k, r) z z ˆ z +e−ik·a2 p0,A (k, r) H p−a2 ,B (k, r) z = −γ0 (1 + e−ik·a1 + e−ik·a2 ), (1.11) where γ0 is the transfer integral between the nearest-neighbor C atoms Substituting Eq 1.10 and 1.11 into Eq 1.7 and setting its determinant to 0, we obtain the dispersion relations for the π electrons of graphene: E ± (k) = ± |HAB | = ±γ0 + 2cos(k · a1 ) + 2cos[k · (a2 − a1 )] + 2cos(k · a2 ); (1.12) ˆ ˆ and equivalently, for k = kx x + ky y, E ± (kx , ky ) = ±γ0 3acc kx + 4cos cos √ 3acc ky + 4cos2 √ 3acc ky (1.13) By plotting Eq 1.13, we obtain the electronic dispersion relations – or E-k relations – of graphene, which are shown in Fig 1.3 Given that we have two electrons per PUC, the lower-energy π-band is completely filled, whereas the higher-energy π ∗ -band is completely empty Moreover, it is observed from Fig 1.3 that the π and π ∗ bands meet at the K and K’ points of the Brillouin zone (BZ); hence graphene does not have a band gap and is best described as a zero-gap semiconductor Another observation from Fig 1.3 is that the energy dispersions at K and K’ are cone-like, which implies that the E-k relationship is linear at these points We can a 1st order Taylor series expansion of Eq 1.12 for the E-k relations at these points, leading to this final expression: κ E ± (κ ) = ±¯ νF |κ | , h κ (1.14) where κ = k − K; with K being the position vector of the K or K’ points of the BZ, CHAPTER INTRODUCTION TO GRAPHENE-RELATED MATERIALS 10 Figure 1.3: The electronic dispersion relations of graphene, which is obtained by plotting Eq 1.13 Figure reproduced with permission from Ref 21 and νF = 3γ0 acc 2¯ h ≈ 106 m/s is the Fermi velocity of the electrons Due to the linear E-k relationship, the electrons in graphene behave like massless Dirac fermions – hence the K and K’ points are often called Dirac points – and exhibit some rather extraordinary properties For example, high electron mobility of up to 200,000 cm2 /Vs has been measured for mechanically-exfoliated graphene, 19 and the mean free path of the electrons in graphene can reach up to mm for annealed samples 20 The long mean free path is κ the result of the suppression of backscattering between κ and −κ Furthermore, since the cones of the π and π ∗ bands are symmetrical, electronic conduction is ambipolar in nature; Novoselov et al showed that Fermi level of graphene can be tuned with an electric field field, making it n- or p-type doped The electrons also exhibit exotic properties like pseudospin, chirality, and the integer and fractional quantum hall effect However, these properties are beyond the scope of this thesis and will not be described in detail Phonon band structure and thermal conductance of graphene – The total thermal conductance K of a material can be divided into contributions from the electrons (Ke ) and from phonons (Kph ) Although graphene has zero band gap and behaves like a semimetal, Ke Kph due to the strong C–C bonds in graphene; thus Kph is a large com- ponent of K in graphene 22 Since graphene has two basis atoms per PUC, there are a total of × modes in the phonon dispersion relations of graphene, as shown in Fig 1.4 Phonon frequency (cm -1 ) CHAPTER INTRODUCTION TO GRAPHENE-RELATED MATERIALS LO TO 1500 1000 ZO 500 11 LA TA ZA Γ Κ Wavevector, q Μ Figure 1.4: The phonon dispersion relations of graphene along the high-symmetry points of the Brillouin zone For the meaning of the symbols, please see text (for details on how to calculate phonon dispersion relations and the ballistic thermal conductance, see Chap 3) Three of the phonon modes are acoustic (A) modes and the rest are optical (O) modes, and they can be further classified as longitudinal (L), transverse (T), or out-of-plane (Z) depending on the direction that the C atoms are moving The low-frequency acoustic modes are responsible for most of the heat conductance of graphene since they are easily excited At small wavevector q, the TA and LA phonon modes of graphene have linear phonon dispersions with calculated group velocities of 13.6 and 21.3 km/s, respectively 23 These values are − times higher than in Si or Ge 24 Theoretically, the ballistic thermal conductivity of graphene was predicted to be ∼ 4000 Wm-1 K-1 at room temperature, which makes graphene one of the materials with the highest thermal conductivity around 22,25 So far, the highest thermal conductivity value measured experimentally is in the range of ∼ 2000 − 5000 Wm-1 K-1 (at room temperature) for suspended mechanically-exfoliated graphene 22,26 1.1.1 Finite size effects The properties of infinite graphene will be modified if we cut graphene into smaller sizes for use in nanosize electronic devices, where quantum confinement and edge effects will come into play In this section, we summarize the properties of nanosize graphene that have been discovered thus far Commonly, strips of graphene called graphene nanoribbons (GNRs) are studied, which CHAPTER INTRODUCTION TO GRAPHENE-RELATED MATERIALS b3 (a) (b) a3 Y X 30 o (c) a4 b1 Κ' Y a1 12 X a2 Μ Γ b4 Κ b2 Figure 1.5: (a) Directions to cut graphene to yield armchair (blue solid line) and zigzag (red solid line) edge graphene nanoribbons (b) The orthogonal unit cell for graphene, with lattice vectors a3 and a4 This unit cell has four C basis atoms and is twice the size of the hexagonal primitive unit cell (a1 and a2 ) that is shown in grey (c) The corresponding reciprocal lattice vectors b3 and b4 for the orthogonal unit cell, with the red rectangle demarcating the boundaries of the 1st Brillouin zone (BZ) Complete E-k information of the orthogonal 1st BZ can be obtained from the hexagonal 1st BZ via the ‘zone-folding’ technique as shown by the black arrows With the zone-folding, the Dirac points now lie along b3 are infinitely long in one dimension and finite in length in the other By cutting graphene along the blue or red dotted line shown in Fig 1.5(a), we obtain armchair edge graphene nanoribbons (AGNRs) or zigzag edge graphene nanoribbons (ZGNRs), respectively The blue and red dotted lines are separated by 30◦ , and slicing the graphene at an angle between these two lines will yield chiral GNRs, whose edges are a mix of the armchair and zigzag edges For examples of chiral GNRs, see Ref 27 To facilitate the discussion of the electronic band structure of GNRs, 28,29 we redefine the unit cell of graphene in terms of orthogonal lattice vectors a3 and a4 , as indicated in Fig 1.5(b) For the initial discussion, assume that each C atom at the edge is bonded to a single H atom to saturate their sp2 orbitals The corresponding reciprocal lattice vectors b3 and b4 , as well as the rectangular 1st BZ are shown in Fig 1.5(c) Since the orthogonal unit cell is twice as large as the primitive hexagonal cell, the 1st BZ of the former is half the size of the latter The E-k information calculated for the 1st BZ of the primitive hexagonal unit cell can be translated or folded into the 1st BZ of the orthogonal cell to yield complete E-k data for the orthogonal unit cell From this zone-folding, information at the Dirac points are now mapped onto ± b3 In terms of the orthogonal unit cell, AGNRs (ZGNRs) are infinitely long in direction CHAPTER INTRODUCTION TO GRAPHENE-RELATED MATERIALS 13 a4 (a3 ), but are finite in width along a3 (a4 ) AGNRs (ZGNRs) of larger widths can be obtained by increasing the number of hexagon rings along a3 (a4 ), such that their rectangular 1st BZ becomes increasingly narrower along b3 (b4 ) The electronic band structure of AGNRs (ZGNRs) can be approximated by zone-folding the rectangular 1st BZ of infinite graphene along b3 (b4 ) into the smaller 1st BZ of the GNRs Due to the electrons being confined in the y-direction for AGNRs, the y-component of the electron wavevector can only take on certain discrete values given by ky = 2πm , (1.15) A where A is the finite width of the AGNR and m ∈ Z The same argument holds for ZGNRs, except that the discretization happens for |kx | Fig 1.6(a,b) shows the electronic band structure of AGNRs and ZGNRs along the main axis of the ribbon [which is along b3 (b4 ) for ZGNRs (AGNRs)], that is obtained from zone-folding the electronic band structure of graphene ZGNRs not have band gaps since the features of the Dirac cones are mapped onto |k| = 2π For AGNRs, the ap- pearance of a band gap depends on the ribbon width: for AGNRs with N = 3p + 2, where N is the number of carbon dimer lines across the AGNR width (see Fig 6.1) and p ∈ Z, the Dirac points are mapped onto the Γ point of the BZ and those AGNRs will have zero band gaps 28 In reality, the actual band structures of the GNRs [Fig 1.6(c,d)] are different from the ones predicted by zone-folding due to edge effects It turns out that a band gap is present even for AGNRs with N = 3p + 2, because the C atoms at the truncated edges have a different chemical environment than the C atoms within the bulk interior of the GNRs 30 In ZGNRs, a flat band appears at the Fermi level, which extends from |k| = 2π to π The flat band is due to ‘edge states’, 30 the origin of which is explained below Depending on the topology of the GNRs, the presence of a truncated edge affects the π-conjugation of the graphene system since some pz electrons might be unable to pair up and form a π-bond, resulting in non-bonding, unpaired electronic states called edge CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 49 in the material are also broadened to a greater extent The magnitude of this energy broadening γi is given by Heisenberg’s uncertainty principle τi γi = h, ¯ (3.12) where h is the Planck’s constant h divided by 2π When steady-state charge current ¯ flows through the junction, N(E) does not change with time t, i.e., dN(E) = dt dN(E) dN(E) + = dt dt 1 (N1 (E) − N(E)) = (N(E) − N2 (E)) τ1 τ2 (γ1 + γ2 )N(E) = γ1 N1 (E) + γ2 N2 (E) N(E) = γ1 N1 (E) + γ2 N2 (E) γ1 + γ2 (3.13) The electron flux across the junction at steady-state is dN(E) dF = dt dt =− dN(E) , dt (3.14) which simply means that the number of electrons flowing into the material from contact must be equal to the electronic charge flowing out from the material and into contact Thus, dF dt dN(E) dN(E) − dt dt γ1 γ2 = [N1 (E) − N(E)] + [N(E) − N2 (E)] 2¯ h 2¯ h = [γ1 N1 (E) − γ2 N2 (E) + (γ2 − γ1 )N(E)] , 2¯ h = (3.15) CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 50 and if we assume that contacts and are the same, i.e., γ = γ1 = γ2 , then dF dt h = h = h ∞ = −∞ ∞ −∞ ∞ −∞ πγD(E)( f1 − f2 )dE πγ D(E) ( f1 − f2 )dE θe (E)( f1 − f2 )dE (3.16) In the above derivation, we have assumed that there is negligible scattering of the electrons during the conduction; in other words, the electron transport is ballistic In diffusive transport where the collisions between the electrons is inelastic, Eq 3.16 becomes 160,162 dF = dt h where T (E) = λ λ +L , ∞ −∞ T (E)θe (E)( f1 − f2 )dE, (3.17) λ is the mean free path of the electrons, and L is the length of the junction It can be proven that θe = πγD(E) is the number of conduction channels for the electrons across the junction at E per electron spin 160,162 If contact 1, contact 2, and the material are entirely uniform – composed of the same substance and are of the same dimensions – θe (E) is easily obtained by first calculating electronic band structure of the entire junction along the direction of electron transport, and subsequently counting the number of bands 162 that cross each E, as shown in Fig 3.5 In the more general nonequilibrium Green’s function approach, 163,164 T (E)θe (E) = Trace(G† Γ2 Gr Γ1 ), r (3.18) where Gr is the retarded Green’s function for the material, Γ j = i(Σ j − Σ† ), and Σ j is j the self-energy of contact j CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 51 Energy (eV) -1 -2 -3 -4 -5 Transport Direction Figure 3.5: The counting method to obtain θe (E) from the electronic band structure calculated along the direction of electronic transport: at E = −1.0 eV, θe (E) = since there are two electronic bands available for electron transport; whereas θe (E) = for E = −3.5 eV Derivation of steady-state charge current I Since I=e dF , dt (3.19) the steady-state charge current is I= 2e h ∞ −∞ T (E)θe (E)( f1 − f2 )dE (3.20) Derivation of G If V is small enough such that f2 ≈ f1 ≈ f , we can express f2 as a first-order Taylor series expansion in terms of f1 : f2 = f1 + f1 − f2 = ∂f (µ2 − µ1 ) ∂µ ∂f (e V ) ∂µ (3.21) CHAPTER THERMAL AND ELECTRONIC CALCULATIONS Since the integration in Eq 3.17 is with respect to E, and f1 − f2 = − ∂f ∂µ 52 ∂f = − ∂ E , we have ∂f (e V ), ∂E (3.22) which allows us to rewrite Eq 3.17 as 2e2 h I= ∞ −∞ T (E)θe (E) − ∂f ∂E dE V (3.23) ∂f Note that the − ∂ E term acts like a window function that determines the energy range at which current flows Also, since it is normalized (i.e., ∞ −∞ ∂f − ∂ E dE = 1), we can easily tell at first glance which energies contribute most to the charge current By comparing Eq 3.2 with Eq 3.23, we have G = where 2e2 h 2e2 h ∞ −∞ T (E)θe (E) − ∂f ∂E dE, (3.24) is commonly called the quantum of conductance We can express G in a more concise manner using Lorenz functions, where the nth order Lorenz function is: Ln = h ∞ −∞ T (E)θe (E)(E − µ)n − ∂f ∂E dE; (3.25) therefore, G = e2 L0 (3.26) Derivation of S Similar to what was done for V , we conduct a Taylor series expansion of the Fermi- Dirac function with respect to T : ∂f T ∂T ∂ f (E − µ) = − − T ∂E T f2 = f1 + f1 − f2 (3.27) CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 53 By substituting Eq 3.27 into Eq 3.17, I=− 2e h ∞ −∞ T (E)θe (E) − (E − µ) dE T , T ∂f ∂E (3.28) and by comparison with Eq 3.6, we have ST = − 2e hT ∞ −∞ T (E)θe (E) − ∂f ∂E (E − µ)dE (3.29) In terms of the Lorenz functions, e ST = − L1 T (3.30) Since S= ST , G (3.31) L1 eT L0 (3.32) therefore S=− Derivation of the steady-state heat current IH If µ1 ≈ µ2 ≈ µ, then the heat energy carried per electron across the junction is (E − µ) The steady-state charge current flowing through the junction is thus given by IH = (E − µ) dF , dt (3.33) thus IH = h ∞ −∞ T (E)θe (E)(E − µ)( f1 − f2 )dE (3.34) CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 54 Derivation of Π and Ke f1 − f2 due to V and T from Eq 3.22 and 3.27 is ( f1 − f2 ) = − ∂f ∂E e V− − ∂f ∂E E−µ T T (3.35) Substituting Eq 3.35 into Eq 3.34 gives us IH = 2e h − h ∞ −∞ ∞ T (E)θe (E)(E − µ) − −∞ T (E)θe (E) (E − µ)2 T ∂f dE V ∂E ∂f − dE T ∂E (3.36) Comparing Eq 3.9 and Eq 3.36 yields −ΠG = 2e h ∞ −∞ T (E)θe (E)(E − µ) − ∂f ∂E dE, (3.37) so Π=− L1 eL0 (3.38) Also, by comparing Eq 3.9 with Eq 3.36, ∂f ∞ (E − µ)2 − dE T (E)θe (E) h −∞ T ∂E L1 L1 = L2 − − e2 L0 − , T eL0 eT L0 Ke + ΠGS = Ke therefore, Ke = 3.3 T L2 − L1 L0 (3.39) Heat conductance in materials due to phonons In Sec 3.1, we examined heat conduction in a material due to the electrons Such a picture is incomplete, since lattice vibrations (called phonons) in the material are also capable of conducting heat in the material In this section, we give a physical description of phonons and how to calculate the heat conductance due to them CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 55 Any lattice vibrational mode in a crystal can be calculated from a linear combination of basic modes called normal modes, which are orthogonal to each other To derive the normal modes of phonons, 29,144,156,165 we define a periodic supercell (SC) of a crystal in which the position of each atom in the lattice is given by Rni = Rn + ri + uni , (3.40) where Rn denotes the Bravais lattice vectors of the nth primitive unit cell (PUC) of the crystal in the SC, ri is the position vector of the ith basis atom in the PUC, and uni is a small displacement of the ith atom We denote uniα as the αth component of uni , where α is the Cartesian direction x, y, or z If uniα is small compared to the interatomic spacing of the crystal, the ground state energy E of the system can be calculated using a Taylor series expansion: E = E0 + ∑ n,i,α ∂E ∂ uniα uniα + ∂ 2E ∑ n,i,α;m, j,β ∂ uniα ∂ um jβ uniα um jβ + , (3.41) where the indices m, n = 1, 2, · · · , number of PUCs in the SC; indices i, j = 1, , · · · , number of basis atoms in the PUC; and α, β = Cartesian directions x, y, z The subscript ‘0’ indicates that those terms are evaluated when the atoms in their equilibrium positions Since ∂E ∂ uniα = 0, we are left with the second-order term if we consider the higher-order terms to be negligible This is known as the harmonic approximation If we assume that the motions of the atoms obey classical mechanics, we can write the following based on Newton’s second law: Mi d2 uniα dt = − = − ∂E ∂ uniα ∑ Cniα,m jβ um jβ , (3.42) m, j,β where the force constant is Cniα,m jβ = ∂ 2E ∂ uniα ∂ um jβ (3.43) CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 56 Since the solutions to Eq 3.42 must be periodic in space and time, we assume the following ansatz uni (q,t) = Ai (q, ω)ei(q·Rn −ωt) , (3.44) where Aiα (q, ω) are the polarization vectors of the atomic vibration, q is the wavevector of the vibration, and ω is the frequency of the vibration Substituting the ansatz into Eq 3.42, we get a series of linear equations ∑ ∑ Cniα,m jβ eiq·(Rm−Rn) j,β A jβ = ω Mi Aiα (3.45) m There are some constraints that limit the values of Cniα,m jβ : • Cniα,m jβ is a real value and symmetric; i.e., Cniα,m jβ = Cm jβ ,niα • Since a crystal possesses translational symmetry, it does not matter relative to which unit cell n the summation ∑Cniα,m jβ eiq·(Rm −Rn ) is taken, which allows us m ˜ ˜ to set Rn = We now have Ciα, jβ (q) = ∑C0iα,m jβ eiq·Rm , where Ciα, jβ (q) is the m discrete Fourier transform of the force constant in real space 165 • Depending on the point group symmetry of the crystal, there might be some space ˜ group operations that Ciα, jβ (q) has to obey See Sec 3.3.1.1 for more details When the series of linear equations in Eq 3.45 are expressed in matrix form, we have ˜ C(q)A(q, ω) = ω (q)MA(q, ω) , (3.46) where each element of the mass matrix M jβ ,iα = Mi δi j δαβ , and δi j , δαβ are Kronecker deltas The above equation is not a standard eigenvalue problem since the mass matrix changes the eigenvalue ω for each row Aiα (q, ω) We can renormalize the matrix CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 57 equations, 165 ˜ C(q)A(q, ω) = ω (q)MA(q, ω) 1 1 ˜ C(q)M − M A(q, ω) = ω (q)M M A(q, ω) 1 1 ˜ M − C(q)M − M A(q, ω) = ω (q)M A(q, ω) ˜ D(q)A (q, ω) = ω (q)A (q, ω) , ˜ ˜ where D(q) is the dynamical matrix with matrix elements Diα, jβ (q) = (3.47) ˜ √ Ciα, jβ (q), mi m j and A (q, ω) is the phonon eigenvector It is related to the original polarization vectors √ by Aiα (q, ω) = mi Aiα (q, ω) For nontrivial solutions of Eq 3.47 to exist, we solve the following determinantal equation ˜ D(q) − ω I = 0, (3.48) where I is the identity matrix 3.3.1 Supercell method for calculation of phonon dispersion relations There are several methods to calculate the dynamical matrix in Eq 3.47, e.g., using perturbation theory 166 For the work in this thesis, we calculate the dynamical matrix using the supercell method 165,167,168 From Sec 3.3, the matrix elements of the dynamical matrix is given by ˜ Diα, jβ (q) = √ ∑C0iα,m jβ eiq·Rm , mi m j m (3.49) CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 58 and (cf Eq 3.43) ∂ 2E ∂ u0iα ∂ um jβ ∂ ∂E = ∂ u0iα ∂ um jβ ∂ Fm jβ = − ∂ u0iα C0iα,m jβ = This means that the dynamical matrix for each wavevector q is obtained from the force constant matrix in real space In the supercell method, we compute the force constant matrix by first constructing a SC that consists of a certain number of PUCs as shown in Fig 3.6 Next, we displace the ith atom in the 0th PUC from its equilibrium position along direction α by a small displacement ±δ0iα , and then calculate the forces Fm jβ (±δ0iα ) acting on the jth atom in the mth PUC along direction β using the Hellmann-Feynman theorem (see Sec 2.4) Finally, we use a finite central-difference scheme to evaluate the force constant matrix C0iα,m jβ = − Fm jβ (+δ0iα ) − Fm jβ (−δ0iα ) 2δ0iα (3.50) In theory, the construction of the dynamical matrix requires an infinite number of PUCs in the SC However, since the forces between the atoms decay quickly as the distance between them increases, we can limit the number of PUCs in the SC to some cutoff mcutoff ; i.e., ∞ 1 mcutoff C0iα,m jβ eiq·Rm ≈ √ eiq·Rm √ ∑ ∑C mi m j m=0 mi m j m=0 0iα,m jβ (3.51) The value of mcutoff is system dependent and checks should be conducted to ensure that the force constants for the atoms in the SC most distant to the ith atom are close to zero 3.3.1.1 Reducing number of calculations in supercell method with space group operations Without exploiting any symmetry in the system, a PUC with N atoms will require × CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 59 Y X Figure 3.6: In the supercell method for calculating the force constant matrix of the phonon eigenvalue problem, a SC cell is first constructed; which in this example consists of × primitive unit cells (PUCs) and is outlined by the red dashed line Each primitive unit cell (PUC) is denoted by the dashed black line and contains one atom (grey ball) The atom in the 0th PUC outlined by the dashed blue line is displaced by a small distance in the x and y directions in turn, and the forces generated on the other atoms in the SC are tracked The force constant matrix is constructed from these force values using the finite central-difference method × N = 6N static DFT calculations where the factor of two is required for positive and negative displacements (±δ0iα ) of the finite central-difference method If some space group operations map the ith atom to other atoms in a PUC, it is possible to reduce the 6N calculations by transferring the interatomic force constants obtained from the ith atom to other equivalent atoms To this, we assume that xf and xf are the fractional coordinates of the ith and i th atom in the PUC, respectively, and are related through xf = Rf xf + tf , where Rf and tf are the rotational and translational parts of a space group ˆ operation O = {Rf |tf } in the Seitz notation Equivalently, if Cartesian coordinates x and x are used instead, x = ARf A−1 xf + Atf , where the kth column (k = 1, 2, 3) of the matrix A is formed from the primitive lattice vector ak = a1k i + a2k j + a3k k of the PUC If Ciα, jβ is known for the ith atom in the PUC and the jth atom in the SC, then Ci α , j β = −1 ∑ ∑ Uα sCis, jtUtβ s=1 t=1 (3.52) CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 60 ˆ ˆ where O maps the ith atom to another equivalent i th atom in the PUC, and O maps the jth atom to j th atom in the SC The × matrix U is ARA−1 3.3.2 Derivation of the heat conductance due to phonons Kph In Sec 3.2, we derived equations for the charge and heat current across a junction due to the electrons based on the Landauer approach The derivation of the heat current across a junction due to phonons is the same, with the exception that each phonon mode is quantized with energy hω, and phonons states are populated according to the Bose¯ Einstein distribution instead of the Fermi-Dirac distribution since they are bosons The heat current across the junction due to phonons is thus (cf Eq 3.34) ph IH = h ∞ (¯ ω)Tph (¯ ω)θph (¯ ω)(n1 − n2 )d(¯ ω), h h h h h where ni (¯ ω, Ti ) = e(¯ ω)/kTi − h If (3.53) −1 is the Bose-Einstein distribution T is small enough such that n2 ≈ n1 ≈ n, we can express n2 as a first-order Taylor series expansion in terms of n1 : ∂n T ∂T ∂n = − T ∂T n2 = n1 + n1 − n2 (3.54) Now ∂n ∂T h hω e(¯ ω)/kT ¯ = , h kT (e(¯ ω)/kT − 1)2 (3.55) and h ∂n e(¯ ω)/kT = − ; h ∂ (¯ ω) h kT (e(¯ ω)/kT − 1)2 (3.56) which means that n1 − n2 = − hω ¯ T − ∂n ∂ (¯ ω) h T (3.57) CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 61 Substituting Eq 3.57 into Eq 3.53 gives us ph IH = − where ∞ k2 T h Tph (¯ ω)θph (¯ ω) h h hω ¯ kT ∂ − ∂ (¯n hω) hω ¯ kT ∞ − ∂n ∂ (¯ ω) h d(¯ ω) T , h (3.58) d(¯ ω) is the window function for the heat current due to h phonons (cf Eq 3.23 for the window function of the charge current due to electrons) ∞ Since hω ¯ kT ∂ − ∂ (¯n hω) d(¯ ω) = h π2 , we normalize this window function to make it so that it is easy to tell which phonon wavenumbers are responsible for most of the heat current Therefore, Eq 3.58 now becomes ph IH π k2 T =− 3h ∞ Tph (¯ ω)θph (¯ ω) h h π hω ¯ kT − ∂n ∂ (¯ ω) h d(¯ ω) T h = −Kph T , where π k2 T 3h (3.59) (3.60) is called the quantum of heat conduction, and π k2 T Kph = 3h ∞ Tph (¯ ω)θph (¯ ω) h h π hω ¯ kT − ∂n ∂ (¯ ω) h d(¯ ω) h (3.61) is the heat conductance due to the phonons 3.4 Thermoelectric figure of merit The thermoelectric figure of merit (TFOM) Z= GS2 , Ke + Kph (3.62) or more commonly, the unitless TFOM ZT = GS2 T , Ke + Kph (3.63) CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 62 I Tc Cold p-type n-type Th r, K, S Hot Qh R Load Figure 3.7: A prototypical thermoelectric generator with one side of the n-type and p-type legs at a high temperature of Th and the other side at a low temperature of Tc , driving a load of resistance R The rate of heat energy loss at the hot side is Qh and the electric current generated by the Seebeck effect is I The total internal resistance, heat conductance, and Seebeck coefficient of the entire device is r, K, and S, respectively is used as a measure of the efficiency of thermoelectric materials: the higher ZT is, the more efficient the material Its origin is from the derivation of the efficiency of a thermoelectric generator 162,169,170 In the thermoelectric generator shown in Fig 3.7, the electric current I generated from the Seebeck effect is I= S∆T , r+R (3.64) and the power to the external load is W = I2R = S2 ∆T R (r + R)2 (3.65) For the meaning of the symbols in the equations, please refer to the caption of Fig 3.7 The rate of heat energy loss at the hot side of the thermoelectric generator Qh is Qh = K∆T + STh I − I2r , (3.66) where the first term on the right-hand-side of the equation is the heat loss due to thermal CHAPTER THERMAL AND ELECTRONIC CALCULATIONS 63 conductance, the second term is the heat loss due to the Seebeck effect, and the third term is due to the resistive heating within the generator Note that since heat generated from resistive heating does not travel in any specific directions, half of that heat produced goes back towards the hot side The efficiency of the thermoelectric generator is thus η = W Qh (3.67)  =  ∆T  Th + m m+1 , Kr(m+1) ∆T − 2T (m+1) S2 Th h (3.68) where m = R r By substituting Eq 3.62, into Eq 3.68, we get  η =  ∆T  Th + m m+1 , (m+1) ∆T − 2T (m+1) ZTh h (3.69) and by maximizing Eq 3.69 with respect to m, the resistance of the external load is Rη,max = rm = r where T = (Th +Tc ) (1 + ZT ), (3.70) Substituting Eq 3.70 into Eq 3.69, we get ηmax = ∆T Th (1 + ZT ) − Tc (1 + ZT ) + Th (3.71) Therefore, from Eq 3.71, we can see that achieving maximum efficiency for the thermoelectric generator is a matter of maximizing Z ... 3acc √ x−y , ˆ ˆ (1. 2) CHAPTER INTRODUCTION TO GRAPHENE- RELATED MATERIALS b1 (b) a2 a2 Κ'' a1 A a1 b2 (a) Μ B Y Γ Κ X Figure 1. 1: (a) The atomistic model of the hexagonal network of graphene; the... 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