Symmetries in crystals
Basic lattice properties
A 3D crystal is described by an infinite set of points in space, the so called « crystal lattice ». The position of these points can be reconstructed by the discrete span of three basis vector. Given the basis, each point of the lattice can be uniquely identified by a linear combination of these vectors and the application of the translation symmetry over a discrete group manifold. The lattice vectors, here defined as ~a1, ~a2, ~a3, define a parallelepiped containing a single point of the lattice, and thus identifiable as primitive unit cell: thence, any point defined by ~ ;n2;n3 ~ (2.1) Vn1 = n1~a1 + n2~a2 + n3~a3 VN ;
with ni integers belongs to the crystal, which can be clearly presented as the repetition in space of the primitive cell. Let’s consider now a generic L2-integrable function, f (~r), that possess the same symmetry of the lattice and let’s write it as Fourier series: f (~r) = å fkeiQk ~r:
Symmetries in crystals
It is straightforward to notice that, for the function to fulfill the equality (~) = (~ +~ ), f r f r VN a condition on the ~ has to be imposed, specifically Qk Qk VN = 2pN with N integer. The ~ vectors that satisfy such condition possess the same periodicity Qk of the original lattice and are defined in Fourier space. They form what is called « reciprocal lattice ». Whereby in reciprocal space it is possible to define a transformed function that is invariant under translations and periodic.
We can use eq. 2.3 to construct the reciprocal lattice in a similar fashion as for the direct one. Given the direct lattice unit-cell volume, V =~a1 (~a2 ^~a3), to satisfy eq. 2.3 we can define bi V ~a j ^~ak:
Thus, in the Fourier space the three vectors b1 , b2 and b3 rocal lattice and a unit cell in Reciprocal space, the latter being conventionally named « Brillouin zone » (BZ); as in the direct lattice, a span of the basis vector with integer coeﬃcients allows to reach every point of the transformed lattice. Having functions periodic in reciprocal space is clearly helpful for the calculations, as it permits to focus on the BZ only, as for the other regions they can be reached by translations.
As for the unit cell types, there are several ways the lattice vector can orient and thus several possible geometries. As far we are concerned in this thesis, we are going to study zincblende materials, such that the three lattice vectors are taken equal in modulus and forming an orthogonal triad and such that the basis is constituted by two atoms in the unit cell. A choice of the lattice vector for this structure in the monatomic case is as follows: ~a1 = ( 1;0;1) a ; (2.5) a being the lattice constant of the material under consideration. By mean of eq. 2.4 the basis vector in reciprocal space are ~ 1; 1;1) 2p (2.8) which correspond to a body-centered cubic lattice. It is also customary to identify the center of the Brillouin zone (0;0;0) as G – point.
Bloch’s theorem, electron bands and energy gap
Let’s consider a crystal (lattice vectors ~a1, ~a2, ~a3) in a periodic potential V, such that: V (~x0) V (~x0 + R); ~ +ny~a2 +nz~a3 ~n ~a, and nx,ny,nz integers. We will show now that the wave- with R nx~a1 function in a material where the translation symmetry holds must have the following form: y(~r) = eikr~u(~r); where (~) = (~ + ~ ) has the same periodicity of the lattice and the wave-vector ~ is u r u r R k generated by the reciprocal basis. This is the Bloch’s theorem.
It is clear that the Hamiltonian is invariant under the translation operator ˆ (~! ~ +~ ): T~a x x R
Thus, either operator can be diagonalised with the same basis. We can also note that T~ay(~r) = t~ay(~r) = y(~r + R);
~ ~ y(~r) = t ~ y(~r); (2.15)
T~ T~ay(~r) = t~ t~ay(~r) = y(~r +~a + b) = T meaning that = ik~n ~a, with ~ real vector generated by the reciprocal basis of the t~a e k crystal. If we define now a function u(~r) = e ikry(~r), we can easily see that it must have the periodicity of the lattice and this completes our proof. Let’s consider now a periodic single particle Hamiltonian: h¯ 2Ñ2 y +V y = Ey: (2.16)
Because of the translation symmetry and the Bloch’s theorem, the potential can be re-written as V (~r) = å~ ~ ~ ~ V~ eiG ~r and the wavefunction as y(~r) = å~ y~ ei(G+k)~r, with G~ belonging to the reciprocal lattice, meaning that h¯2m E!yG~ + å~ VG~yG~ ~k = 0; (2.17) where the approximation Ñ2u(~r) k2u(~r) has been used. Even considering the V ! 0 limit, the solution of the previous equation cannot be the free-electron model one, namely eik r, as the latter and clearly neglects the periodicity of the lattice and the degeneracy at the edges of the BZ. Therefore, we shall include a certain number of Fourier coeﬃcients – let’s say n – which characterise the state. This n – index is the « band index », and labels the Bloch’ state along with~ . In presence of a weak but non-k zero periodic potential cos ~ ~ , first order perturbation theory straightforwardly V0 G r shows how the degeneracy is removed and an energy-gap is created, that is E+; = h¯2k2=2m V0. Considering the spin degeneracy, for an N – electron system (in the BZ) we can accommodate them from the lowest energy state to the highest according to the Pauli’s exclusion principle. These considerations are of paramount importance to predict whether a material has insulating or metallic properties. In absence of strong electronic correlations, the crystal possesses metallic properties if the band with higher energy is only partially unoccupied. If that band is fully occupied, then the system is an insulator or a semiconductor, and energies of the order of Egap are required to excite its electrons.
Density functional theory
The accurate modeling of the ground state properties of materials requires the use of the principles of quantum mechanics (QM) and the solution of the time-independent Schrödinger equation å h¯2 Ñ2y +V y = Ey: (2.18) i 2mi
However, due to the exponential-like scaling of the Hilbert space for the ground state search, finding the exact QM solution is practically unfeasible even for systems with relatively few atoms. This has produced a plethora of methods to cope with the numerical hurdles and approximately but realistically find the quantum properties of a system in its ground state. We can shortly mention the Hartree-Fock method, tight-binding and variational approaches or Monte Carlo based techniques. It can be proven, however, that we don’t need the exact ground state wave-function y to model the ground state properties of a material: we just need the single particle density function r. This is the basis for what is called density functional theory (DFT): we will now highlight the basic theorems behind DFT that make it, at least in principle, an exact ground state theory. We can consider a system defined by H=H0 +Vext ; where H0 includes the contribution from kinetic energies and inter-particle interactions, ˆ ˆ fixed, it is clear that the while Vext represents an external potential. By considering H0 ground state wave-function and thus all the system properties, single particle density included, are a functional of the external potential. At the core of DFT lies the fact that there is a 1:1 correspondence between the external potential (by a constant factor) and r. To prove it, we will follow the standard argument of Kohn and Hohenberg .
Let us assume we have two external potentials, ˆ and ˆ s.t. V1 V2 constant but s.t. they lead to the same single particle density.
That is, we assume the two potentials, i.e. two diﬀerent ground states, may produce the same electronic density. For each case we can use the variational theorem:
< hy j ˆ jy i = + hy j ˆ ˆ jy i = + r (~)[ (~) (~)] 3 ; (2.20)
E1 2 H1 2 E2 2 V1 V2 2 E2 2 r V1 r V2 r d r
< hy j ˆ jy i = hy j ˆ ˆ jy i = r (~)[ (~) (~)] 3 : (2.21)
E2 1 H2 1 E1 1 V1 V2 1 E1 1 r V1 r V2 r d r
If we now take r1 = r2, we obtain E1 + E2 < E1 + E2 which completes our proof. Up to a constant factor the external potential is consequently fully determined by the single particle density and therefore the knowledge of r can in principle be used to infer all the ground state properties of the system, with the true ground state density function being the one that minimises the density functional energy. We can split the total energy of the system as a sum of kinetic and potential terms: E[r] = T [r] + EH [r] + Exc[r] + vext (~r)r(~r)d3r F[r] + vext (~r)r(~r)d3r; (2.22) where the external potential contribution is separated from the internal part, dubbed F, which constitutes a universal functional. The elements T , EH and Exc are the kinetic, Hartree and exchange-correlation (XC) energies respectively. A standard expression is known for the Hartree term, of Coulombic origin: vH (~r;[r]) = Z d3r0 ~r ~ d dHrr : (2.23)
However, an exact expression for the kinetic and exchange-correlation functionals T [r] and Exc[r] respect to the density is unknown. Within the called orbital-free density functional approach  explicit expressions of T with respect to the electronic density can be derived. This approach resembles closely the Thomas-Fermi model , a density-based theory for interacting many-body electron systems that predates DFT. We will explore in more details the strategy devised by Kohn and Sham . To estimate T consists into splitting r into single particle contributions ffa g1::N , with a shorthand for~ ; (wave-vector and band index, respectively); this way and considering k n the a-state occupancy fa we can define density and kinetic operator as: r(~r) = å fa jfa (~r)j2 (2.24)
Density functional theory
where fa is either 1 or 0 depending on whether the a state is occupied or not. Finally by minimising the energy functional we get, for each of the N degrees of freedom in our system, the so called Kohn-Sham (KS) equations: h¯ 2 fa + v[r]fa = ea fa (2.26) 8a 2 f1:::Ng where v[r] = vext + vH (~r;[r]) + vxc(~r;[r]): (2.27) Eq. 2.26 are non linear with respect to the Kohn-Sham eigenvectors and they have to be solved self-consistently. The numerical procedure generally involves the expansion of the KS eigenstates with respect to a known basis. The choice of this basis includes plane waves, although it is possible to use more localised states like Gaussians, Slater or hydrogen atom basis. Due to the requirement of orthogonality between diﬀerent wavefunctions, the behaviour of the states close to the nuclei is usually characterised by strong oscillations, which require a considerable number of basis set coeﬃcients in order to be properly described. Often to speed up the calculations it is preferable to continuously smoothen the description of the states near the core (within a certain cutoﬀ) while retaining their true behaviour out of the core region. Moreover, since inner states are mostly localised, they do not play a relevant role in typical solid state and chemical phenomena and therefore can be treated within a « frozen core » approximation, which means they can be computed for isolated atoms and transferred to crystals and molecules. This is the basic philosophy behind pseudopotential theory (PT) , with the internal electrons absorbed into the ionic potential so that it is possible to construct smooth valence (pseudo)wavefunctions.
We also mention the projector augmented wave method (PAW), conceptually similar to PT and in the context of plane waves . Since this is going to be the approach used in our calculations, it is worth giving it a short introduction. PAW introduces a linear operator tˆ such that: jfa i tˆ f˜a ; (2.28) where jfa i is the all-electron one-particle KS wavefunction and is a pseudo-state that has a smoother behaviour near the atom nuclei. The original and the pseudo wavefunctions diﬀer only in a region close to the atomic nucleus and this prompts to define: t = 1 + åtR; ˆ vanishing out of a sphere near the nuclei and defined by the radius R. If we with tR expand f˜a as a linear combination with respect to a pseudo-basis q˜i we can write: tˆ = 1ˆ + åi (jqii q˜i )hpij; (2.30) ˆ are the all electron wavefunctions for an isolated atom and jpii where jqii t qi i pseudostates. Then operators can are « projector functions » orthonormal to the ˜ be written in terms of this representation that preserve the all-electron eigenvalue spectrum , and again, the frozen core approximation is usually employed. It should be noted that the computational cost of solving the KS equations for a system of N 1023, typical for a solid, is still obviously too high. Nevertheless, in a crystalline material the periodicity comes in help: we can enormously reduce the number of degrees of freedom by considering only those belonging to the first Brillouin zone. We are left with the problem of evaluating Exc. Although DFT is in principle an exact theory, the reliability of its prediction is intertwined with the accuracy of the chosen exchange-correlation functional. This term has been introduced as an improvement of the single particle approximation – previously used to define the Hartree and kinetic part of the KS equations – which neglects the eﬀect of particle exchange and the many-body correlations present in the real systems. An electronic propriety that, for example, is subtly sensitive to the choice of the XC functional is the band gap, whose determination is pivotal for the understanding of several physical properties – like the dielectric functions, refraction indices and doping to mention a few – where the physics is largely dominated by the electronic behaviour around the Fermi surface. Functionals that make use of the LDA  or GGA  approximations often fail to properly represent the many-body electronic correlations inside a material, while other methodologies like hybrid functionals , the random phase approximation [25, 26, 27, 28] and other Green’s function based techniques like the Hedin equations and GW [29, 30] perform better while still preserving the single particle nature of the KS equations, although with a higher numerical cost.
Finally, the stability of the Fermi surface is of paramount importance to get the iterative solution of the Kohn-Sham equations properly converged. This turns to be of extreme importance when studying metals, where the close proximity of the occupied bands respect to the unoccupied ones prompts the use of fractional « smeared » occupations for the KS – states even at zero temperature [31, 32, 33].
Table of contents :
2 First principles phonon theory
2.1 Symmetries in crystals
2.2 Density functional theory
2.3 Lattice dynamics
2.4 Inter-atomic force constants
2.5 Lattice impurities
2.6 Chapter overview
3 Theory of thermal transport
3.1 Linear response theory
3.2 Boltzmann transport equation
3.3 Measuring the thermal conductivity
3.4 Chapter overview
4 Introduction to boron arsenide
4.1 Phonon band structure and pristine conductivity
4.2 Intrinsic defects
5 Substitutional impurities in boron arsenide
5.1 VM – scattering
5.2 VK – scattering
5.3 Total scattering
5.4 Frequency shift
5.5 Thermal conductivity
5.6 Effect of compensation
6 Introduction to half-Heusler compounds
7 Substitutional impurities in half-Heusler compounds
7.1 Classification of the compounds and the considered substitutions
7.2 Phonon-defect scattering
7.3 Thermal conductivity