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MIT 3 23 - Homework 5

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MIT OpenCourseWare http://ocw.mit.edu 3.23 Electrical, Optical, and Magnetic Properties of MaterialsFall 2007For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � 1 Homework # 5 October 15, 2007 Homework is due on Wednesday October 17th, 5pm Density of states We saw both in recitation and in class, that we can calculate the 3D density of state corresponding to a given dispersion formula E(�k), by using the following formula: V dS g(E) = (2π)3 E(�k)=E ��E(�k)� for the 3D case ��k where the integral is carried out over the ”isosurface” of energy E, i.e the sur-face deﬁned by the equation E(kx, ky, kz ) = E. The inﬁnitesimal surface ele-ment is dS and V is the volume of the 3D crystal. The 2D equivalent is the following: � g(E) = A dl for the 2D case (2π)2 E(�k)=E ��E(�k)���k where the integral is carried out over the ”isosurface” of energy E, i.e the curve deﬁned by the equation E(kx, ky ) = E. The inﬁnitesimal element of length along the isosurface is dl and A is the area of the 2D crystal. Finally for the 1D case we have: g(E) = (2Lπ) k0 | (1 k=k0)| for the 1D case dE (k) dk where k0 are all the k points such that E(k0) = E, and L is the length of the 1D crystal. 1.1 Free electron gas in 1D, 2D and 3D Now we will apply this to the 1D, 2D and 3D free electron gas. 1) Write down the dispersion relations E(�k) for a free electron gas in 1D, 2D and 3D. From those relations, describe the corresponding isosurfaces, i.e draw a typical isosurface for each case (1D, 2D, 3D). 2) For the 2D and 3D cases , ﬁnd the appropriate set of coordinates to de-scribe an isosurface. Using this set of coordinates, give a mathematical expres-sion for the ”surface” element dS. Finally give an analytical form for the density of states. 3) For the one dimensional case, use the third formula above to calculate the density of states. 4) Plot the 1D, 2D and 3D density of states as a function of the energy E. 1 Nicolas Poilvert & Nicola Marzari� 1.2 The case of tightly bound electrons in 1D Let’s consider tightly bound electrons described by the following band dispersion (for a 1D linear chain of atoms): E(k) = �0 − 2γ cos(ka) 1) Deﬁne the interval of all possible values for E(k) when k is in the ﬁrst Brillouin Zone. Plot the band dispersion for k in the ﬁrst Brillouin Zone. 2) Using the formula for the 1D case, calculate the density of states. Plot this density of states as a function of the energy in the appropriate interval (deﬁned in question 1)). 2 Tight-binding method 2.1 geometrical description of the crystal Let us consider a simple cubic lattice (3D) of lattice parameter a, with one atom per unit cell centered at the origin of the coordinate system. Each atom has one valence electron and in the atomic limit (i.e in the limit where the atoms are far from each other) this electron is described by an s-type atomic orbital φs(�r). 1) Give a mathematical expression for the primitive basis vectors �a1,�a2,�a3 that describe this simple cubic lattice. Calculate the expression for the recipro-cal primitive vectors �b1,�b2,�b3. What is the volume of the unit cell in real space? What is the volume of the unit cell in reciprocal space, i.e the ﬁrst Brillouin Zone? Relate those two volumes by an equation. 2) Express the equilibrium position vectors R�for the atoms in this simple cubic crystal in terms of the �a1,�a2,�a3 basis. Sketch the ﬁrst Brillouin Zone in reciprocal space. 2.2 calculation of the band dispersion The tight-binding method is an extension of the LCAO method, used in Chem-istry, to periodic systems like solids. The spirit of the technique is the following: If one looks at the wavefunction for an electron in the atomic limit (when the atoms of the solid are far from each other), then this wavefunction looks very much like the atomic orbital φs(�r). Now if we bring the atoms closer and closer to each other, the wavefunction for the electron will be perturbed and at suﬃ-ciently small distances between atoms , the electron will be able to ”hop” from atom to atom, and it will delocalize over the entire solid. Nevertheless when the electron is localized around one particular atom, let s ay the atom centered around position R�, the wavefunction must still look very much like φs(�r − R�). So the idea of the tight-binding metho d is to write down the electronic wave-function as a linear combinaison of all the atomic orbitals φs(�r − R�), centered around each atom R�in the crystal (the summation over R�below concerns all atoms in the crystal): ψs(�r) = �c�φs(�r − R�)R R2� � � � � � � � � � |� � � � � � � � But in order to be an ”acceptable” wavefunction, ψs(�r) must satisfy Bloch’s theorem. This is a big constraint for ψs(�r), and as a consequence, we can write down an expression for ψs(�r) for each wavevector �k: ψ s�k(�r) = √1N � R�ei�kR�φs(�r − R�) The √1N is just a normalization factor (N being the number of unit cells in our crystal), and the coeﬃciants cR�are given by ei�kR�. We are interested in the eigenenergy Es(�k) of the periodic one-electron eﬀec-tive hamiltonian Hˆ. To calculate this eigenenergy we use the following result: |Hˆ|ψ(�k) = �ψ �ψ �ks�ksEs|ψ�ks�ks1) Give the mathematical expression for ψ s�k|ψ s�k in terms of integrals over the volume of the crystal. In this expression you should obtain terms of the form: crystal φ∗ s (�r − R�)φs(�r − R��)d�r Because of the Born-Von Karman boundary conditions, one can show that this integral is equal to the following one: crystal φ∗ s (�r)φs(�r − (R�� − R�))d�r So you see that this integral only depends on the relative distance R�� − R�. Using this, simplify the expression for ψ �ψ �? If the set of atomic orbitals � � sk|sk φs(�r − R�) form an orthonormal basis, what is the mathematical express ion � �R�for ψ s�k|ψ s�k ? We will now consider that the set of atomic orbitals form an or-thonormal basis � � 2) Now that we have expressed ψ s�kψ s�k , we need to focus on the numer-ator ψ s�k|Hˆ|ψ s�k . Because Hˆis a periodic

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