h is planck's constant
a is lattice constant
t is time.
Problem 4. In a simple 1-d tight binding model, the electron energy is given by an...
Consider a one-dimensional tight binding model of electrons hopping between atoms. Let the distance between atoms be called a, and here let us label the atomic orbital on atom ln) for n-1,..,N (you may assume orthonormality of orbitals, ie., (1m)- nm). n as Suppose there is an on-site energy e and a hopping matrix element -t. In other words, suppose (IH|m) = E for n-m and (1비m)=-t for n=m±1. (a) Derive and sketch the dispersion curve for electrons. (b) How...
Consider a free electron, empty lattice model with effective mass m* in a simple cubic crystal with direct lattice distance a, and reciprocal lattice vectors of length a. Find the energies at the high symmetry points Г, X, M and R and indicate the zone boundary rsion along TX, TR, Г b. Find the expression for the lowest energy band in the XM direction. Sketch the Energy band diagram along RIXM「 c.
Part 2 will rate thanks in advance
1. Phonon density of states and specific heat. Assuming that phonons of a three-dimensional crystal obey the following isotropic dispersion relation К ka w2 sin where a is the lattice constant, K the spring constant, and k the wave vector (1) Please derive an expression for the phonon density of states; (2) Please derive an expression for the phonon intermal energy and specific heat
1. Phonon density of states and specific heat. Assuming...
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This is a solid state physics problem. ( I use this book : kittel
introduction to soild state physocs ) thank you !
5. [Graphene] Consider graphene, a hexagonal lattice of carbon atoms as shown in the figure. The distance between neighboring carbon atoms is a0.143 nm (a) B A (a) Write down the unit lattice vectors ā, and a and unit reciprocal lattice vectors b, and b,. [5] a2 (b) The two sites A and B are not...
B3 (a) Assume that the T = 0 version of the Fermi-Dirac distribution, namely 1 f (E) exp [E E)/(kBT) +1 in the usual notation, with Ep the Fermi energy, applies for T> 0. Sketch, on the same axes, the distribution for T = 0 and for T> 0, marking the Fermi energy and indicating the thermal energy kBT 5 Marks (b) In the Sommerfeld model (free electron quantum gas), each electron occupies (n/L)3 of k-space volume. Remembering that we...
Question 21 Consider a free electron in one dimension (i.e. an electron free to move along say the x-direction on (a) The time-independent Schrödinger equation is Αψη (x)-Εηψη (x), where is the Hamiltonian (total energy) operator, and ψη (x) are the electron wave functions associated with energies En Assuming the electron's energy entirely comprises kinetic energy (as it is 'free' there is no potential energy term), write down the Schrödinger equation given that the momentum operator in one- dimension is...
Consider a one-dimensional tight binding model of electrons hopping between atoms. Let the distance between atoms be called a, and here let us label the atomic orbital on atom n as In) for n-1,..N (you may assume orthonormality of orbitals, i.e, (n|m) -8nm) Suppose there is an on-site energy є and a hopping matrix element-t. In other words, suppose 〈nlH1m)=ε for n-m and (IH1m)--t for n-m±1. (e) If each atom is monovalent (it donates a single electron) what is the...
Problem 1 (25 points). According to the Bohr's model of the hydrogen atom, the total energy of the electron in the nth orbital _ mg is E. =- 13.6(en) 16) where n=1,2,...and K = 4Tt€ in MKS units and m is the electron 2nK?? ? mass=9.11x10 kg; leV=1.6x10-19Joules. a) n=1 is the ground state of the Hydrogen atom and has value E= -13.6 eV. Explain why this value is negative. Define the ionization energy and calculate it for Hydrogen atom...
2. Based on the Kroing-Penney model, the periodic potential energy of an electron in a solid is shown aside. Ass urne that for a given allowed band αα<< 1 and ka<<1. Psin(aa) (a) Prove that the E -k relatione)+ cos(aa) - cos(ka)) may be put in the form: E -Ak' +B Write the constants A and B in terms of P, a, h, m, and h (b) If U-3 eV, a-0.3 nm, and b 0.025 nm i- Find numerical values...
4. A particle moves in a periodic one-dimensional potential, V(x a)-V(x); physically, this may represent the motion of non-interacting electrons in a crys- tal lattice. Let us call n), n - 0, +1, t2, particle located at site n, with (n'In) -Sn,Let H be the system Hamiltonian and U(a) the discrete translation operator: U(a)|n) - [n +1). In the tight- binding approximation, one neglects the overlap of electron states separated by a distance larger than a, so that where is...