ECE 535 Theory of Semiconductors and Semiconductor Devices Fall 2015 Homework # 5 Due Date: 11/17/2015

Size: px

Start display at page:

Download "ECE 535 Theory of Semiconductors and Semiconductor Devices Fall 2015 Homework # 5 Due Date: 11/17/2015"

Marilynn Craig
5 years ago
Views:

1 ECE 535 Theory of Semiconductors and Semiconductor Devices Fall 2015 Homework # 5 Due Date: 11/17/2015 Problem # 1 Two carbon atoms and four hydrogen atoms form and ethane molecule with the chemical formula C 2 H 4. It is the most manufactured hydrocarbon in the world. The structure of the ethane molecule is shown below: (a) The 2p-orbitals on the C atoms oriented out of the plane of the paper do not have a non-zero energy matrix element with any of the 1s-orbitals, 2s-orbitals, or 2p-orbitals that are in the plane of the paper. Explain why this is so? You may ignore the 2p-orbitals on C atoms that are oriented out of the plane of the paper for part (b) and (c). You are then left with 4 1s-orbitals - one on each H atom, 2 2s-orbitals one on each C atom, and 4 2p-orbitals two on each C atom that are in the plane of the paper. (b) What is the hybridization scheme (sp 2 or sp 3 ) for each C atom? (c) Write an appropriate trial solution for the wavefunction of the electron using the LCAO. Use the results from (b) as a guide. (d) Using your result from part (c), solve for the eigenenergies. Show that all the eigenenergies and corresponding degeneracies going from the atomic orbitals to the molecular orbitals in an energy level diagram similar to the ones that we drew in class. Use values of energy matrix elements given the lectures (e.g. V ssσ, V spσ, E s etc.). Now let us consider the 2p-orbitals on the C atoms oriented out of the plane of the paper. (e) Write an appropriate trial solution for the wavefunction of the electron using LCAO using only the 2p-orbitals oriented out of the plane of the paper. (f) Using your solution in part (e), solve for the eigenenergies. Show that all of the eigenenergies and the corresponding degeneracies going from the atomic orbitals to the molecular orbitals in an

2 energy level diagram using the same representation scheme as in part (d). What type of bond is formed by the 2p-orbitals on the C atoms oriented out of the plane of the paper (sigma or pi)? (g) Combine your results from part (d) and part (f) and draw an energy level diagram that shows all the energies and the corresponding degeneracies going from the atomic orbitals to the molecular orbitals (remember that we are considering ALL of the orbitals). On the energy level diagram indicate using black dots the electron occupation of each molecular orbital (you have a total of 12 electrons that participate in bonding). Problem # 2: Consider the graphene crystal shown below: The trial tight-binding solution including the s, p x and p y -orbitals, with corresponding energies E s, E p, and E p, can be written as: ψ = e N c φ r R d + c φ " r R d + c φ " r R d e + c φ r R d + c φ " r R d N + c φ " r R d which results in a 6x6 matrix that can be written as:

3 In your responses to the questions, please use the standard expressions for the matrix elements (e.g. V ssσ, V spσ, E s etc.). (a) Find the 6 diagonal elements of the matrix: H 11, H 22, H 33, H 44, H 55, H 66 (b) Find the matrix elements: H 14, H 15, and H 16 (c) Find the matrix elements: H 24, H 34 (d) Find the matrix elements: H 25, and H 36 (e) Describe how many bands will result from the above calculation and how many will be completely filled with electrons and how many will be partially filled with electrons at zero temperature. Problem # 3: Consider the following 2D hexagonal lattice along with the primitive lattice vectors: The corresponding reciprocal lattice is shown below:

4 Suppose that the potential is: V r = V cos b r + V cos b r + V cos b + b r + V cos b b r where in 2D: r = x x + y y. Within this problem, please use the following constants: Lattice constant a = 2.5 angstroms and V = V = V = V = 2.0. (a) As a result of the periodic potential, the free electron state with k-vector equal to (0, 4π/3a), which is the K-point within the FBZ, is coupled very strongly to two other free electron states. What are the k-vectors of the other free-electron states? Note that the K-point is located at the intersection of 2 different Bragg planes and, therefore, the free electron state becomes coupled to three other free electron states. (b) Write the solution for the actual electron wavefunction at the K-point as a superposition of three degenerate free-electron states, obtain a 3x3 matrix energy eigenvalue equation, and then solve it to obtain: i. The energies of the 3 energy bands at the K-point. ii. The actual wavefunction of the electron at the K-point corresponding to the 3 energy bands. (c) As a result of the periodic potential, the free electron state with k-vector equal to (2π/ 3a, 0), which is the M-point within the FBZ, is coupled very strongly to other free electron states. What are the k-vectors of the other free-electron states? Note that the M-point is located at the

5 intersection of one Bragg planes and, therefore, the free electron state becomes coupled to two other free electron states. (d) Write the solution for the actual electron wavefunction at the M-point as a superposition of three degenerate free-electron states, obtain a 2x2 matrix energy eigenvalue equation, and then solve it to obtain: i. The energies of the 2 energy bands at the M-point. ii. The actual wavefunction of the electron at the M-point corresponding to the 2 energy bands. Note/Lesson: If you did part (b) correctly you would have noticed that the 3-fold degeneracy of the lowest 3 free-electron bands at the K-point is only partially lifted when the periodic potential is turned on; two bands remain degenerate at the K-point but the third one has a higher energy. The hexagonal Bravais lattice under examination here describes graphene, as shown below. The underlying hexagonal Bravais lattice is shown by the small black dots. The two carbon atoms per primitive cell are shown by the blue and red filled circles in the figure below. The fact that in part (b) two of the bands remain degenerate at the K-point even in the presence of a periodic atomic potential is interesting. As I mentioned, it has something to do with the presence of inversion symmetry in the graphene crystal.

6 Now let us explore more about the role of inversion symmetry in graphene. A 2D crystal will have inversion symmetry with respect to the y-axis if V(-x, y) = V(x, y). Similarly, a 2D crystal will have inversion symmetry with respect to the x-axis if V(x, -y) = V(x, y). (e) Verify that the potential: V r = V cos b r + V cos b r + V cos b + b r + V cos b b r is inversion symmetric with respect to both the x and y axes. Now turn the graphene into boron nitride and use the following constants: V = 1.0, V = 3.0, V = V = 2.0. (f) Verify that the following potential does not have inversion symmetry: V r = V cos b r + V cos b r + V sin b + b r + V cos b b r (g) Write the solution for the actual electron wavefunction at the K-point as a superposition of three degenerate free-electron states, obtain a 3x3 matrix energy eigenvalue equation, and then solve it to obtain: i. The energies of the 3 energy bands at the K-point. ii. The actual wavefunction of the electron at the K-point corresponding to the 3 energy bands. If you did this correctly, you should see that the 3-fold degeneracy at the K-point has been completely lifted. Problem # 4: In the first homework, you looked at the problem of free electrons in a magnetic field. In this situation, the electrons moved in circular orbits in real-space with a frequency, ω c, normally referred to as the cyclotron frequency. For free electrons, ω = In this problem, you will look at electrons in the conduction band of a solid. Suppose that the energy band dispersion near the conduction band minimum is described by: E k = E k + ħ k k M k k The motion of each electron in k-space is described by the equation:

7 and in real space by the equation: ħ " = e v k B " = v k. Needless to say, the motion is complicated in both real and k-space. The exploration of this motion is the purpose of this problem. (a) Show that the component of the crystal momentum of an electron parallel to the magnetic field is independent of time. For the remainder of this problem, we will refer to this component as k. (b) Show that the electron energy is independent of time. (c) Argue that the results from (a) and (b) imply that in k-space the orbit of an electron with initial energy E 0 is given by the intersection of the constant energy surface corresponding to the energy E 0 with a plane that passes through the point k and is perpendicular to the direction of the magnetic field, as shown below. This shows that the motion of the electron in k-space is periodic. (d) In real space, the electron motion is described by the position vector r t. The projection of the electron motion in a plane perpendicular to the magnetic field is given by r t. Show that,

8 r t = r t B (e) The orbit of the electron in k-space is given by the time-dependent vector k t and the projection of the electron orbit in real space in a plane perpendicular to the magnetic field is given by r t. Show that the two orbits are related by: r t r t = 0 = B k t k t = 0 HINT: Start by taking the vector cross-product of an equation on both sides by the magnetic field and then integrating. The above relation shows that the projection of the motion of the electron in real space in a plane perpendicular to the magnetic field will be periodic since the motion in k- space is periodic, as we saw earlier in part (c). For parts (f) and (g) assume that the magnetic field is applied in the z-direction and is given by B = B z. The inverse effective mass tensor is given by: ħ M = 1/m 1/m " 1/m " 1/m " 1/m 1/m " 1/m " 1/m " 1/m From part (e) it follows that the motion of the electrons in the x-y plane (and also in k-space) is periodic and we suppose that the period has a frequency of ω c. (f) Find an expression for ω c in terms of the components of the inverse effective mass tensor. To help get you started, the answer can be written in terms of the determinant of a sub-matrix of the inverse effective mass matrix. This is not intended to be an algebra intensive problem if you do it with some thought. HINT: Try starting with this equation from the notes: ħ = e v k B (g) The frequency ω c can now be written as ω = where m e is the cyclotron effective mass. Find an expression for the cyclotron effective mass. Note that the cyclotron effective mass depends on the direction in which the magnetic field is applied. This is how effective mass tensors are measured experimentally. (h) Let s assume that the effective mass tensor is now completely diagonal and given by, " M = 1/m /m /m.

9 The magnetic field is applied in the direction of the unit vector n = n, n, n and is given by B = B n. Show that now the cyclotron effective mass is given by the expression M =. HINT: There are several approaches to solve this problem. I used a rotation matrix to solve it, for example.

Physics 211B : Problem Set #0

Physics 211B : Problem Set #0 These problems provide a cross section of the sort of exercises I would have assigned had I taught 211A. Please take a look at all the problems, and turn in problems 1, 4,