Nanowire band structure Hampus Mårtensson Jönsson Andreas Malmgren Project supervisor: Florinda Vinas FFF042 The Physics of Low-Dimensional Structures and Quantum Devices Lund University December 15, 2015 Contents Nanowire band strucure i 1 Introduction 1 2 Theory 1 2.1 Schrödinger equation (SE) in an infinite 2-D potential well . .1 2.2 Density of States (DOS) . .2 2.3 Conductivity . .3 3 Results & Discussion 3 3.1 Probability density . .3 3.2 Sub band structure . .4 3.3 DOS . .5 3.4 Conductivity . .6 4 References 8 i 1 Introduction The purpose of this project is to calculate the band structure for a quasi 1-D cylin- drical nanowire. A quasi 1-D system is a system where the electrons are confined in two dimensions and are free to move in the third dimension. In this model the electrons are confined in the radial direction and free to move along the axis of the wire. Along with the band structure, the density of states (DOS) and the conductivity of different materials will also be considered. This can be done by varying physical parameters such as effective mass and the radius of the wire. The approach for this project is to treat the nanowire as being of infinite length and as a 2-D infinite potential well in the radial direction. 2 Theory 2.1 Schrödinger equation (SE) in an infinite 2-D potential well In order to realise the band structure and the energy levels in a nanowire the SE, (1), must be solved for the system. 2 ~ 2 H = − r + V = E (1) 2m Since the wire is cylindrical the SE must be rewritten with the Laplace operator in cylindrical coordinates with (r; θ; z) as in equation (2). 2 2 2 2 ~ @ 1 @ 1 @ @ − + + + + V (r; θ) (r; θ; z) = E (r; θ; z) (2) 2m @r2 r @r r2 @θ2 @z2 Equation (2) represents a cylindrical system with a potential that is independent of the position along the wire axis (z-axis). By assuming a solution of separable products as in equation (3) with plane wave solutions in the z direction, the z dependence can by eliminated and equation (2) becomes equation (4), where " is related to the total energy E as in equation (5). Note that we can not assume the same type of plane wave solutions in the r and θ direction since we expect some type of confinement in the allowed energies in those directions. (r; θ; z) = u(r)Θ(θ)eikzz (3) 2 2 2 ~ @ 1 @ 1 @ − + + + V (r; θ) (r; θ) = " (r; θ) (4) 2m @r2 r @r r2 @θ2 2 2 ~ kz " = E − (5) 2m Equation (4) now represents a 2-D potential well in polar coordinates with radius R. In our model the potential is zero inside the well and infinite outside the well as stated below. 1 of 8 ( 0 when r < R V (r; θ) = 1 when r > R This requires the wave functions to vanish at the boundaries of the well, i.e (r = R; θ) = 0. The wave functions must also be single valued which requires (r; θ = 0) = (r; θ = 2π). Since θ only appears as a second derivative in equation (4), eilθ is a solution to the radial part in equation (3) where l is restricted to integers l = 0; ±1; ±2;::: due to the boundary conditions. This further reduces equation (4) into a radial part only as in equationp (6) inside the well which is known as Bessels equation and where k = 2m" . ~ d2u du r2 + r + ((kr)2 − l2)u = 0 (6) dr2 dr The solutions to equation (6) are the non-elementary Bessel functions Jl and Yl. Since the Yl solutions diverge at the origin they can not be normalized and we will not consider them. The eigenvalues to the Bessel eigenfunctions are the square of the roots to the corresponding Bessel function with quantum number l. Since 2 a Bessel function has several roots we must label the eigenvalues αl;n where n is the n:th positive root from the origin. Given the eigenfunctions and eigenvalues to equation (6) we can construct the wave functions and the energy eigenvalues to equation (4) as follows in equation (7) and (8). αl;nr ilθ (r; θ) = Jl e (7) R 2 2 ~ αl;n "l;n = (8) 2mR2 By combining equation (5) and (8) we can solve for the total energy E in the system which involves transport in along the z-axis and thus leads to parabolic sub bands in the conduction band. 2 2 ~ kz El;n(kz) = "l;n + (9) 2m 2.2 Density of States (DOS) The 1-D density of states as a function of the energy for a quantum wire can be described by equation (10) where El;n are the energy eigenvalues from equation (9) and Θ is the Heaviside Step Function. s X 1 2m n1D(E) = Θ(E − El;n) (10) π E − El;n l;n ~ Where m is the effective mass. 2 of 8 2.3 Conductivity The conductivity of a material is a measurement of the ability to conduct a current in response to an applied electric field. The conductivity can be calculated by equation (11) σ = qnµ (11) where n is the concentration of electrons in the conduction band, q is the electron charge and µ is the electron mobility. The mobility can be found in tables for different materials and n can be calculated by equation (12) together with the DOS and the Fermi-Dirac distribution function. Note that n is a function of the Fermi energy EF . Z 1 n(EF ) = n1D(E)F (E; EF )dE (12) 0 3 Results & Discussion 3.1 Probability density The probability density for a quantum wire versus the wire radius is shown in figure 1. Here, the probability density of the nanowire is the square modulus of the wave function given in equation (7). The angle, θ, dependence disappears in the square modulus, since the angular dependency comes in the form of eilθ, where l = 1; 2; 3;::: is an integer. Figure 1 shows the probability density arising from 5 different orders of the bessel functions Jl which in this case are the solutions to the 2-D infinite well in the radial direction. From the figure one can see that the lowest order of the wave function (J0) has the highest probability density of being in the middle of the wire, and the 1st order has the second to highest probability density, and so forth. A possible reason for why the probability density peaks shift to a higher radius for a higher order, might follow from that a higher order correlates with a higher electron energy, which yields a greater tendency for the electrons to be closer to the edge of the wire. 3 of 8 Figure 1: Probability density as a function of radial distance r for a nanowire with fixed radius of 7 nm. The curves represent different orders of the wave function as indicated in the legend. 3.2 Sub band structure Figure 2 shows the energy band structure for a GaAs nanowire of radius 2 nm plotted against kz in the first Brillouin zone. This interval is chosen since it is periodically reoccurring over the entire lattice due to the crystal structure of the material. The band structure in figure 2 displays the the different energy sub band levels which are obtained by combining equation (8) and (9) and plotting against kz. Notice that the arrangement of the sub bands from low to high energy are not in order according to the quantum numbers l and n which is a result of the 2-D confinement. Another important property of the eigenvalues is that all the eigenvalues αl;n are degenerate when l 6= 0. This can not be seen in figure 2 but will important when considering the DOS. Increasing the radius of the GaAs nanowire from 2 nm to 9 nm yields the result shown in figure 3. In comparison to figure 2 one can see that the energy differ- ence between the sub bands decreases and almost merges into a continuum. This means that the nanowire loses its quasi 1-D properties as the radius becomes larger. This result is what we could expect since we are removing the confine- ment in the radial direction by increasing the radius. 4 of 8 Figure 2: The energy-band structure of sub bands for a GaAs nanowire with a radius of 2 nm. Figure 3: The energy-band structure of sub bands for a GaAs nanowire with a radius of 9 nm. 3.3 DOS The density of states for a 1-D system can be calculated by equation (10) when the energy eigenvalues from equation (8) are known. This can then be plotted for different materials by varying parameters such as the effective mass and the radius of the nanowire. Figure 4 and 5 are plots of the DOS for GaAs and InAs nanowires with a radius of 7 nm. The effective mass for electrons in GaAs is 0:067 and 0:032 in InAs which obviously affects the DOS as can be seen in the plots. 5 of 8 It seems that a larger effective mass contributes to more available states in the energy interval [0; 3] eV as shown in the plots. Figure 4: Density of states of a GaAs nanowire. Figure 5: Density of states of a InAs nanowire. 3.4 Conductivity The conductivity is a property that gives concrete information about a materials functionality in form of current conduction.