Density of States: 2D, 1D, and 0D Lecture Prepared by: Calvin R. - PowerPoint PPT Presentation

Density of States: 2D, 1D, and 0D Lecture Prepared by: Calvin R. King, Jr. Georgia Institute of Technology ECE 6451 Introduction to Microelectronics Theory December 17, 2005 ECE 6451 Georgia Institute of Technology

Introduction The density of states function describes the number of states that are available in a system and is essential for determining the carrier concentrations and energy distributions of carriers within a semiconductor. In semiconductors, the free motion of carriers is limited to two, one, and zero spatial dimensions. When applying semiconductor statistics to systems of these dimensions, the density of states in quantum wells (2D), quantum wires (1D), and quantum dots (0D) must be known. ECE 6451 Georgia Institute of Technology

Derivation of Density of States (2D) We can model a semiconductor as an infinite quantum well (2D) with sides of length L. Electrons of mass m* are confined in the well. If we set the PE in the well to zero, solving the Schrödinger equation yields   2 h  − ∇  ψ = ψ 2 E   2 m   ∂ ψ ∂ ψ 2 2 + + ψ = 2 k 0 (Eq. 1) ∂ ∂ 2 2 x y 2 mE = where k 2 h ECE 6451 Georgia Institute of Technology

Derivation of Density of States (2D) Using separation of variables, the wave function becomes ψ = ψ ψ ( x , y ) ( x ) ( y ) (Eq. 2) x y ψ x ψ Substituting Eq. 2 into Eq. 1 and dividing through by y yields ∂ ψ ∂ ψ 2 2 1 1 + + = 2 k 0 where k= constant ψ ∂ ψ ∂ 2 2 x y x y This makes the equation valid for all possible x and y terms only if terms including are individually equal to a constant. ψ ψ ( x ) and ( y ) x y ECE 6451 Georgia Institute of Technology

Derivation of Density of States (2D) Thus, ∂ 2 ψ 1 = − 2 k ψ ∂ 2 x x 2 2 = + where 2 k k k x y ∂ ψ 2 1 = − 2 k ψ ∂ 2 y y The solutions to the wave equation where V ( x ) = 0 are sine and cosine functions ψ = + A sin( k x ) B cos( k x ) x x Since the wave function equals zero at the infinite barriers of the well, only the sine function is valid. Thus, only the following values are possible for the wave number (k): π π n n = = y = ± k x , k for n 1 , 2 , 3 .... x y L L ECE 6451 Georgia Institute of Technology

Derivation of Density of States (2D) Recalling from the density of states 3D derivation… k-space volume of single state cube in k-space:   π π π π       3   = =       V   − sin gle state  a   b   c  V   k-space volume of sphere in k-space: π 3 4 k = V sphere 3 V is the volume of the crystal. V single-state is the smallest unit in k-space 2 mE = where k and is required to hold a single electron. 2 h ECE 6451 Georgia Institute of Technology

Derivation of Density of States (2D) Recalling from the density of states 3D derivation…     π π π π π       3 3 k-space volume of single state cube in k-space:     = = = V           − sin gle state 3  a   b   c  V L     π 3 4 k k-space volume of sphere in k-space: = V sphere 3 V   1 1 1 Number of filled states in a sphere: = Sphere × × × ×   N 2 V  2 2 2  − sin gle state 4 Correction factor for π 3 k A factor of two is added π 3 3  1  4 k L 3 to account for the two redundancy in counting = × × = N 2   possible electron spins identical states +/- n x , +/- π 3 π 2  8  3 n y , +/- n z of each solution. 3 L ECE 6451 Georgia Institute of Technology

Derivation of Density of States (2D) For calculating the density of states for a 2D structure (i.e. quantum well), we can use a similar approach, the previous equations change to the following:     π π π π 2 2     k-space volume of single state cube in k-space:     = = = V         − sin gle state 2  a   b  V L     k-space volume of sphere in k-space: = π 2 V circle k Number of filled states in a sphere:   V 1 1 = × × × N circle 2   V  2 2  − sin gle state π 2 2 2 k  1  k L = × × =   N 2 π 2 π  4  2 2 L ECE 6451 Georgia Institute of Technology

Derivation of Density of States (2D) 2 mE 2 2 continued…… k L k = Substituting yields N = , 2 h π 2 2   2 mE   2 L   2 2 h mL E   = = N π π 2 2 h The density per unit energy is then obtained using the chain rule: 2 dN dN dk L m = = π 2 dE dk dE h ECE 6451 Georgia Institute of Technology

Derivation of Density of States (2D) The density of states per unit volume, per unit energy is found by dividing by V (volume of the crystal). 2 g ( E ) 2D becomes: L m m π 2 h = g ( E ) = 2 D 2 π 2 L h As stated initially for the electron mass, m m * . Thus, * m It is significant that the 2D D = g ( E ) density of states does not depend 2 π 2 h on energy. Immediately, as the top of the energy-gap is reached, there is a significant number of available states. ECE 6451 Georgia Institute of Technology

Derivation of Density of States (1D) For calculating the density of states for a 1D structure (i.e. quantum wire), we can use a similar approach. The previous equations change to the following: π π π       = = =       V k-space volume of single state cube in k-space: − sin gle state a V L       V line = k k-space volume of sphere in k-space:   V 1 = × × N line 2   V 2   − sin gle state Number of filled states in a sphere: k kL = = N π π L ECE 6451 Georgia Institute of Technology

Derivation of Density of States (1D) kL 2 mE Continued….. N = , k = Substituting yields π 2 h 2 mE L L 2 h = = N 2 mE π π h L ( ) Rearranging…… = 1 / 2 N 2 mE π h The density per unit energy is then obtained by using the chain rule: 1 ( ) − 1 / 2 ⋅ 2 mE 2 mL ( ) − 1 / 2 ⋅ dN dN dk 2 mE mL 2 = = = π π dE dk dE h h ECE 6451 Georgia Institute of Technology

Derivation of Density of States (1D) The density of states per unit volume, per unit energy is found by dividing by V (volume of the crystal). g ( E ) 1D becomes: ( ) − 1 / 2 ⋅ 2 mE mL ( ) − 1 / 2 ⋅ 2 mE m m π h = = = g ( E ) 1 D π π L 2 mE h h Simplifying yields… m m ⋅ g ( E ) = 1 D π 2 mE m h 1 m ⋅ g ( E ) = 1 D π 2 E h ECE 6451 Georgia Institute of Technology

Derivation of Density of States (1D) As stated initially for the electron mass, m m * . Also, because only kinetic energy is considered E Ec . 1 m * Thus, = ⋅ g ( E ) 1 D π − 2 ( E E ) h c ECE 6451 Georgia Institute of Technology

Derivation of Density of States (0D) When considering the density of states for a 0D structure (i.e. quantum dot), no free motion is possible. Because there is no k-space to be filled with electrons and all available states exist only at discrete energies, we describe the density of states for 0D with the delta function. Thus, = δ − g ( E ) 2 ( E E ) 0 D c ECE 6451 Georgia Institute of Technology

Additional Comments The density of states has a functional dependence on energy. ECE 6451 Georgia Institute of Technology

Additional Comments ECE 6451 Georgia Institute of Technology

Practical Applications Quantum Wells (2D) - a potential well that confines particles in one dimension, forcing them to occupy a planar region Quantum Wire (1D) - an electrically conducting wire, in which quantum transport effects are important Quantum Dots (0D) - a semiconductor crystal that confines electrons, holes, or electron-pairs to zero dimensions. ECE 6451 Georgia Institute of Technology

Quantum Dots • Small devices that contain a tiny droplet of free electrons. • Dimensions between nanometers to a few microns. • Contains single electron to a collection of several thousands • Size, shape, and number of electrons can be precisely controlled ECE 6451 Georgia Institute of Technology

Quantum Dots • Exciton: bound electron- hole pair (EHP) • Attractive potential between electron and hole • Excitons generated inside the dot • Excitons confined to the dot – Degree of confinement determined by dot size – Discrete energies ECE 6451 Georgia Institute of Technology

Fabrication Methods • Goal: to engineer potential energy barriers to confine electrons in 3 dimensions • 3 primary methods – Lithography – Colloidal chemistry – Epitaxy ECE 6451 Georgia Institute of Technology