Lecture 3: Density of States

Transcription

1 ECE-656: Fall 2011 Lecture 3: Density of States Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA 8/25/11 1 k-space vs. energy-space N 3D (k) d 3 k =! 4" d 3 k = D 3 3D ( E)dE N(k): independent of bandstructure D(E): depends on E(k) N(k) and D(E) are proportional to the volume,!, but it is common to express D(E) per unit energy and per unit volume. We will use the D 3D (E) to mean the DOS per unit energy-volume. 2

2 about the limits of the integrals f 0! 0 BW >> k B T E F 3 outline 1) Density of states 2) Example: graphene 3) Discussion 4) Summary This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License. 4

3 example: 1D DOS 5 example: 1D DOS for parabolic bands independent of E(k) parabolic E(k) E = E C +!2 k 2 2m * D 1D (E) = 1!! 2m * E " E C! = 1! de dk = 2 ( E " E C ) m * 6

4 density of states in a nanowire 7 2D density of states 8

5 density of states in a film 9 effective mass vs. tight binding T Si = 3 nm sp 3 s*d 5 tight binding calculation by Yang Liu, Purdue University,

6 effective mass vs. tight binding near subband edge well above subband edge sp 3 s*d 5 tight binding calculation by Yang Liu, Purdue University, exercise 12

7 how does non-parabolicity affect DOS(E)? non-parabolicity increases DOS (E) 13 alternative approach 14

8 proof in k-space, we know: can also work in energy-space: $ n L = f 0 E ( ) 1 L n L = 1! f L 0 ( E)" E # E k k $ #! E " E k de k ( ) ( ) de 15 interpretation counts the states between E and E +de 16

9 outline 1) Density of states 2) Example: graphene 3) Discussion 4) Summary 17 graphene Graphene is a one-atom-thick planar carbon sheet with a honeycomb lattice. source: CNTBands 2.0 on nanohub.org Graphene has an unusual bandstructure that leads to interesting effects and potentially to useful electronic devices. 18

10 graphene E(k) Brillouin zone Datta: ECE 495N fall 2008: (Lecture 21) (Lecture 22) 19 simplified bandstructure near E = 0 We will use a very simple description of the graphene bandstructure, which is a good approximation near the Fermi level. k y (valley degeneracy) k x neutral point ( Dirac point ) We will refer to the E F > 0 case, as n-type graphene and to the E F < 0 case as p-type graphene. 20

11 D 2 D DOS for graphene: method 2 ( ) ( E) = 1 #! E " E A k = 1 A k A 2$ & ( ) % 2 '!(E " E )2$k 2 k dk 0 D 2 D g V % ( E) =!! 2 2 " &#(E $ E k )E k de k F 0 D 2 D ( E) = 2E E > 0!! 2 2 " F D 2 D ( E) = 2 E!! 2 2 " F 21 DOS for graphene: method 1 22

12 DOS for graphene: method 1 kdk N(k) dk = Ag V! EdE = Ag V!!" F = AD 2 D ( E)dE ( ) 2 D 2 D ( E) = 2 E!! 2 2 " F 23 outline 1) Density of states 2) Example: graphene 3) Discussion 4) Summary 24

13 density of states D 1D E D 2D E D 3D E 25 density of states for bulk silicon 6 DOS (10 22 cm 1 ev 1 ) ENERGY (ev) The DOS is calculated with nonlocal empirical pseudopotentials including the spin-orbit interaction. (Courtesy Massimo Fischetti, August, 2011.) 26

14 computing the density of states 6 DOS (10 22 cm 1 ev 1 ) ENERGY (ev) no. of states = (!k) 3 2" # ( ) $ 2 Courtesy Massimo Fischetti, August, density of states for bulk silicon (near the band edge) 10 conduction band 8 valence band DOS (10 21 cm 1 ev 1 ) ELECTRON KINETIC ENERGY (ev) 8 m e,d1 = m e (g c =6) α 1 = 1.0 ev 1 m e,d2 = m e (g c =6) α 2 = 0.0 ev 1 DOS (10 21 cm 1 ev 1 ) m h,d1 = m e (g v =1) α 1 = 0.5 ev 1 m h,d2 = m e (g v =1) α 2 = 0.25 ev HOLE KINETIC ENERGY (ev) 28 (Courtesy Massimo Fischetti, August, 2011)

15 outline 1) Density of states 2) Example: graphene 3) Discussion 4) Summary 29 summary 1) When computing the carrier density, the important quantity is the density of states, D(E). 2) The DOS depends on dimension (1D, 2D, 3D) and bandstructure. 3) If E(k) can be described analytically, then we can obtain analytical expressions for DOS(E). If not, we can compute it numerically. 30

16 questions 1) Density of states 2) Example: graphene 3) Discussion 4) Summary 31