PH575 Spring 2019 Lecture #10 Electrons, Holes; Effective mass Sutton Ch. 4 pp 80 -> 92; Kittel Ch 8 pp 194 – 197; AM p. <-225->
Thermal properties of Si (300K)
3# 2#
∆T 1# 0# !1# 0# 1# 2# 3# !2# !3# p!Si#<111># Vs !4# n!Si#<100># !5# Seebeck#Voltage#(mV)# Temperature#Difference#(K)#
• p-type <111> Si • n-type <100> Si • p = 6.25×1018 cm-3 • n = 9.5 ×1017 cm-3 • ρ = 0.014 Ωcm • ρ = 0.021 Ωcm • S = + 652 µV K-1 • S = - 872 µV K-1 • κ = 148 W/mK • κ = 148 W/mK Rodney Snyder -3 2 2 -3 2 2 • PF = 3 x 10 V /K Wm • PF = 3 x 10 V /K Wm Dan Speer • ZT = 0.006 • ZT = 0.006 Josh Mutch Dispersion relation for a free electron, where k is the electron momentum: 2 k 2 E(k) = mass 2m Group velocity −1 −1 ! 1 d 2 E k $ d k 1 dE k k p ( ) ! ( + $ ( ) 2 2 = * - = m = = = v # dk & #dk ) m, & dk m m " % " %
These are generalizable to the periodic solid where, now, k is NOT the individual electron momentum, but rather the quantity that appears in the Bloch relation. It is called the CRYSTAL MOMENTUM
1 −1 m* 2 2 E k v = ∇k E(k) = #"∇ k ( )%$ Problem on hwk Change in energy on application of electric field: δ E = qε ⋅vδt (F ⋅d) Change in energy with change in k on general grounds: δ E = ∇k E ⋅δk 1 v = ∇k E(k) d(k ) qε ⋅vδt = vk ⋅δk ⇒ qε = dt
Get an equation of motion just like F = dp/dt ! hk/2π acts like momentum ….. We call it crystal momentum, because it's not the true electron momentum. Crystal momentum is what is changed by the external force. Now define m* by: d k dv ( ) ≡ m * k dt dt 2 d k dvk dk m * d E ( ) m * = 2 2 dk dt dk dt
m * d 2 E 2 = 1 ⇒ m* = 2 dk 2 d 2 E 2 dk
2 In 3-d: m* = 2 ∇k E http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/illustr/effective_mass.gif E(k) 2
1-dimensional chain (& β<0): ka -π π E(k) = α + 2β cos(ka) 0
1 v(k) v = ∇k E(k) 2
ka v(k) = −2βasin(ka) -π π
2 mass m* = 10 d 2 E 2 dk 2 − ka m* = 2 2βa cos(ka) Holes • If one electron in state k is missing from an otherwise filled band, all the other ≈ 1023 electrons can be described by the concept of a single hole. What is this hole's momentum, energy, velocity, mass ?
• Momentum: The hole's momentum is -k (in units of h/2π); in other words the opposite of the momentum of kh = −ke the missing electron. E The completely filled band has zero total momentum (as many CB states k as -k). The loss of electron with momentum k k changes total band momentum VB to -k. This is the momentum of k k the equivalent hole. e h Holes - Energy • The energy of the hole is the negative of the energy E k = −E k of the missing electron (E=0 h ( h ) e ( e ) is top of VB). • In order to excite an electron to create an electron-hole pair, energy is added to the system, e.g. optically. E CB • Of the added energy, E is assigned to the e' ω Ee' electron in the CB, and k E VB Eh is assigned to hole h in the VB. ke kh Holes E CB • These properties can be described by replacing the entire Hole” Band valence band (with many k electrons and 1 missing one), by VB a band with a single particle of ke kh momentum kh=-ke and energy E = -Ee (where subscript e refers to the state of the MISSING electron, not the the state in the CB the electron has gone to (e'). E CB • This is the green band in the picture. “Hole” Bandk ω = E + E k k e' h e h >0 >0 Holes - velocity
• The (instantaneous) velocity of E the hole is equal to the CB (instantaneous) velocity of the unoccupied electron state (same Hole Band slope of E(k). k VB v k = v k k k h ( h ) e ( e ) e h
• But note that the the velocity of the excited electron in the CB is not relevant here! Holes
• Effective mass: The effective mass of the hole is mh kh = −me ke opposite to the effective mass of ( ) ( ) the missing electron. (Curvature of inverted dispersion relation is opposite). E CB