The BTE with a B-field: Simple Approach

Transcription

1 ECE 656: Electronic Transport in Semiconductors Fall 017 The BTE with a B-field: Simple Approach Mark Lundstrom Electrical and Computer Engineering Purdue University West Lafayette, IN USA 10/11/17

2 Introduction Solving the BTE with a small B-field gives: J n = σ S E -σ S µ H E B ( ) σ S = n S qµ n µ H = µ n r H ω c << 1 Hall mobility µ n q m * r H Hall factor

3 A question Solving the BTE with a B-field can get mathematically complicated. Is there a simpler way? 3

4 A simpler (approximate) approach F e = qe - q υ B = d p dt Assume that scattering makes p = 0 after a scattering time, τ d p dt p = q E - q υ B p = ( q E - q υ B ) 4

5 A simpler (approximate) approach p = ( q E - q υ B ) = m * υ υ = q m * E - q m * υ B (Can be solved exactly for the velocity. See prob. 4.18, Lundstrom.) υ q m * E υ qτ me + q m * ( m * ) E B (Low B-fields) 5

6 B-field dependent current equation υ qτ me + q m * ( m * ) E B υ q τ m E + q m * ( m * ) E B Average velocity of all electrons with different scattering times. J n = nq υ = nq q m * E - nq q ( m * ) E B 6

7 Current equation J n = nq q m * E - nq q ( m * ) E B J n = nqµ n E - nq q m * q m * E B J n = σ n E -σ n µ n E B J n = σ n E -( σ n µ H ) E B µ H = r H µ n r H = τ m 7

8 Summary The simple approach gives the same current equation as the BTE: J n = nqµ n E -( σ n µ H ) E B But not a prescription for computing the averages: µ H = r H µ n r H = τ m 8

9 Comment We have assumed a low B-field. What does low mean? What happens if the B-field is large? 9

10 B-field dependent DD equation: again F e = qe - q υ B = d p dt p = ( q E - q υ B ) = m * υ υ = qτ me - q m * m * υ B Consider a D geometry. Electric field in the x-y plane with B-field normal to the plane. 10

11 D solution with a z-directed B-field υ = qτ me - q m * m * υ B υ x = q m * E x - q m * υ y υ y = q m * E y + q m * υ x D problem z-directed B-field 11

12 Solution in D with z-directed B-field υ x = q m * E x - q m * υ y υ y = q m * E y + q m * υ x υ x = q m * E x + q m * E y q m * υ x υ x 1+ q m * = q m E * x + q m * E y υ x = µ ne x + µ n E y 1+ ( ω c ) ω c = q m * cyclotron frequency 1

13 Cyclotron frequency Low B-field means that electrons scatter many times before completing an orbit. ω c << 1 T >> F B υ High B-field means that electrons can complete an orbit without scattering. ω c >> 1 T << y B-field points out ( ) ω c = π T 13 x

14 Cyclotron frequency ω c = q m * = µ n 14

15 Some numbers assume silicon n 0 = cm -3 J n = 10 A/cm µ n = 1000 cm V-s r H = 1 =,000 Gauss ( 10 4 Gauss = 1T) µ H 0.0 << 1 For moderate mobilities and typical B-fields, we are usually in the low B-field regime. W y = 1 µm 15

16 Solution υ x = µ ne x + µ n E y 1+ ( ω c ) υ y = µ ne y µ n E y 1+ ( ω c ) Assume T = 0 K, so there is no need to average the scattering time. J x = nqυ x = J y = nqυ y = σ n ( ) E x µ n E y 1+ µ n σ n 1+ µ n ( ) ( ) ( ) E x + µ n E y µ n = q m * = ω c τ 16

17 Magneto-conductivity tensor J x J y = σ n 1 µ n ( ) µ n 1 1+ µ n E x E y ( ω c = µ n ) J i = σ ij ( )E j 17

18 Comments σ ij 1 µ n 1+ µ n µ n 1 ( B ) z = nqµ n 1) A magnetic field affects both the diagonal and offdiagonal components of the magneto-conductivity tensor. ) Small magnetic field means: µ n << 1 ω c τ << 1 18

19 Magneto-resistivity tensor J x J y = σ n 1 µ n ( ) µ n 1 1+ µ n E x E y J i = σ ( ij )E j J x J y = σ L σ T σ T σ L E x E y ρ L = σ L σ L + σ = 1 T σ n E x E y = ρ L ρ T ρ T ρ L J x J y ρ T = σ T σ L + σ T = µ n σ n 19

20 Comparison J x J y = σ n 1 µ n ( ) µ n 1 1+ µ n E x E y E x E y = 1 1 µ n σ n µ n 1 J x J y 1) Longitudinal magneto-resistance is independent of B ) Hall voltage is proportional to B 0

21 Strong B-fields For strong B-fields: µ n >> 1 ω c τ >> 1 Interesting quantum effects can occur. 1

22 Hall and integer quantum Hall effect Hall effect V x V H Quantum Hall effect V H V x V H ~ B V x independent of B Longitudinal magnetoresistance M.E. Cage, R.F. Dziuba, and B.F. Field, A Test of the Quantum Hall Effect as a Resistance Standard, IEEE Trans. Instrumentation and Measurement, Vol. IM-34, pp , 1985