Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013

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1 Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering times 7 IV. Hall Factors 13 I. Introduction We write the mobility as µ q m * (1) and the Hall factor as r H. () What are these factors in 1D, D, and 3D for parabolic energy bands, E E C + m, (3) * and power law scattering? τ ( ) B τ s. (4) II. Solution of the BE Begin with the general BE: f t + υ i r f + F e i p f Ĉf, (5) and mae the Relaxation ime Approximation to the collision integral, f ( p) p Ĉf f p δ f, (6) to write the steady- state, spatially homogeneous BE as 1

2 p qe i p f δ f. (7) Now assume p f p f (8) to find δ f ( p) +q E i p f. (9) Recognizing that f is a function of energy, we use the chain rule p f ( E) f p E f υ, (1) which can be used to write (9) as δ f p f υ i( qe ). (11) Equation (11) is the solution to the steady- state, spatially uniform BE in the Relaxation ime Approximation. o find the current, we evaluate 1 J n ( r ) ( q) υ ( )δ f ( r, ), (1) L d where d 1,, or 3 depending on the dimension. For the x- directed current: 1 J nx ( r ) ( q)υ x ( )δ f J nx L d 1 ( r ) L d ( r, ) ( q)υ x ( ) f υ i( qe ). (13) For an x- directed electric field 1 J nx ( r ) q υ L d x ( ) f υ x E x 1 J nx ( r ) q υ L d x ( ) f E σ x ne x, so we have J nx σ xx E x σ xx 1 q υ L d x ( ) f. (14)

3 Equation (14) is valid in 1D, D, or 3D for arbitrary bandstructures If we write σ n nqµ n, then we obtain from (14) µ n 1 L d q υ x ( ) f nq qυ x ( ) f E Now let s assume parabolic energy bands and write the mobility as in (1) µ n q τ qυ x f m m * f from which we obtain m * υ x f, (15) f which is the definition of the average scattering time and is valid for parabolic energy bands in 1D, D, or 3D. f. Case i): 3D with parabolic energy bands Assuming that υ is equally distributed between the three degrees of freedom, υ x υ y υ z υ So (15) becomes 3. m * υ 3 f f E, (16) and for parabolic energy bands 1 m* υ ( ) (17) so (16) becomes ( ) f. (18) 3 f 3

4 Equation (18) is analogous to Eq. (7.76) in Near- Equilibrium ransport (Lundstrom and Jeong). If we now assume non- degenerate conditions, then f 1 B f, and (18) becomes ( ) f ( 3 B ) f Recognizing that the average thermal energy is W 3 n B n we can express (19) as. (19) ( ), ( ) ( ) where the average of a quantity, X E according to X X ( E) f ( E) f ( E), () is over the equilibrium distribution function. (1) Equation () is valid for a 3D semiconductor with parabolic energy bands under non- degenerate conditions. It is Eq. (3.6) in Fundamentals of Carrier ransport (Lundstrom). Case ii): D with parabolic energy bands Assuming that υ is equally distributed between the three degrees of freedom, υ x υ y υ. So (15) becomes m * υ f f E, () and for parabolic energy bands, we use (17), so () becomes 4

5 ( ) f. (3) f Equation (3) is Eq. (7.76) in Near- Equilibrium ransport (Lundstrom and Jeong). If we now assume non- degenerate conditions, then f 1 B f, and (3) becomes ( ) f. (4) B f Recognizing that the average thermal energy in D is W n B n ( ), we can express (4) as ( ) ( ), (5) which is the same as (). Equation (5) is valid for a D semiconductor with parabolic energy bands under non- degenerate conditions. Case iii): 1D with parabolic energy bands In 1D, υ x υ So (15) becomes m * υ f, (6) f and for parabolic energy bands, we use (17), so (6) becomes ( ) f. (7) 1 f Equation (7) is analogous to Eq. (7.76) in Near- Equilibrium ransport (Lundstrom and Jeong). 5

6 If we now assume non- degenerate conditions, then f 1 B f, and (7) becomes ( ) f ( B ) f. (8) Recognizing that the average thermal energy in 1D is W n B n ( ), we can express (8) as ( ) ( ), (9) which is the same as () and (5). Equation (9) is valid for a 1D semiconductor with parabolic energy bands under non- degenerate conditions. Summary: he general expression for the average scattering time for a semiconductor with parabolic energy bands is ( ) f d f, (3) where d 1,, or 3 depending on the dimension. For a nondegenerate semiconductor we find (independent of dimension) ( ), (31) ( ) where the average is over the equilibrium distribution as defined in (1). 6

7 III. Exercises: Woring out average scattering times 1) For power law scattering in D Γ s + τ Γ, where s is the characteristic exponent in the expression τ ( ) B τ s, and nondegenerate conditions are assumed. Prove this result. Solution: Begin with the definition of average momentum relaxation time: ( ) f ( E E B η C ) f F d num denom f f (i) Consider the numerator first. In D, we have: num 1 ( E E B η C ) f 1 τ F π Assuming parabolic energy bands: m * d g V de so (ii) becomes num τ g V m * π B s+1 f de B s+1 f π d (ii) (iii) (iv) Change variables to η E E C B E E F C B so num τ g V m * B π η s+1 1+ e η dη (v) (vi) 7

8 num τ g V m * B π Γ s + Now wor on the denominator: denom f 1 denom 1 π f d denom g V m * π denom g V m * B π f de π F η s+1 F f π d dη 1+ e η (vii) m * denom g B V Γ( 1)F π ( ) (viii) Using (viii) and (vii), we find num denom τ Γ s + Γ 1 F s F ( ) (ix) Assuming non- degenerate conditions, we find Γ s + τ Γ 1 Γ s + τ Γ. ) Wor out the corresponding result in 1D. Solution: he solution proceeds much as in problem 1). ( ) f d f 1 B ( E E η C ) f F 1 f num denom 8

9 num 1 B 1 B τ ( ) f E E ( )τ C B B s+1 f d s f d Assuming parabolic energy bands: m* num τ E C B s+1 d m* f m * ( ) 1/ de ( ) 1/ de num τ m * B Change variables: E C B s+1/ f de num τ m * B E C η s+1/ 1+ e η dη num τ num τ denom 1 m * B m * B f B Γ( s + 3/ )F η s+1/ ( ) F B Γ( s + 3/ )F s 1/ ( ) (x) 1 f d denom 1 f f denom m* denom m* B m * 1/ ( 1+ e E E C ) de B ( ) 1/ de 1/ B ( 1+ e E E C ) de B 9

10 denom m* B denom m* B Using (x) and (xi) num denom τ η 1/ dη 1+ e η Γ( 1/ )F 1/ ( ) (xi) Γ( 3/ ) Γ s + 3/ ( 1 )Γ 1/ num denom τ Γ s + 3/ F s 1/ F 1/ F s 1/ F 1/ ( ) ( ) ( ) For non- degenerate conditions power law scattering, and 1D, (xii) Γ s + 3/ τ Γ 3/. 3) Wor out the corresponding result in 3D. Solution: he solution proceeds much as in problem 1) and ). ( ) f 1 ( E E B η C ) f F ( d ) f 3 f num 1 B 1 B 4πτ ( ) f E E ( )τ C B B s+1 f d s f 4π d num denom Assuming parabolic energy bands: m* d m* ( ) 1/ de d m* 3/ 1/ 3 de 1

11 num 4πτ E C B num πτ 3/ m* 3 num πτ 3/ m* B 3 E C Change variables: num πτ ( m* B ) 3/ 3 num πτ ( m* B ) 3/ 3 s+1 3/ f m * B E C η s+3/ dη 1+ e η s+1 B 1/ 3 de 1/ f de s+3/ Γ( s + 5 / )F η s+3/ ( ) F num πτ ( m* B ) 3/ Γ( s + 5 / )F 3 s+1/ Now the denominator: denom 3 f 3 f 4π d denom 3 ( f 4π d 3π f m ) * 3/ 3 denom 3π ( m* ) 3/ 3 f 3/ 1/ de denom 3π m* B 3 Γ( 3/ )F 1/ Using (xiii) and (xiv) num denom τ Γ s + 5 / F s+1/ Γ s + 5 / τ Γ 5 / 3Γ ( 3/ ) F s+1/ F 1/ f de (xiii) 1/ de (xiv) F 1/ ( ) τ Γ s + 5 / ( 3 )Γ 3/ For non- degenerate conditions power law scattering, and 3D, Γ s + 5 / τ Γ 5 /. F s+1/ F 1/ (xv) 11

12 Summary of non- degenerate results: For power law scattering, parabolic energy bands, and non- degenerate carrier statistics. the transport average momentum relaxation time in 1D, D, and 3D are: Γ( ) Γ( 5 / ) Γ s + 3/ 1D : τ Γ 3/ D : τ Γ s + 3D : τ Γ s + 5 /. Knowing these times, we get the mobility from µ q m * he analogous procedure in the Landauer approach is to relate the mobility to the mean- free- path according to µ D n B q υ λ B q o compute λ, we assume power law scattering λ ( ) B λ E r where the characteristic exponent for the mfp is r rather than s. From the definition of the average mean- free- path λ de de λ( E) M ( E) f f M E assuming parabolic bands and nondegenerate conditions, we could express λ in terms of Gamma functions. 1

13 IV. Hall Factors he definition of the Hall factor was given in () as r H. We have wored out the denominator in 1D, D, and 3D. For the numerator, we can recognize from (4) that τ m is also in power law form. From (4), we write τ B s, (xx) so we can evaluate by using the results for with s s. Hall Factor in 1D: (non- degenerate) Γ s + 3/ τ Γ 3/ Γ( 3/ ) Γ s + 3/ τ r H τ Γ( 3/ ) Γ( 3/ ) Γ s + 3/ Γ s + 3/ τ Γ( s + 3/ )Γ 3/ Γ( s + 3/ ) Hall Factor in D: (non- degenerate) Γ( ) Γ s + τ Γ τ Γ s + 13

14 r H Γ( ) τ Γ s + Γ Γ ( s + )Γ Γ s + τ Γ( s + ) Hall Factor in 3D: (non- degenerate) Γ s + 5 / τ Γ 5 / r H Γ( 5 / ) τ Γ ( s + 5 / ) Γ( 5 / ) Γ( s + 5 / ) τ Γ( 5 / ) τ Γ s + 5 / Γ ( s + 5 )Γ 5 Γ( s + 5 ) Summary of Hall factors: (non- degenerate) Γ( s + 3/ )Γ 3/ 1D : r H Γ( s + 3/ ) D : r H Γ ( s + )Γ( ) Γ( s + ) 3D : r H Γ ( s + 5 )Γ( 5 ) Γ( s + 5 ) 14