(A6.8) 



1022 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



We see that for thermal equilibrium, with T = Te and v = Ve , this equation 

 gives correctly the result that there is no net interchange of energy be- 

 tween electrons and phonons. 

 The equation for mobility for the case of phonon scattering is 



/xo = ^qt/diy/irVim. (A6.7) 



This expression may be used to reduce the steady state equation: 



= (cte/i/)field — {d8>/dt) phonons 



= {vT/ve)qfJLoE^ + {S/'\/T){mchT/-C){Ve/vT) [1 — {Vc/VtY 



to 



(ve/vr)' - (ve/vr)' = (St/ S2) (ji^/ cf . (A6.9) 



This equation may be solved for Ve/vri 



(ve/vr = a/(4)(1 + [1 + (STr/SKnoE/cWy' (A6.10) 



The drift velocity then becomes 



Vi = nE = Mofi V2/(l + [1 + {d>ir/^){n,E/cni''Yi\ (A6.11) 



For E <<C c/mo, we have 



Vd = Mo£. (A6.12) 



For E ^ c//xo, we have^^ 



Vd = (32/37r)i/^ (mo£c)i/2 = iM(noEcy'\ (A6.13) 



These equations define a critical field Ec at which the two limiting cases 

 would give the same mobility: 



Ec = {32/3Try'\/fio = 1.84c//xo. (A6.14) 



At this critical field, the drift velocity is less than the value on either limit- 

 ing form by a factor of 



Vd/noEc = \/2 / [1 + 5^/2]i/2 = 0.785. (A6.15) 



This reduction in mobility corresponds to an electron temperature of 



Te = r/(0.785)2 = 1.62r. (A6.16) 



Expressed in terms of c the drift velocity given by the limiting forms is 



Vc = Vd (extrapolated) = 1.84c (A6.17) 



"This relationship has been j)ul)lishe(i in Bulletin of Am. Phys. Soc, Vol. 26, No. 1, 

 paper 55. 



