By Hari G. Garg, Charles A. Hales

Advances in warmth move is designed to fill the knowledge hole among on a regular basis scheduled journals and collage point textbooks by means of supplying in-depth evaluate articles over a broader scope than is allowable in both journals or texts.

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In those equations assumed a scalar effective mass for the electrons m*. The electric ﬁeld is determined using the Gauss law equation written in terms of electric potential (E : 9 ), · ( ) : 9e(N 9 N 9 n ; p) : 9eNC L N NC : (N 9 N 9 n ; p), L N (123) volume averaging theory 43 where is the dielectric constant of Si, N is the n-doping concentration, N L N is the p-doping concentration, and p is the hole number density. B. VAT-Based Two-Temperature Conservation Equations Conservation equations derived using VAT enable one to capture all of the physics associated with transport of heat at the micro scale with more rigor than any other method.

There are other possible forms of the left-hand side of the momentum equation VAT equations that will not be pursued at this time. volume averaging theory 49 The homogeneous volume averaged electron gas energy equation for a heterogeneous polycrystal becomes *W C *t K ; 1 · (W V )2 : 9e1nV · E2 9 k 1 · (nV T )2 C C K C K C C K (W 9 (3/2)k T ) C M , ; 1 · (k T )2 9 C C K ^ ^ C\M (147) or *1W 2 1 C K ; · 1W V 2 ; C C K *t * 1KQ (W V ) · ds C C K k (nV T ) · ds C C K 1KQ ; · [+k , (1s 2+T , )] ; · [1s 2+k T , ] C K K C K C C K +k , *T 1 C K C · ds .

Assuming constant thermal properties, the two equations can be combined, yielding a onetemperature equation *T *T 1 *T 1 *T J; C J : J; J, (109) *x C *x*t *t C *t 2 2 2 where the thermal diffusivity of electron gas , equivalent thermal diffusivC ity , and thermal wave speed C are deﬁned by 2 2 K K K G C , C : C . : C, : (110) 2 C C C C 2 C ;C C J C J C Tzou [96] later proposed a uniﬁed two-ﬂuid model to derive the general hyperbolic equation with two relaxation times and , 2 O 1 *T * *T ( T ) : ; O , (111) T ; 2 *t *t *t which he argues is the same equation derived from two-step models in metals.