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Extra resources for Semiconductor Physics: An Introduction
Sample text
For,,~1 38 3. 16) is obtained 2. For equal effective masses of electrons and holes, the Fermi level is independent of the temperature and is located in the middle of the gap. For m n ~ mp , the Fermi level approach es the conduction band edge with increasing temperature. 7 meV. 20) For the entropy density, a rather lengthy expression is obtained [Ref. 1, Chap. 8 h] which is simplified by neglecting all donors and acceptors: Sn = - «(IT) n + UnlT + k B Ne F 312 (1'/n) = (5 Unl3 - n OIT. 21) From F= U - TS, we now find for the free energy F n = n (- 2 Un13.
D(11 k z). According to quantum statistics, phase spaee ean be thought to consist of eells of volume h3 = (27t 11)3 with up to 2 electrons of opposite spins ............ 0 E: fIEl E: j 9j E:3 93 E: 2 92 E:, 9, " .... '':\, \ .......... \ \ " "\ i . ". 2. Energy levels EJ, E2, ••. , in energy bands, with gl, g2, ... 3. Fermi-Dirac distribution function for vari· ous values of the temperature. 1 Fermi Statistics 37 per cello Integration over coordinate space yields the crystal volume V. Hence, the number of states in an energy range de is given by Vh 3tP k V g(e)de=2 (21th)3 =2 41t2 (27 m,,)312 (e-ee) l12 de.
The two atoms are therefore centers of cells a and b. 2 Energy Bands by Tight Binding ofElectrons to Atoms 19 the ",-function in cell b centered at e = (a/4; a/4; a/4) is given by "'b = exp (2 n i k· e) . 2) a/4) + Cb(Y - a/4) + Db(z - a/4)]} p(r')/r', where r' is the distance from the center of cell b. The continuity condition of '" and d",/dr at the halfway intermediate points between the atom and its 4 nearest neighbors yields 8 equations for the 8 unknown coefficients A a ••• D b • A solution of the homogeneous system of linear equations exists if the determinant vanishes, yieldingp2 = 0,p'2 = (dp/dr)2 = 0 and sp' + s' P sp' - s' p ~-~ = + 1 [1 + cos(n a k -=2 + cos(n a k z) x) cos(n a k y) + cos(n a k y) cos(n a k z) cos(n a k x )]ll2.