TAILIEUCHUNG - Fundamentals of Semiconductors Part 3

Tham khảo tài liệu 'fundamentals of semiconductors part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 84 2. Electronic Band Structures 52 exP ri-rr - j X i mik r - r 7 I H Ị ộrrípk r - rjr . Instead of summing j over all the unit cells in the crystal we will sum over the nearest neighbors only. In the diamond and zinc-blende crystals this means i will be summed over the atom itself plus four nearest neighbors. These atoms will be denoted as i 1 2 3 4 5. With these approximations the collection of matrix elements of the form in forms an 8x8 matrix. Applying symmetry arguments allows the number of nonzero and linearly independent matrix elements of 3Ễjnt to be greatly reduced. As an example we will consider the matrix element H 1_S2- From this matrix element is given by HS1_S2 exp i d 4- exp ik 4 2 exp i 4 3 exp ik 4 4 X 51 r .n 52 where we have assumed that atom 1 is located at the origin and da a 1 to 4 are the positions of its four nearest neighbors with 4 1 1 1 l ữ 4 4 2 1 4 4 3 -1 1 -l 4 and 4 4 1 1 l a 4 . The matrix element 511 ẵ int I 52 is basically the same overlap parameter vssa as we have defined for molecules. The other matrix elements . 2 and Hpj. 2 etc. can also be expressed in terms of the overlap parameters vspo Vppo and vppĩt. For example 2 can be shown to contain four terms involving the four phase vectors exp ik da and the matrix element 51 I â int I 2f2 . However for each nearest neighbor 51 pfjnt I 2 has to be decomposed into Ơ and -T components as shown in Fig. . This decomposition introduces a factor of COS0 l 3 . The or sign depends on whether the 5 orbital lies in the direction of the positive or negative lobe of the px orbital. As a result it is convenient to introduce a new set of four overlap parameters appropriate for the diamond lattice 4V vsp ự3 4V w 3 8V 3 vxy 4Vwơ 3 - 4 3 With this notation the matrix element 51 r I 9 iĩlt I X2 r - dị is given by VSpo V3 Vjp 4. The remaining three matrix elements are related to Tight-Binding or LCAO Approach to the Band .

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