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Unformatted text preview: 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 6 Quick review of Lecture 5 In the last lecture, we approximate the potential field as rectangular wells in the crystal. From periodicity, the Bloch theorem gives additional equations ( ( ) ikn a + b ) [ x ) ] = Ψ x e , Ψ + ( a + b n Ψ = Ae − iKx which is used to determine the coefficients in + Be iKx . We have also determined the value of the wavector k in the Bloch theorem, using the Born-von Karman periodic boundary condition Ψ [ x + a N + b )] = Ψ ( x ) . This yields allowed k ( values as n 2 2 π n π k = N(a + b) = L (n=0, ± 1, ± 2,...), where L is the length of the crystal. For each k n , there are two quantum states denoted by Ψ k s (spin up, spin down). When n n , goes from 1 to N, k n varies from 0 to 2 / π a . Therefore, in the following E-k figure we totally have 2N quantum states Ψ k s . n , k Note: (1) For big crystals, N is very big (on the magnitude of 10 23 ) and ∆ = 2 π is also L small. The following curve can be regarded as quasi-continuous. (2) The Born-von Karman periodic boundary condition is no longer valid when N becomes very small in nanomaterials. 15 10 Normalized Electron Energy E/Eo 5 0 - 1 1 k / ( π /a) 2.57 Fall 2004 – Lecture 6 1 3.1.3 Consequences of solid energy levels we just obtained: (1) Electrons wave function extends through the whole crystal, they belong to all atoms collectively. Recall the splitting of waves discussed in last lecture. According to Pauli’s exclusion principle, the wavefunctions of adjacent wells cannot overlap. A continuous wave extending through the whole structure will be formed in this situation. The wavefunction no longer corresponds to an individual atom. NOT allowed by Pauli exclusion principle Continuous wavefunction across the structure -13.6 eV -13.6 eV/n 2 . . . Similar argument also exists for atomic energy levels. When two atoms become closer, the overlap of electron wave functions will cause band split. (2) Filling of electrons As mentioned before, in E-k figure every band (k changes for 2 / π a ) can accommodate 2N quantum states. At zero temperature, the filling rule for the electrons is that they always fill the lowest energy level first, as required by thermodynamics. If one atom only has one electron, the band is half filled since there are only N valence electrons in this case, as shown in the next figure. The topmost energy level that is filled with electrons at zero Kelvin is called the Fermi level . The electron energy and momentum can be changed (almost) continuously within the same band because the separation between...
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- Fall '04
- Energy, Condensed matter physics