Ssp_14 - 5 Electrons in Solids Adiabatic approximation(see Chapter 4 no interaction between electrons and moving nuclei Extension nuclei core

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5-1 5 Electrons in Solids Adiabatic approximation (see Chapter 4) : no interaction between electrons and moving nuclei. Extension: nuclei + core electrons atomic core calculation of electronic states in approximation of rigid atomic cores. One electron approximation: Schrödinger eq. for > 10²³ interacting electrons in periodic potential of atomic cores! Simplification: one electron in electrostatic potential of atomic cores and all other electrons. Other electrons largely screen potential of atomic nuclei . Electron -electron interaction neglected: important for magnetism and superconductivity! Fig. 5.1 Sketch of the potential for one electron in a periodic lattice of positive cores (+). The vacuum level E vac is the level to which the electon has to be promoted in order for it to leave the crystal and to escape to infinity. The simplest approximation of this system is an electron in a square potential well (- ---) with infinitely high walls at the surface of the crystal.
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5-2 5.1 Free Electron Model 5.1.1 Free Electron Gas and Fermi Statistics Additional approximation: interaction of electron with atomic (cores is neglected). Model of free electron gas in square wall potential Metal = cube of side L Infinite walls (actual work function typ. 5 eV) Schrödinger eq. for a particle in an infinite square wall potential: −+ = h ² () () () ' () 2 m rV r r E r ψψ ψ Vxyz xyz L (,,) ,, = ≤≤ V = const. for 0 otherwise, 0 E = E‘ - V 0 −= h ² 2 m rE r in cube (5.1) Infinite walls r = 0 outside cube (5.2) (Fixed boundary conditions) Normalization condition: * rr d r box = 1 (5.3)
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5-3 Solution of (5.1) with (5.3): ψ ( ) sin r L kx ky kz xyz = 2 3 2 (5.4) with () E k mm kkk xyZ == + + hh ²² ² 22 222 (5.5) (5.2) (0, y , z ) = (L, y , z ) = 0 0 ≤≤ yz L , = = = k L n k L n k L n nnn n n xx yy zz i i π ,, . = 1,2,3. . = 0: not square integrable < 0: no linearly independent solution (5.6) L macroscopic quasicontinuous descrete k values ππ La << often dk ks t a t e s
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5-4 Fig. 5.2 Spatial form of the first three wavefunctions of a free electron in a square well potential of length L in the x -direction. The wavelengths corresponding to the quantum numbers n x =1,2,2,. .. are λ =2 L , L , 2 L /3, . ..
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5-5 Density of States Fig. 5.3 a), b). Prepresentation of the states of an infinite square well by means of a lattice of allowed wave vector values in k -space. Because of the two possible spin orientations, each
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This note was uploaded on 01/19/2010 for the course MATERIALS M504 taught by Professor Adelung during the Spring '02 term at Uni Kiel.

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Ssp_14 - 5 Electrons in Solids Adiabatic approximation(see Chapter 4 no interaction between electrons and moving nuclei Extension nuclei core

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