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2 EE Fall 2007 Class 5 slides 1 Outline Review of the behavior free electrons in a solid Band theory Periodic potential wells Energy bands : E vs k diagram Extended and reduced zone representation Effect of an externally applied electric field Steady state electron distribution Band structure in an insulator and in a conductor 2 3 Band Theory Ref: (pp26-35) In the free electron model we assumed that the potential energy in the solid (three dimensional box) was constant and therefore there was no force acting on the electrons. U(x) = U0 where U0 is a constant and does not vary with x In reality there is a force acting on the electrons due to the charge on the ionized atoms of the solid. Since the atoms are periodically located in the solid the potential energy varies within the solid (box) periodically as shown in the figure. U(x) = U(x + a) a is the inter-atomic distance. U(x) is periodic in x with as the period. 4 We have chosen an onedimensional space for conceptualization process. Atoms are located at intervals of a along the x-axis. U(x) = U(x+ na) where n is an integer positive or negative. The electron experiences a force due to the periodically located (positively ionized) atoms and this gives rise a periodically varying potential energy U(x). Under this condition the wave function is not just a plane wave solution but a plane wave whose amplitude varies periodically with the same periodicity. 5 6 7 The energy versus k diagram shows a split into bands of allowed energy separated by bands of forbidden energy as shown in the figure in the next slide. E vs k diagram is still quasi-continuous but the energy plot has gaps at kx = n/ a where n can be a positive or negative integer. (Do not confuse this with n the quantum numbers n x , ny , nz .) In other words, the energy gap arises at kx = + or - / a, + or - 2 / a, + or - 3 / a, + or - 4 / a etc. The energy varies quasi-continuously from 0 to a certain value until kx becomes equal to / a. If kx is increased beyond / a, there is a discontinuity in energy, the energy suddenly jumps to a higher value leaving a gap in energy. The energy range from its value at k x = 0 and its value at kx = / a is called the first energy band and the range of energy between kx = / a and kx = 2 / a is called the second energy band. The difference in energy between the lowest band and the second band is called the forbidden gap or simply bandgap. The consequence is that although the momentum states exist in the infinite range of kx values from negative infinity to positive infinity, energy exists only in certain bands. 8 If we recall that the electron velocity is the group velocity of the wave packet equal to (h/2p)-1dE/dk, we see that the velocity varies from zero in the bottom of the band to a maximum in the middle of the band and becomes zero as the top of the band is reached. Electron states in the positive region of k x values are positive and those in the negative region of kx values are negative. 9 10 11 12 Effect of an applied electric field 13 14 15 16 17 As electrons go to higher energy states, electrons get scattered by phonons and impurities in the crystal. According to the solid state physics theory, the distribution of electrons under steady state conditions is the same as what would exist in the distribution if the electric field had acted on the electrons for a period of time c c where is called the collision (or scattering) relaxation time. 18 19 20 21 22 23

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