Phys463.nb. Many electrons in 1D at T = 0. For a large system (L ), ΕF =? (6.7) The solutions of this equation are plane waves (6.

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1 â â x Ψn Hx Ε Ψn Hx 35 (6.7) he solutions of this equation are plane waves Ψn Hx A exphä n x (6.8) he eigen-energy Εn is n (6.9) Εn For a D syste with length and periodic boundary conditions, Ψn Hx Ψn Hx, we now that the wavevector (oentu) is quantized n n Π (6.0) with n being an integer, n, -, -, 0,,, Or say, Π n n (6.) he eigen-energy here is also quantized n Εn n Π (6.) For a large syste, the energy and oentu becoe continuous variables. Many electrons in D at 0 At 0, the electrons want to stay in the lowest energy states to iniize the energy. But electrons are ferions, which eans that they cannot all just go to the lowest energy state Ε0. Due to the Pauli exclusive principle, one quantu state can host at ost one electron. hese eans that for each n, we can have at ost two electrons (one with spin up and one with spin down). If we have electrons, at 0, the electron occupies the lowest states. he energy of the highest filled state is nown as the Feri energy ΕF. he oentu of this state is nown as the Feri oentu. he wavevector of this state is nown as the Feri wavevector F. Obviously, F ΕF and F (6.3) For a large syste ( ), ΕF? he nuber of quantu state between - to is (the factor coes fro the spin degrees of freedo): - - (6.4) uber of electrons in these states e (6.5) So Π e Π e (6.6) Here e is the density of electrons (nuber of electrons per length). ΕF Π e 8 he botto line:, ΕF and F are deterined by the density of electrons. (6.7)

2 Free electrons in D at finite teperature Distribution function he distribution function f HΕ easures the average nuber of electrons on a quantu state with energy Ε. At zero teperature, the nuber of electrons on a quantu state is if Ε < ΕF and 0 if Ε > ΕF. In other words, this function is a step function f HΕ : Ε < ΕF 0 Ε > ΕF (6.8) At finite, the electron nuber on a quantu state is O fixed. Soeties this state is occupied ( electron) and soeties it is epty (0 electron). So the average electron nuber is soe nuber between 0 and. 0 < f HΕ < at finite (6.9) Feri distribution: For non-interacting ferions, at finite teperature, the distribution function taes this for f HΕ Ε- B (6.0) where is nown as the Feri-Dirac distribution. et s copare it with the Planc distribution (for phonons) we learned in the previous chapter. fphonon HΕ Ε B - (6.) We found two differences: () - turns into and () we have an extra paraeter for electrons. he ± is deterined by the nature of the particles. For bosons (photons, phonons, etc.), it is -. For ferions (electrons, protons), it is always. he paraeter is called the cheical potential. It is a very iportant paraeter in therodynaics and statistical physics (it is as iportant as the concept of teperature). It controls the density of particles in a syste. Here, for a Feri gas, we can consider it as a generalization of ΕF at finite. In fact, we will show below that when 0, ΕF. How to deterine? he cheical potential can be deterined by counting the total nuber of electrons e - f HΕ - exp (6.) - B Divided by on both sides: e - exp (6.3) - B If we now the density of electrons e, we can solve this equation to find. In general, depends on and the electron density. As 0, ΕF. Soe properties of f HΕ (a) f HΕ turns into a step function at 0. At low, Ε- B becoes very large. Depending on the sign of Ε - Ε- i0 herefore, B : - Ε < Ε > (6.4)

3 i0 f HΕ i0 Ε- B : exph - exph 0 37 Ε< (6.5) 0 Ε> his result recovers the zero teperature liit we studied above. And it is easy to notice that the cheical potential plays the role of ΕF here. In other words, ΕF at 0. (b) At any, f H f H - B exph0 (6.6) (c) At < B, f HΕ is close to a step function. Only the part with Ε~ shows strong deviation fro the step function f0 HΕ PlotB: Ε- Ε- E E ,.., Ε- E Ε- E.,..0, Ε- E. 00>, 8Ε, 0, <, PlotStyle hic, Axesabel 8"Ε ", "fhε"<, abelstyle Mediu, Plotegends :"0", " ", " ", " ", "00 ">F 00 B 0 B B B fhε B 0 B B B Ε Free electrons in 3D at zero teperature For free electrons, the Hailtonian in 3D is px p y pz p H One electron in 3D B -ä -ä x -ä y z F- x y z (6.7) he Schrodinger equation is - x y z Ψ Hx, y, z Ε Ψ Hx, y, z (6.8) he solutions for this equation are 3D plane waves Ψ Hx, y, z A expaä Ix x y y z zme (6.9)

4 38 Considering periodic boundary conditions Ψ Hx x, y, z Ψ Hx, y, z (6.30) Ψ Ix y, y, zm Ψ Hx, y, z (6.3) Ψ Hx z, y, z Ψ Hx, y, z (6.3) We find that the oenta are quantized lπ Π x and y nπ and z x y (6.33) z where l,, and n are integers. he eigen-energy is also quantized: Εn x y z 4 Π x l 4 Π y 4 Π z n (6.34) For a very large syste H, the discrete energy and oenta turn into continuous variables. Many electrons in 3D at 0 At 0, the electrons occupy the lowest quantu states to save energy. In other words, the quantu states with Ε ΕF are occupied, while states with Ε > ΕF are epty. Since Ε, this eans that states with oentu F are occupied and states with > F are epty. Here F ΕF. In other words, in the -space, the occupied states for a sphere with radius F. his sphere is nown as the Feri sphere (or the Feri sea). Q: ΕF? HF? A: It is deterined by the density of electrons. he total volue of the Feri sphere is 4Π 3 the total nuber of quantu states is 4Π 3 F 3 F 3. Each quantu state occupies the volue H Π3, which coes fro the uncertainty relation. So F 3 H Π3 (6.35) 6 Π here are two electrons per state (because we have electrons with spin up and spin down), so the total nuber of electrons is F 3 F 3 (6.36) 6 Π 3 Π So F 3 Π 3 (6.37) F is deterined by the electron density. he Feri energy ΕF F 3 Π 3 (6.38) he Feri velocity vf â F Here, the forula v group velocity. F â 3 Π 3 (6.39) is nothing but the definition of the group velocity. otice that Ε Ω and v â Ω â is precisely the definition of the

5 39 Another equivalent definition of the Feri velocity is pf vf (6.40) If the energy Ε is a quadratic function of HΕ, these two definitions are identical. If Ε is O a quadratic function of (which could happen as will be discussed in the next chapter), we use the first definition vf â F he botto line: the first definition is ore generic. he second one H pf only wors for Ε Conductivity Under a static electric field (E), electrons feel a constant force. (6.4) F -e E he second aw of ewton tell us that (6.4) F ât If we have a constant force, P increases linearly as a function of t PHt PHt 0 F t (6.43) In quantu echanics, P, so we have Ht Ht 0 F t Ht 0 - e E (6.44) t Here, if we turn on the E field at t 0, at t 0, the Feri sea is a sphere centered at 0. At t > 0, the Feri sphere is centered at Ht - e E t, because the wavevector of every electron is shifted by this aount at tie t. For free electrons in a perfect crystal, Ht as t. However, in a reals solid, Ht will O diverge, because there are collisions between electrons, between electrons and phonons, and between electrons and ipurities. hese scatterings will reduce the oentu towards 0. hey will introduce viscosity to the Feri liquid, which reduces the velocity and oentu of the liquid. Eventually, the scatterings and the Efield will balance each other and will turn to a fixed value he total force can be written as F -e E - p (6.45) v -e E - Here the first ter is the electric force and the second ter is a viscosity ter, which is proportional to -v, reducing the speed. he coefficients and are the electron ass and the collision tie (average tie between two collisions). Using the second law of ewton: âp p - p0 (6.46) -e E ât where p0 is the oentu before we turn on the electric field and we assue the initial condition pht 0 p0. It is easy to find that the solution to this equation is p p0 - e E I - ã-t M (6.47) his eans that the change of oentu p p - p0 -e E I - ã-t M In the static liit Ht, (6.48)