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Today’s Objectives

Last Two Lecture Summary: Drude : Applied simple model to understand current density and the Hall coefficient in simple metals. Required knowledge/determination of carrier density.

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Today’s Objectives

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  1. Last Two Lecture Summary: Drude: Applied simple model to understand current density and the Hall coefficient in simple metals. Required knowledge/determination of carrier density. Sommerfeld: Showed that Fermi-Dirac statistics can be added to improve the Drude model and defined the Fermi energy. Today’s Objectives Briefly discuss other Fermi terms Derive Fermi-Dirac distribution Review heat capacity with Sommerfeld model

  2. Other Fermi Terms kz Fermi surface kF ky kx • Fermi temperature EF=kBTF(when it acts like a classical gas) • Fermi momentum • And Fermi velocity • These are the momentum and the velocity values of the electrons at the states on the Fermi surface of the Fermi sphere.

  3. Example: potassium metal This temperature will come back up later.

  4. To Determine Thermal Properties, We Need the Fermi-Dirac Distribution At T=0, all levels up to EF are filled. What happens when T is greater than zero? Is this still possible? T=0

  5. It’s a little confusing. You don’t need to know how to derive. Deriving the Fermi-Dirac Distribution(see pages 40-42 Ashcroft, Kittel just gives it) From statistical mechanics, equilibrium properties should be calculated by averaging over all N-particle states, assigning to each state of energy E a weight: More precisely E for an energy level EN=energy at level   Partition function Z, related to… Helmholtz free energy F One possible state N U=internal energy, S=entropy

  6. Combining ; Will use soon… fiN is the probability of one of N electrons being in a particular electron level i at temp T. It equals the sum of probabilities of any system of N electrons with an electron in that state Or one minus states  with no electron in state i

  7. That sounds complicated. Let’s use a trick. What if we add one more electron? Add one here We can build a 35 electron system (N+1) and remove whatever energy we want to consider Remove the one in question Or more generically…

  8. Some algebra manipulation Chemical potential

  9. Solving for fi Seems like we are back to square one! The probably should not change that much from one electron when N large

  10. Temperature affect on probability At the chemical potential, there is a 50% chance of finding an electron

  11. How does this help us understand the problem with Drude’s heat capacity? • When a metal is heated, electrons are transferred from below EF to above EF. The rest of the electrons deep inside the Fermi level are not effected. But it’s not just f, it’s also related to D(E).

  12. Writing Density of States in terms of number of orbitals N V

  13. The free electron gas at T < Fermi temp N(E,T)number of free electrons per unit energy range is just the area under N(E,T) graph. N Big difference at high temps k

  14. N(E,T) g(E) T=0 T>0 E EF The shaded area shows the change in distribution between absolute zero and a finite temperature. N Fermi-Dirac distribution function is a symmetric function, meaning: At low temperatures, the same number of levels below EF is emptied and same number of levels above EF are filled by electrons.

  15. Heat capacity of the free electron gas From the diagram of N(E,T) the change in the distribution of electrons can be resembled into triangles of height ½ g(EF) and a base of 2kBT so the area gives that ½ g(EF)kBT electrons increased their energy by kBT. The difference in thermal energy from the value at T=0°K ~ For an exact calculation:

  16. Differentiating with respect to T gives the heat capacity at constant volume: ~ From earlier: While this works great for metals, we will find that it’s very different for other materials. Heat capacity of Free electron gas

  17. Comparison with DataIt does appear linear at medium temps Phonons dominate at high temperatures but this works well for metals at medium temperatures Classical Model suggests one constant value and same value for all materials

  18. Why Specific Heat?Profile of Frances Hellman Physics Professor at University of California, Berkeley Previous chair of the physics department “My research group is concerned with the properties of novel magnetic and superconducting materials especially in thin film form. We use specific heat, magnetic susceptibility, electrical resistivity, and other measurements as a function of temperature in order to test and develop models for materials which challenge our understanding of metallic behavior. Current research includes: effects of spin on transport and tunneling, including studies of amorphous magnetic semiconductors and spin injection from ferromagnets into Si; finite size effects on magnetic and thermodynamic properties…”

  19. Measuring specific heat on a budgetStep 1: Set-up the calorimeter Energy conserved: qmetal = qwater + qcoffeecup qcup can be ignoredcmetalmmetalTsystem = cwatermwaterTwater Step 2: Boil the water containing metal, Pour Step 3: Stir while measuring temperature

  20. Modern calorimetry works on the same principles, just looks more fancy. constant volume or 'bomb' calorimeter

  21. Finding the 3 dimensional density of states D(E) g(k) g(E) g(E) g(k)

  22. Free electrons in 3D Group: Find D(E) = D(k) dk/dE More than one way to approach D(E)

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