lesson 12 Sommerfeld model ii Solid State Physics

lesson # 12: Sommerfeld model ii Solid State Physics M. Casalboni 2018/’ 19 SOMMERFELD MODEL (1928) free electron model with parabolic dispersion curve Arnold Sommerfeld 1868 -1951 it describes conduction electron in metals usefull for showing thermodynamic thermodinamic implications 1/16

lesson # 12: Sommerfeld model ii thermodynamic implication i. Solid State Physics M. Casalboni 2018/’ 19 electrons can be seen as a GAS in thermal equilibrium BUT Pauli esclusion principle are FERMIONS 2/16

lesson # 12: Sommerfeld model ii Solid State Physics M. Casalboni 2018/’ 19 deviation from «T=0» -behavior μ 0 distribution function 3/16

lesson # 12: Sommerfeld model ii in order to formalize we introduce the first derivative with the property Solid State Physics M. Casalboni 2018/’ 19 4/16

lesson # 12: Sommerfeld model ii how does μ change with temperature? Solid State Physics M. Casalboni 2018/’ 19 it is necessary a mathematical tool called «Sommerfeld expantion» or for demonstration see the textbook pagg 83 -84 as homework using the itegration by parts method and taking into account that the behavior of the functions G(E) and f(E) and in particular that for 5/16

lesson # 12: Sommerfeld model ii how does μ change with temperature? Solid State Physics M. Casalboni 2018/’ 19 it is necessary a mathematical tool called «Sommerfeld expantion» or for demonstration see the textbook pagg 83 -84 as homework 6/16

lesson # 12: Sommerfeld model ii with these tools let us try to calculate how μ depends on temperature Solid State Physics M. Casalboni 2018/’ 19 note that to see what appens changing the temperature let us diffrentiate with respect to it 7/16

lesson # 12: Sommerfeld model ii Solid State Physics M. Casalboni 2018/’ 19 8/16

lesson # 12: Sommerfeld model ii Solid State Physics M. Casalboni 2018/’ 19 why ? A d. E 0 B if density of states was constant near μ 0 the number of state that are empty for a certain T in A would be equal to the number the state with electron in B 9/16

lesson # 12: Sommerfeld model ii Solid State Physics M. Casalboni 2018/’ 19 why ? A d. E>d. E 0 d. E<d. E 0 B if density of states increased with E near μ 0 the number of state that are empty for a certain T in A would stay in a larger interval of E with respect o that in B 10/16

lesson # 12: Sommerfeld model ii Solid State Physics M. Casalboni 2018/’ 19 why ? A d. E>d. E 0 d. E<d. E 0 B if density of states increased with E near μ 0 the number of state that are empty for a certain T in A would stay in a larger interval of E with respect o that in B μ is the energy value where distribution function equals 1/2 11/16

lesson # 12: Sommerfeld model ii thermodynamic implication ii. ELECTRONIC SPECIFIC HEAT Solid State Physics M. Casalboni 2018/’ 19 heat capacity of a solid (extensive quantity!) in solids volume does not change* with temperature at constant V 12/16

lesson # 12: Sommerfeld model ii Solid State Physics M. Casalboni 2018/’ 19 Standard Sommerfeld expansion deriving with respect to the temperature and retaining the significative terms and taking into account that D(E) is almost constant with respect to E 13/16

lesson # 12: Sommerfeld model ii Solid State Physics M. Casalboni 2018/’ 19 14/16

lesson # 12: Sommerfeld model ii Solid State Physics M. Casalboni 2018/’ 19 fraction of electrons equal to T/TF 15/16

lesson # 12: Sommerfeld model ii assignment: Solid State Physics M. Casalboni 2018/’ 19 read the paragraph read the appendices of the chapter 3 of the textbook 16/16