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Advanced Solid State Physics | |
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Current densitiesThe electrical current density is the charge −e associated with an electron state →k, times the group velocity of that state →v→k, times the density of states per unit volume D(→k), times the probability density function f(→k) which gives the probability that state →k is occupied, summed over all →k states, →jelec=−e∫→v→kD(→k)f(→k)d3k.The group velocity is given by, →v→k=∇→kE(→k)ℏ.The density of states tell us how many states with wave number →k there are per unit volume, D(→k)=2(2π)3.In the relaxation time approximation in the absence of a magnetic field, the probability density function is, f(→k,→r)≈f0(→k,→r)−τ(→k)ℏ∂f0∂μ∇→kE(→k)⋅(e→E+∇→rμ+E(→k)−μT∇→rT).The electrical current density is thus, →jelec=−e4π3ℏ∫∇→kE(→k)(f0(→k,→r)−τ(→k)ℏ∂f0∂μ∇→kE(→k)⋅(e→E+∇→rμ+E(→k)−μT∇→rT))d3k.The term that integrates over the Fermi function vanishes because in thermal equilibrium the occupation of a state →k is the same as the occupation of state −→k. ∫1ℏ∇→kE(→k)f0(→k,→r)d3k=∫→v(→k)f0(→k,→r)d3k=0.Thus the current density is, →jelec=e4π3ℏ2∫τ(→k)∂f0∂μ∇→kE(→k)(∇→kE(→k)⋅(e→E+∇→rμ+E(→k)−μT∇→rT))d3k.Similarly, the particle current is, →jn=∫→v→kD(→k)f(→k)d3k, →jn=−14π3ℏ2∫τ(→k)∂f0∂μ∇→kE(→k)(∇→kE(→k)⋅(e→E+∇→rμ+E(→k)−μT∇→rT))d3k.The energy current density is, →jU=∫→v→kE(→k)D(→k)f(→k)d3k. →jU=−14π3ℏ2∫τ(→k)∂f0∂μE(→k)∇→kE(→k)(∇→kE(→k)⋅(e→E+∇→rμ+E(→k)−μT∇→rT))d3k.Finally, the electronic component to the thermal current is, →jQ=∫→v→k(E(→k)−μ)D(→k)f(→k)d3k. →jQ=−14π3ℏ2∫τ(→k)∂f0∂μ(E(→k)−μ)∇→kE(→k)(∇→kE(→k)⋅(e→E+∇→rμ+E(→k)−μT∇→rT))d3k.The relaxation time τ(→k) can be calculated using Fermi's golden rule of time dependent perturbation theory. However, often it is assumed that the same relaxation time can be used for all →k states and τ is pulled out of the integral. |