Question

a) Consider the above equation, 2T obtained for three dimensions. Following the same lines of reasoning above, show that in two dimensions, this equation becomes: g,A where A is the two-dimensional area of the two-dimensional sample b) Find kF as a function of N and A for such a two-dimensional system e) Find the Fermi energy for such a two-dimensional system d) At nonzero temperature, we want to compute the chemical potential (T). By imposing that the average number of electrons must remain constant as a function of t that μ(T)-kBT In[exp(rnh2 /(nıkBT))-1] where n N/A

Answer 1

Given: Let us consider the given equation g_sV/2 pi^2 integral F (k) K^2 dk, which is obtained for three dimensions. a) Now, we need to show that the same above equation in two dimensions. We know that: N = sigma_K F(k) N = g_s A/(2 pi)^2 integral^infinity^infinity_0 k dk integral^2pi_0 d theta (F(k). N = g_s A/(2 pi)^2 integral^k_0 k dk [theta]^2pi_0 F(k) N = g_s A/(2 pi)^2 integral^k_0 k dk [2 pi] F (k) N = g_s A/(2 pi)^2 2 pi integral^k_0 k dk F(k) N = g_s A/2 pi integral^k_0 k dk F(k) N = A g_s/2 pi integral F(k) k dk. Where: A = two dimensional area for two -dimensional sample. b) Now, we determine the k_F expression as a function of N and A \r\n For the above 2-dimensional system. We know that: N = Ag_s/2 pi integral F(k) kd k when k = k_F F(k) Vertical-bar _k = k_f = 1 Substitute these values in above equation.

Answer 2

Given: Let us consider the given equation g_sV/2 pi^2 integral F (k) K^2 dk, which is obtained for three dimensions. a) Now, we need to show that the same above equation in two dimensions. We know that: N = sigma_K F(k) N = g_s A/(2 pi)^2 integral^infinity^infinity_0 k dk integral^2pi_0 d theta (F(k). N = g_s A/(2 pi)^2 integral^k_0 k dk [theta]^2pi_0 F(k) N = g_s A/(2 pi)^2 integral^k_0 k dk [2 pi] F (k) N = g_s A/(2 pi)^2 2 pi integral^k_0 k dk F(k) N = g_s A/2 pi integral^k_0 k dk F(k) N = A g_s/2 pi integral F(k) k dk. Where: A = two dimensional area for two -dimensional sample. b) Now, we determine the k_F expression as a function of N and A \r\n For the above 2-dimensional system. We know that: N = Ag_s/2 pi integral F(k) kd k when k = k_F F(k) Vertical-bar _k = k_f = 1 Substitute these values in above equation.