BCS 발표 2
1. free & isotropic system에서 고체의 band 구조
\mathcal{H}=\sum_p \epsilon_p c^\dagger_pc_p,\quad \epsilon_p=\frac{p^2}{2m}
FD statistics -> filled inside fermi surface.
\epsilon_p =\begin{cases}\frac{p^2}{2m}-E_F,& p>p_F\\ E_F-\frac{p^2}{2m},& p<p_F\end{cases}
Define hole \hat{h}_{p,\sigma} = - \hat{c}^\dagger_{-p,-\sigma}
\mathcal{H}=\sum_{p>p_F} \epsilon_p c^\dagger_p c_p+\sum_{p<p_F}\epsilon_p h^\dagger_p h_p
hole의 spin은 어떻게 되는가?2. Quantization of Phonon
For quantized electron, phonon field and electron field interact H=\sum_{k,k',q,\nu}g(q,\nu)c^\dagger_{k+q}c_k(b^\dagger_{q,\nu}+b_{-q,\nu})
(momentum conservation)
g의 스케일은 무엇이 결정할까?
저 term이 나오는 물리학이 뭘까?
3. Phonon-mediated electron attraction
\hat{H}=\hat{H}_0+\hat{H}_I
\hat{H}_0 makes energy eigenstates, \hat{H}_I makes transition energy -> transition happen!
T_{fi}=\langle f |\hat{H}_I|i\rangle + \sum_n \langle f|\hat{H}_I|n\rangle\langle n|\hat{H}_I|i\rangle \left(\frac{1}{E_i-E_n+i\epsilon}+\frac{1}{E_f-E_n-i\epsilon}\right)\cdots
Electron-electron scattering is in opposite momentum:
(왜? 대칭적 운동량: 질량중심 고정 -> 에너지/운동량 교환해주는 포논 에너지가 낮음)
E_I=\epsilon_{p_1}+\epsilon_{p_2}
E_{II}=\epsilon_{p'_1}+\epsilon_{p'_2}
E_1=\epsilon_{p'_1}+\epsilon_{p_2}+\hbar \omega_q
\langle II|H_{e-ph-e}|I\rangle\\ \nonumber = \frac{1}{2}\sum_{i=1,2}\langle II|H_{e-ph-e}|i\rangle \left(\frac{1}{E_{II}-E_i}+\frac{1}{E_I-E_i}\right)\langle i|H_{e-ph-e}|I\rangle\\ \nonumber =W^*_\mathbf{q}\left(\frac{1}{\epsilon_{\mathbf{p}'_1}-\epsilon_{\mathbf{p}_1}-\hbar\omega_\mathbf{q}}+\frac{1}{\epsilon_{\mathbf{p}'_2}-\epsilon_{\mathbf{p}_2}-\hbar\omega_\mathbf{q}}\right)W_\mathbf{q} \\ \nonumber =\frac{|W_\mathbf{q}|^2}{\hbar}\left(\frac{1}{\omega-\omega_\mathbf{q}}-\frac{1}{\omega+\omega_\mathbf{q}}\right)=\frac{2|W_\mathbf{q}|^2}{\hbar}\frac{\omega_\mathbf{q}}{\omega^2-\omega^2_\mathbf{q}}
W_q is the matrix element for emission of the photon with momentum q
\hbar\omega=\epsilon_{p'_1}-\epsilon_{p_1}=-(\epsilon_{p'_2}-\epsilon_{p_2})
When |\omega|<\omega_q, reults negative, attractive.
운동량 보존은 항상 성립(interaction term 형태).
에너지 보존은 (중간에만) 위배 가능.
\omega_q는 뭐가 결정? 음수가 되면 끌어당기는 물리학?
방향에 상관없는 끌어당김 -> L=0, orbital wave function symmetric
-> spin wave function antisymmetric -> up/down 끼리 끌림.
4. Cooper problem
Two electron
\psi(r_1,r_2)
\left[ \hat{H}_e(r_1)+\hat{H}_e(r_2)+W(r_1-r_2)\right] \Psi(r_1,r_2)=E\Psi(r_1,r_2)
이 때 \hat{H}=\frac{\hat{p}^2}{2m}=-\frac{\hbar^2}{2m}\nabla^2이고, 정확히 하 \hat{H}_e(\nabla_1)\psi_{p\uparrow}(r_1)=\epsilon_p\psi_{p\uparrow}(r_1)임.
\psi_{p\uparrow}(r_1)\propto e^{ipr_1/\hbar},\quad \psi_{-p\downarrow}(r_2)\propto e^{-ipr_2/\hbar}
\Psi(r_1,r_2)=\sum_p c_p\psi_{p\uparrow}(r_1)\psi_{-p\downarrow}(r_2)=\sum_p a_pe^{ipr/\hbar}=\Psi(r)
Fourier transform
2\epsilon_p a_p+\sum_{p'}W_{p,p'}a_{p'}=Ea_p
Surface 근처에서 -W인 W_{p,p'} 대입, \epsilon_p는 0부터 E_c=\hbar \omega_D까지. E_c\ll E_F (정확히 안따라감)
상호작용하는 p의 범위, 왜 포논의 속도가 아닌 v_F가 들어가는지.
Ground state energy
|E_b|=\frac{2E_c}{e^{1/N(0)W}-1}
Strong coupling N(0)W\gg 1, |E_b|=2N(0)WE_c
두개 묶인 상태는 spin 0이네? 보존이네? 온도 0이면 BEC하네?
결합 에너지가 커지면 그만큼 Pair가 많이 생길 것.
이걸 깨면 exited state가 되는 것.
5. Mean Field
H=\sum_{k\sigma}\xi_kc^\dagger_{k\sigma}c_{k\sigma}+\sum_{kk'}W_{kk'}c^\dagger_{k\uparrow}c^\dagger_{-k\downarrow}c_{-k'\downarrow}c_{k'\uparrow}
superconductor order parameter( Cooper pair 개수)가 평균장이 된다고 가정.
fluctuation 0이라는 가정?
quadratic Hamiltonian -> 다시 Band (H를 diag)를 보겠다!
self-consistent 하다: 적어도 틀리지는 않음. (모르겠다. 직접 해봐야 감이 잡힐듯)
E_k=\sqrt{\xi^2_k+|\Delta_k|^2} (\xi는 interaction 없을 때 energy.)
6. Bogoliubov Quasiparticle
Cooper Pair는 hole끼리, particle끼리 pair이룸.
Bogoliubov quasiparticle은 hole과 particle을 함께 생각함.
\gamma_{k\uparrow}=u^*_k c_{k\uparrow}+v^*_kh_{k\uparrow}=u^*_k c_{k\uparrow}+v^*_kc^\dagger_{-k\downarrow}\\ \gamma^\dagger_{-k\downarrow}=u_kc^\dagger_{-k\downarrow}-v_k h^\dagger_{-k\downarrow}=u_kc^\dagger_{-k\downarrow}-v_kc_{k\uparrow}
c입장에서 보면,
c_{k\uparrow}=u_k \gamma_{k\uparrow}-v^*_k\gamma^\dagger_{-k\downarrow}
c^\dagger_{-k\downarrow}=u^*_k\gamma^\dagger_{-k\downarrow}+v_k\gamma_{k\uparrow}
따라서 k를 +랑 - 다 세지 말고 한쪽으로만 세야함.
Constraint: \{\gamma_{k\uparrow},\gamma^\dagger_{k\uparrow}\}=|u_k|^2+|v_k|^2=1
This make transformation unitary.
|u_k|는 \Delta 없을 때 surface 밖에만 있어서 particle fraction
|v_k|는 \Delta 없을 때 surface 안에만 있어서 hole fraction
\Delta 생기면 주황 부분 값이 약간 달라짐.
7. BdG Equation
We obtain the following commutation relations: \begin{align}[c_{i\uparrow},\mathcal{H}_{eff}] &=& \sum_{j\sigma'}\tilde{h}_{i\uparrow,j\sigma'}c_{j\sigma'}+\sum_j \Delta_{ij}c^\dagger_{j\downarrow},\\ \nonumber [c_{i\uparrow}^\dagger,\mathcal{H}_{eff}] &=& -\sum_{j\sigma'}\tilde{h}_{j\sigma',i\uparrow}c_{j\sigma'}^\dagger-\sum_j \Delta_{ij}^*c^\dagger_{j\downarrow},\\ \nonumber[c_{i\downarrow},\mathcal{H}_{eff}] &=& \sum_{j\sigma'}\tilde{h}_{i\downarrow,j\sigma'}c_{j\sigma'}-\sum_j \Delta_{ji}c^\dagger_{j\uparrow},\\ \nonumber[c_{i\downarrow}^\dagger,\mathcal{H}_{eff}] &=& -\sum_{j\sigma'}\tilde{h}_{j\sigma',i\downarrow}c_{j\sigma'}^\dagger+\sum_j \Delta_{ji}^*c^\dagger_{j\uparrow}\end{align}
Electron field operator can be expressed as a linear combination of electron- and hole-liker quasiparticle excitations, which enables us to perform a Bogoliubov canonical transformation: \begin{align} c_{i\sigma}=\sum^{'}_n (u^n_{i\sigma} \gamma_n -\sigma v^{n*}_{i\sigma}\gamma^\dagger_n),\quad c_{i\sigma}^\dagger=\sum^{'}_n (u^{n*}_{i\sigma} \gamma_n^\dagger -\sigma v^{n}_{i\sigma}\gamma_n)\end{align}
where \sigma=\pm 1 denotes the up an down spin orientations. The operators \gamma_n^\dagger create a Bogoliubov quasiparticle at state n. The prime sign above the summation in the transformation means only those states with positive energy are counted.
we compare the coeffecients of the term with \gamma_n and \gamma_n^\dagger and arrive at \begin{align}E_n u^n_{i\uparrow} &=& \sum_{j\sigma'}\tilde{h}_{i\uparrow,j\sigma'}u^n_{j\sigma'}+\sum_j \Delta_{ij}v^n_{j\downarrow},\\ E_n u^n_{i\downarrow} &=& \sum_{j\sigma'}\tilde{h}_{i\downarrow,j\sigma'}u^n_{j\sigma'}+\sum_j \Delta_{ji}v^n_{j\uparrow},\\ E_n v^n_{i\uparrow} &=& -\sum_{j\sigma'}\sigma'\tilde{h}^{*}_{i\uparrow,j\sigma'}v^n_{j\sigma'}+\sum_j \Delta_{ij}^{*}u^n_{j\downarrow},\\ E_n v^n_{i\downarrow} &=& -\sum_{j\sigma'}\sigma'\tilde{h}^{*}_{i\downarrow,j\sigma'}v^n_{j\sigma'}+\sum_j \Delta_{ij}^{*}u^n_{j\uparrow} \end{align}
The set of BdG equations can be case into a matrix form: \begin{align}\sum_j \hat{M}_{ij}\hat{\phi}_j =E_n\hat{\phi}_i\end{align} where \begin{align} \hat{M}_{ij}=\begin{bmatrix} \tilde{h}_{i\uparrow j\uparrow}& \tilde{h}_{i\uparrow j\downarrow}&0 & \Delta_{ij}\\ \tilde{h}_{i\downarrow j\uparrow}& \tilde{h}_{i\downarrow j\downarrow}&\Delta_{ji} &0 \\ 0& \Delta^*_{ij}& -\tilde{h}^*_{i\uparrow j\uparrow}& \tilde{h}^*_{i\uparrow j\downarrow}\\ \Delta^*_{ji} &0& \tilde{h}^*_{i\downarrow j\uparrow}& -\tilde{h}^*_{i\downarrow j\downarrow} \end{bmatrix} \end{align} and \begin{align}\hat{\phi}_i=\begin{pmatrix}u_{i\uparrow}\\ u_{i\downarrow}\\ v_{i\uparrow}\\ v_{i\downarrow}\end{pmatrix}\end{align}
Particle-Hole Symmetry
If (u^n_{i\uparrow},v^n_{i\downarrow},u^n_{i\downarrow},v^n_{i\uparrow}) is the solution to the BdG equations with eigenvalue E_n, then (-v^{n*}_{i\uparrow},u^{n*}_{i\downarrow},v^{n*}_{i\downarrow},-u^{n*}_{i\uparrow}) is the solution to the same equations with eigenvalue -E_n.
Hole 만드는거랑 Particle 없애는게 같음.
찐연구시작
Real space에서 BdG equation을 self-consistent하게 풀어봄. -> gap, supercurrent 봄
피보나치는 1000개
AB는 2000개 (r=30) t=1, 10-3를 error로 잡고 대충 해보고 코드 맞으면, 그다음에 줄임.
목표: 2D honeycomb transformation into dodecagonal quasicrystals driven by electrostaticforce