Superconductivity in Graphene Induced by the Rotated Layer

TL;DR

This paper explains how rotating layers in graphene bilayers induces superconductivity by transforming repulsive interactions into attractive ones through modifications in the electronic spinor structure.

Contribution

It provides a theoretical explanation for superconductivity in rotated graphene bilayers by analyzing the effects of layer rotation on electronic spinors and interactions.

Findings

Rotation modifies spinors, turning repulsive interactions attractive.

Superconductivity arises at 'magic' angles due to lattice commensuration.

Theoretical framework explains recent experimental observations.

Abstract

Recent discoveries in graphene bilayers revealed that when one of the layers is rotated, superconductivity emerges. We provide an explanation for this phenomenon . We find that due to the layer rotations, the spinors are modified in such way that a repulsive interaction, becomes attractive in certain directions. This result is obtained following a sequence of steps: when layer $2$ is rotated by an angle $\theta$ ,this rotation is equivalent to a rotation of an angle $-\theta$ of the linear momentum .Due to the discreet lattice, in layer $1$, the Fourier transform conserves the linear momentum $modulo$ the hexagonal reciprocal lattice vector . In layer $2$, due to the rotation, the linear momentum is conserved $modulo$ the $Moire$ reciprocal lattice vector . Periodicity is achieved at the $magical $ angles obtained from the condition of commensuration of the two lattices. We find that the rotations transform the spinors around the nodal points, such that a repulsive interaction becomes attractive, giving rise to superconductivity.

Equations

$$\displaystyle L_{int.}=|U[\bar{\theta}]|\int\,dy\Big{[}\tilde{C}^{\dagger}_{2,R,\uparrow}(p_{x}=0,y)\tilde{C}^{\dagger}_{2,R,\downarrow}(p_{x}=0,y)\tilde{C}_{2,L,\downarrow}(p_{x}=0,y)\tilde{C}_{2,L,\uparrow}(p_{x}=0,y)+$$

$$\displaystyle\tilde{C}^{\dagger}_{2,L,\uparrow}(p_{x}=0,y)\tilde{C}^{\dagger}_{2,L,\downarrow}(p_{x}=0,y)\tilde{C}_{2,R,\downarrow}(p_{x}=0,y)\tilde{C}_{2,R,\uparrow}(p_{x}=0,y)\Big{]},$$

$$a_1(\vec{R})=\frac{1}{\sqrt{L^2}}\sum _{\vec{k}}{a}_1(\vec{k})e^{i\vec{k}\cdot \vec{R}}$$

$$b_1(\vec{R}+{\vec{\delta }}_r)=\frac{1}{\sqrt{L^2}}\sum _{\vec{k}}{b}_1(\vec{k})e^{i\vec{k}\cdot (\vec{R}+{\vec{\delta }}_r)};r=1,2,3$$

$$\displaystyle H_{1}=-\sum_{\vec{R}}\sum_{r=1,2,3}\Big{[}\gamma_{0}a^{\dagger}_{1}(\vec{R})b_{1}(\vec{R}+\vec{\delta_{r}})+H.C.\Big{]}$$

$$\displaystyle=-\frac{1}{N_{1}N_{2}}\sum_{\vec{R}}\sum_{r=1,2,3}\sum_{k}\sum_{p}\Big{[}\gamma_{0}a^{\dagger}_{1}(\vec{k})b_{1}(\vec{p})e^{-i(\vec{k}-\vec{p})\cdot\vec{R}}e^{i\vec{p}\cdot\vec{\delta}_{r}}+H.C.\Big{]}$$

$$\displaystyle=-\sum_{\vec{k}}\sum_{\vec{p}}\Big{[}b^{\dagger}_{1}(\vec{k})a_{1}(\vec{p})\phi_{1}(\vec{p})\delta_{k_{x},p_{x}+G^{(1)}_{x}+G^{(2)}_{x}}\delta_{k_{y},p_{y}+G^{(1)}_{y}+G^{(2)}_{y}}+H.C.\Big{]}$$

$$H_{1}=\sum_{\vec{p}\in B.Z.}\Big{[}a^{\dagger}_{1}(\vec{p})b_{1}(\vec{p})\phi_{1}(\vec{p})+H.C.\Big{]}$$

$$\phi_{1}(\vec{p})=-\gamma_{0}\sum_{r=1,2,3}e^{-i\vec{p}\cdot\vec{\delta_{r}}}=e^{-ip_{x}}\Big{[}1+2e^{i\frac{3}{2}p_{x}}Cos[\frac{\sqrt{3}}{2}p_{y}\Big{]}$$

$$\displaystyle H_{2}=-\sum_{\vec{R}}\sum_{r=1,2,3}\gamma_{0}\Big{[}b{\dagger}_{2}(\vec{R^{\prime}})a_{2}(\vec{R^{\prime}}-\vec{\delta}_{r^{\prime}})+H.C.\Big{]}$$

$$\displaystyle=-\frac{1}{N_{1}N_{2}}\sum_{\vec{R}}\sum_{r=1,2,3}\sum_{\vec{k}}\sum_{\vec{p}}\Big{[}\gamma_{0}b^{\dagger}_{2}(\vec{k})a_{2}(\vec{p})e^{-i(\vec{k}-\vec{p})\cdot\vec{R}^{\prime}}e^{-i\vec{p}\cdot\vec{\delta}_{r^{\prime}}}+H.C.\Big{]}$$

$$\displaystyle=-\frac{1}{N_{1}N_{2}}\sum_{\vec{R}}\sum_{r=1,2,3}\sum_{\vec{k}}\sum_{\vec{p}}\Big{[}\gamma_{0}b^{\dagger}_{2}(\vec{k})a_{2}(\vec{p})e^{-i(\vec{k}[-\theta]-\vec{p}[-\theta])\cdot\vec{R}}e^{-i\vec{p}[-\theta]\cdot\vec{\delta}_{r}}+H.C.\Big{]}$$

$$\displaystyle H_{2}=-\sum_{\vec{k}}\sum_{\vec{p}}\Big{[}a_{2}(\vec{k})b_{2}(\vec{p})\varphi_{2}(\vec{p})\delta_{k_{x},p_{x}+g_{x}[\theta]}\delta_{k_{y},p_{y}+g_{y}[\theta]}+H.C.\Big{]}$$

$$g_x[\theta ]=(G_x^{(1)}+G_x^{(2)})Cos[\theta ];g_y[\theta ]=(G_x^{(1)}+G_x^{(2)})Sin[\theta ]$$

$$H_{2}=\sum_{\vec{p}\in MoireB.Z}\Big{[}b_{2}(\vec{p})a_{2}(\vec{p})\varphi_{2}(\vec{p})+H.C.\Big{]}$$

$${\phi }_2(p)={\varphi }_1(\vec{p}[-\theta ])={\gamma }_0\sum _{r=1,2,3}{e}^{-i\vec{p}\cdot {\overrightarrow{{\delta }^{\prime }}}_r}={\gamma }_0\sum _{r=1,2,3}{e}^{-i\overrightarrow{p[-\theta ])}\cdot {\vec{\delta }}_r}$$

$$\displaystyle\varphi_{2}(\vec{p})=\gamma_{0}e^{-ip_{x}Cos[\theta]}e^{-ip_{y}Sin[\theta]}\Big{[}1+2e^{i\frac{3}{2}p_{x}Cos[\theta]}e^{i\frac{3}{2}p_{y}Sin[\theta]}Cos[\frac{\sqrt{3}}{2}(p_{x}Cos[\theta]-p_{y}Sin[\theta])]\Big{]}$$

$$\displaystyle H_{\perp,3}=-\frac{\gamma_{3}}{\gamma_{0}}\sum_{r=1,2,3}\sum_{\vec{R}}\Big{[}b^{\dagger}_{2}(\vec{R^{\prime}})a_{1}(\vec{R})+a^{\dagger}_{2}(\vec{R^{\prime}})b_{1}(\vec{R}+\vec{\delta}_{r})\Big{]}=$$

$$\displaystyle-\frac{1}{N_{1}N_{2}}\frac{\gamma_{3}}{\gamma_{0}}\sum_{r=1,2,3}\sum_{\vec{R}}\sum_{\vec{k}}\sum_{\vec{p}}\Big{[}b^{\dagger}_{2}(\vec{k})a_{1}(\vec{p})e^{-i(\vec{k}[-\theta]-\vec{p})\cdot\vec{R}}e^{-ik[-\theta]\cdot\vec{\delta}_{r}}+a_{2}(\vec{k})b_{1}(\vec{p})e^{-i(\vec{k}[-\theta]-\vec{p})\cdot\vec{R}}e^{ip\cdot\vec{\delta}_{r}}\Big{]}$$

$$\displaystyle-\frac{1}{N_{1}N_{2}}\frac{\gamma_{3}}{\gamma_{0}}\sum_{r=1,2,3}\sum_{\vec{R}}\sum_{\vec{k}}\sum_{\vec{p}}\Big{[}b^{\dagger}_{2}(\vec{k})a_{1}(\vec{p})e^{-i(\vec{k}[-\theta]-\vec{p})\cdot\vec{R}}\varphi^{*}_{2}(\vec{k})+a_{2}(\vec{k})b_{1}(\vec{p})e^{-i(\vec{k}[-\theta]-\vec{p})\cdot\vec{R}}\phi_{1}(\vec{p})\Big{]}=$$

$$\displaystyle-\frac{\gamma_{3}}{\gamma_{0}}\sum_{\vec{k}}\sum_{\vec{p}}\Big{[}b^{\dagger}_{2}(\vec{k})a_{1}(\vec{p})\delta_{k_{x},p_{x}Cos[\theta]-p_{y}Sin[\theta]+g_{x}[\theta]}\delta_{k_{y},p_{y}Cos[\theta]+p_{x}Sin[\theta]+g_{y}[\theta]}\varphi^{*}_{2}(\vec{k})$$

$$\displaystyle+a_{2}(\vec{k})b_{1}(\vec{p})\delta_{k_{x},p_{x}Cos[\theta]-p_{y}Sin[\theta]+g_{x}[\theta]}\delta_{k_{y},p_{y}Cos[\theta]+p_{x}Sin[\theta]+g_{y}[\theta]}\phi_{1}(\vec{p})\Big{]},$$

$$H_{\perp,3}=-\frac{\gamma_{3}}{\gamma_{0}}\sum_{\vec{p}\in MoireB.Z}\Big{[}b^{\dagger}_{2}(\vec{p})a_{1}(\vec{p})\phi_{1}(\vec{p})+a^{\dagger}_{2}(\vec{p})b_{1}(\vec{p})\phi^{*}_{1}(\vec{p})\Big{]}$$

$${\Psi }_2(\vec{p})=C_2(\vec{p})u_2(\vec{p})+D_2^{†}(\vec{p})v_2(\vec{p})$$

$$H_2=\sum _{\vec{p}}{C}_2^{†}(\vec{p})C_2(\vec{p})|{\phi }_2(\vec{p}))|+D_2^{†}(\vec{p})D_2(\vec{p})|{\phi }_2(\vec{p})|$$

$${\Psi }_1(\vec{p})=C_1(\vec{p})u_1(\vec{p})+D_1^{†}(\vec{p})v_1(\vec{p})$$

$$H_1=\sum _{\vec{p}}{C}_1^{†}(\vec{p})C_1(\vec{p}){\varphi }_1(\vec{p}))|+D_1^{†}(\vec{p})D_1(\vec{p}){\varphi }_1(\vec{p})|$$

$$H_1=\sum _{\vec{p}}{C}_1^{†}(\vec{p})C_1(\vec{p})|{\varphi }_1(\vec{p})|$$

$$H_2=\sum _{\vec{p}}{C}_2^{†}(\vec{p})C_2(\vec{p})|{\phi }_2(\vec{p})|$$

$$H_{\perp ,3}=-\frac{{\gamma }_3}{{\gamma }_0}\sum _{\vec{p}\in MoireB.Z}{C}_2^{†}(\vec{p})C_1(\vec{p})|{\varphi }_1(\vec{p})|Cos[{\alpha }_2(\vec{p})+{\alpha }_1(\vec{p})]+H.C.$$

$$E^{-}(\vec{p})=-\mu+\frac{1}{2}\Big{(}|\phi_{1}(\vec{p})|+|\varphi_{2}(\vec{p})|-\sqrt{(|\phi_{1}(\vec{p})|-|\varphi_{2}(\vec{p})|)^{2}+4((\frac{\gamma_{3}}{\gamma_{0}})|\phi_{1}(\vec{p})|Cos[\alpha_{2}(\vec{p})+\alpha_{1}(\vec{p})])^{2}}\Big{)}$$

$$\displaystyle\sum_{\vec{p}\in B.Z}.\Big{[}a^{\dagger}_{1}(\vec{p})b_{1}(\vec{p})\phi_{1}(\vec{p})+H.C.\Big{]}$$

$$\displaystyle=\sum_{\vec{q}}\Big{[}a^{\dagger}_{1}(\vec{p}=\vec{K^{\prime}}_{1}+\vec{q})b_{1}(\vec{p}=\vec{K^{\prime}}_{1}+\vec{q}_{1})\phi_{1}(\vec{p}=\vec{K^{\prime}}_{1}+\vec{q}_{1})+$$

$$\displaystyle+a^{\dagger}_{1}(\vec{p}=\vec{K}_{1}+\vec{q})b_{1}(\vec{p}=\vec{K}_{1}+\vec{q})\phi_{1}(\vec{p}=\vec{K}_{1}+\vec{q})+H.C.\Big{]}$$

$$\displaystyle=-\sum_{\vec{q}}\Big{[}a_{1,L}^{\dagger}(\vec{q})b_{1,L}(\vec{q})\phi_{1,L}(\vec{q})+a^{\dagger}_{1,R}(\vec{q})b_{1,R}(\vec{q})\phi_{1,R}(\vec{q})+H.C.\Big{]}$$

$$\displaystyle=\sum_{\vec{q}}\Big{[}v_{F}\Phi_{1,L}^{\dagger}(\vec{q})\Big{(}\tau_{1}q_{y}-\tau_{2}q_{x}\Big{)}\Phi_{1,L}(\vec{q})+v_{F}\Phi_{1,R}^{\dagger}(\vec{q})\Big{(}-\tau_{1}q_{y}-\tau_{2}q_{x}\Big{)}\Phi_{1,R}(\vec{q})\Big{]}$$

$$\displaystyle\Phi_{1,L}(\vec{q})=\Big{[}a_{1}(\vec{p}=\vec{K^{\prime}}_{1}+\vec{q}),b_{1}(\vec{p}=\vec{K^{\prime}}_{1}+\vec{q})\Big{]}^{T}=\Big{[}a_{1,L}(\vec{q}),b_{1,L}(\vec{q})\Big{]}^{T}$$

$$\displaystyle\Phi_{1,R}(\vec{q})=\Big{[}a_{1}(\vec{p}=\vec{K}_{1}+\vec{q}),b_{1}(\vec{p}=\vec{K}_{1}+\vec{q})\Big{]}^{T}=\Big{[}a_{1,R}(\vec{q}),b_{1,R}(\vec{q})\Big{]}^{T}$$

$${\Psi }_1(\vec{r})={\Phi }_{1,L}(\vec{r})e^{i{\overrightarrow{K^{\prime }}}_1\cdot \vec{r}}+{\Phi }_{1,R}(\vec{r})e^{i{\vec{K}}_1\cdot \vec{r}}$$

$${\Psi }_1(\vec{p})={\Psi }_1(\vec{p})\mu [-p_x]+{\Psi }_1(\vec{p})\mu [p_x]={\Phi }_{1,L}(\vec{q}+{\overrightarrow{K^{\prime }}}_1)\mu [-q_x-K_{x,1}^{\prime }]+{\Phi }_{1,R}(\vec{q}+{\vec{K}}_1)\mu [q_x+K_{x,1}],$$

$${\Phi }_{1,L}(\vec{q})=C_{1,L}(\vec{q})U_{1,L}(\vec{q})$$

$${\Phi }_{1,R}(\vec{q})=C_{1,R}(\vec{q})U_{1,R}(\vec{q})$$

$$\displaystyle U_{1,L}(\vec{q})=\frac{1}{\sqrt{2}}\Big{[}1,-ie^{\alpha(\vec{q})}\Big{]}^{T},\hskip 7.22743ptU_{1,R}(\vec{q})=\frac{1}{\sqrt{2}}\Big{[}1,ie^{-i\alpha(\vec{q})}\Big{]}^{T},\hskip 3.61371pt\alpha(\vec{q)}=ArcTan\Big{(}\frac{q_{y}}{q_{x}}\Big{)}$$

Full text

I. INTRODUCTION

A commensurate triangular $Moire$ pattern is formed when a top-layer grapene is rotated with respect to the bottom layer at certain angles Crespi ; Morell ; Mac1 . A model with a large amount of atoms in a commensurable unit cell was considered in order to explain the formation of the flat bands that might lead to superconductivity.The model consists of interlayer interaction of $\pi$ orbitals . The fit of the tight binding model was reproduced using a Density Functional Theory ( $DFT$) calculation of stacked bilayers. Further progress has been achieved by Mac1 ; Mac2 ; Sharma who computed the spectrum considering Coulombic interactions and phonon-mediated superconductivity. Moire insulators have been viewed as a surface for a Symetric Protected Topological phases Cenke and by proximity to Mott insulators Senthil .The proximity to Wigner crystallization and Mott insulation has also been considered Philips .The effect of Van Hove singularity was used by Betouras in analogy with high $T_{c}$ superconductivity calculation were double logarithmic singularity was the cause of Superconductivity for bar-repulsive interactions .The uniform rotation generates flat bands. When the rotation angle generates a periodic $Moire$ structure commensurate with the honeycomb lattice , and the unit cell contains a large number of atoms, the Brillouine Zone ($BZ$) becomes small, the fermi velocity vanishes and a flat band appears vafek ; koshino . For simplicity, we consider the rotation of layer $2$ in such a way that the site occupied by atom $B^{\prime}$ is situated directly opposite from atom $A$ in layer $1$ . A commensurate structure is obtained if atom $B^{\prime}$ is moved by the rotation to a position formerly occupied by an atom of the same kind Lopes . Following Shallcross ; Lopes ,we determine the condition for the angles $\theta(n)$ of a commensurate rotation. Our goal is to investigate the uniform rotation which can shine light on the mechanism which is responsible for the attraction and for causing superconductivity.

To achive this goa,l we need to compute the effect of rotation on the spinors. We will use the tight binding model for bilayer graphene Castro ; Jackiw and take into consideration the discreetness of the lattice Marder . A two-dimensional honeycomb array of carbon atoms forming a hexagonal lattice can be viewed as a superposition of two triangular sublattices ,$A$ and $B$.The generators of lattice $A$ are vectors $\vec{a}_{1}$ and $\vec{a}_{2}$.We have three vectors $\delta_{r}$ connecting any site from lattice $A$ to nearest neighbor sites belonging to $B$( layer $1$). Layer $i=1$ is not rotated and the sum over the position of atom $A$ and $B$ give rise to a summation over the reciprocal lattice vectors $\sum_{n,m}e^{i\vec{p}\cdot\vec{R}_{n,m}}=N\delta_{k_{x},G^{(1)}_{x}+G^{(2)}_{x}}\delta_{k_{y},G^{(1)}_{y}+G^{(2)}_{y}}$ where, $\vec{R}_{n,m}=n\vec{a}_{(1)}+m\vec{a}_{(2)}$ with integers $n$ and $m$ .

Layer $2$ is uniformly rotated. In real space the uniform rotation of the coordinates by an angle $\theta$ is equivalent to a rotation $-\theta$ of the momentum space $e^{i\vec{p}[-\theta]\cdot\vec{R}_{n,m}}=e^{i\vec{p}\cdot\vec{R^{\prime}}_{n,m}}$ where $\vec{R^{\prime}}_{n,m}$ is the rotated vector .Performing the dicreete summation in layer $2$, we obtain $\sum_{n,m}e^{i\vec{p}[-\theta]\cdot\vec{R}_{n,m}}=N\delta_{k_{x},g_{x}[\theta]}\delta_{k_{y},g_{y}[\theta]}$ where $g_{x}[\theta]=(G^{(1)}_{x}+G^{(2)}_{x})Cos[\theta]$ and $g_{y}[\theta]=(G^{(1)}_{x}+G^{(2)}_{x})Sin[\theta]$ are the reciprocal lattice vectors which at the magical angle become the new reciprocal lattice vector which emerges in the following way: layer $2$ is rotated about a site occupied by atom $B^{\prime}$ directly opposite of an atom $A$ (layer $1$). A commensurate structure is obtained if a $B^{\prime}$ atom is moved by rotation to a position formerly occupied by an atom $B^{\prime}$ of the same kind. The $Moire$ pattern is periodic and a translation from the center to the position of $B^{\prime}$ is a translation symmetry given by, $cos[\theta_{i}]=\frac{3i^{2}+3i+0.5}{3i^{2}+3i+1}$,i=0,1,2..vectors. A superlattice with basis vectors $\vec{t}_{1}=i\vec{a}_{1}+(i+1)\vec{a}_{2}$; $\vec{t}_{2}=-(i+1)\vec{a}_{1}+(2i+1)\vec{a}_{2}$ is formed Shallcross . The $Moire$ reciprocal lattice vector is given by $g_{x}[\theta]=g_{x}[\theta_{i}]$ and $g_{y}[\theta]=g_{y}[\theta_{i}]$ ($cos[\theta_{i}]=\frac{3i^{2}+3i+0.5}{3i^{2}+3i+1}$). Using the tunneling between the layers at magic angles we obtain flat bands.

We linearize the bilayer Hamiltonian with respect to the nodal position and obtain a Dirac representation .For layer $2$, the nodal position depends on the rotated angles \vec{K}_{2}=\frac{4\pi}{3a}\Big{[}Sin[\theta],\frac{1}{\sqrt{3}}Cos[\theta]\Big{]} and \vec{K^{\prime}}_{2}=-\frac{4\pi}{3a}\Big{[}Sin[\theta],\frac{1}{\sqrt{3}}Cos[\theta]\Big{]}.

As a result, the spinor will depend on the rotated angles. For certain angles and for certain valley components, the repulsive interaction becomes attractive:

[TABLE]

where $p_{x}=0$ corresponds to the nodal component $K_{2,x}$.As a result, the superconductor is one dimensional with periodicity in the transversal direction.

The band satisfy $E(\vec{p})=\epsilon(\vec{p})-\mu<\Lambda$(cut-off).In our case the flat band, could be of the order of the chemical potential , and the condition $E(\vec{p})=\epsilon(\vec{p})-\mu<\Lambda$ is not obeyed and superconducting is not achieved .

The outline of this paper is: in chapters II and III we consider the model in the real space representation.In chapter IV we linearize the model around the nodal points obtaining a Dirac representation for the two valleys. We show that at the magic angles, the low energy bands are flat and the spinors transform the repulsive interaction to an attractive interactions in certain directions. In chapter V, we include the spin degrees of freedom and double the number components of spinor.

III- The real space approach

In order to investigate the effect of the rotation in real space, we introduce the spinors $\Psi_{i}=[a_{i},b_{i}]$, where $a_{i}$ and $b_{i}$ represent the two honeycomb lattices and $i=1,2$ is the index of the two layers. For layer $1$, we have a two dimensional honeycomb array of Carbon atoms forming a hexagonal lattice which can be viewed as a superposition of two triangular sublattices ,$A$ and $B$.The generators of lattice $A$ are vectors $\vec{a}_{1}$ and $\vec{a}_{2}$.We have three vectors $\delta_{r}$ connecting any site from lattice $A$ to nearest neighbor sites belonging to $B$.We have the representation:

[TABLE]

where $a_{1}(\vec{R})=a_{1}(\vec{R}+L)$ and $b_{1}(\vec{R})=b_{1}(\vec{R}+L)$ obey the Born-Von Karman conditionMarder $\vec{k}=\frac{2\pi}{L}\vec{n}$.

The Hamiltonian for layer $1$ is given by Wallace ; Jackiw .

[TABLE]

Using the periodicity of the reciprocal lattice $a_{1}(p_{x},p_{y})=a_{1}(p_{x}+G^{(1)}_{x}+G^{(2)}_{x},p_{y}+G^{(1)}_{y}+G^{(2)}_{y})$,and $b_{1}(p_{x},p_{y})=b_{1}(p_{x}+G^{(1)}_{x}+G^{(2)}_{x},p_{y}++G^{(1)}_{y}+G^{(2)}_{y})$ we obtain:

[TABLE]

where the hexagonal reciprocal lattice vectors are \vec{G}^{(1)}=\frac{2\pi}{3a}\Big{[}1,\sqrt{3}\Big{]} and \vec{G}^{(2)}=\frac{2\pi}{3a}\Big{[}1,-\sqrt{3}\Big{]}, with the two Bravais unit cell vectors \vec{a}_{(1)}=\frac{a}{2}\Big{[}3,\sqrt{3}\Big{]}, \vec{a}_{(2)}=\frac{a}{2}\Big{[}3,-\sqrt{3}\Big{]}.

The discrete sum over the integers $n$ and $m$ ,$\vec{R}_{n,m}=n\vec{a}_{(1)}+m\vec{a}_{(2)}$ determines the position of the lattice atom $A$ Castro . Atom $B$ is given by the relative vectors $\vec{\delta_{r}}$ ,r=1,2,3. with respect to atom $A$ at position $\vec{R}_{n,m}$.The sum over the vectors $\vec{\delta}_{r}$ determines the function $\phi_{1}(\vec{p})$.

[TABLE]

The location of the two nodes in layer $i=1$ is given by $\vec{K}_{1}=[0,\frac{4\pi}{3\sqrt{3}}]$ and $\vec{K^{\prime}}_{1}=[0,\frac{-4\pi}{3\sqrt{3}}]$ which obey $\phi_{1}(\vec{p}=\vec{K}_{1})=0$ and $\phi_{1}(p=\vec{K^{\prime}}_{1})=0$. Layer $2$ is rotated with respect to layer $1$ . In layer $1$ the atoms are $A_{1}$ and $B_{1}$ while in layer $2$ the atoms are $A^{\prime}=A_{2}$ and $B^{\prime}=B_{2}$ . In a stacked $A,B$ bilaye,r $A^{\prime}$ and $B^{\prime}$ have the same horizontal position as atoms $A_{1}$ and $B_{1}$.

The rotation of layer $2$ occurs in such a way that the site occupied by atom $B^{\prime}$ is located directly opposite an atom $A$ (layer 1).A commensurate structure is obtained if a $B^{\prime}$ atom is moved by rotation to a position formerly occupied by an atom $B^{\prime}$ of the same kind. The $Moire$ pattern is periodic and a translation from the center to the position of $B^{\prime}$ is a translation symmetry.

$cos[\theta_{i}]=\frac{3i^{2}+3i+0.5}{3i^{2}+3i+1}$,where the vector $i$ is i=0,1,2…

The superlattice bases are $\vec{t}_{1}=i\vec{a}_{1}+(i+1)\vec{a}_{2}$; $\vec{t}_{2}=-(i+1)\vec{a}_{1}+(2i+1)\vec{a}_{2}$;

In addition, in layer $2$, the vector $\vec{R}_{n,m}$ is replaced by the rotated vector $\vec{R^{\prime}}_{n,m}$: \vec{R^{\prime}}_{n,m}=\vec{R^{\prime}}(\theta)_{n,m}=\Big{[}R^{\prime}_{x},R^{\prime}_{y}\Big{]}_{n,m}=\Big{[}R_{x}Cos[\theta]-R_{y}Sin[\theta],R_{x}Sin[\theta]+R_{y}Cos[\theta]\Big{]}_{n,m} and $\vec{\delta}_{r}$ is replaced by, \vec{\delta^{\prime}}_{r}(\theta)=\Big{[}\delta^{\prime}_{r,x},\delta^{\prime}_{r,y}\Big{]}=\Big{[}\delta_{r,x}Cos[\theta]-\delta_{r,y}Sin[\theta],\delta_{r,x}Sin[\theta]+\delta_{r,y}Cos[\theta]\Big{]} The Hamiltonian for the rotated layer $2$ is given by:

[TABLE]

When we rotate the coordinates of layer $2$ by an angle $\theta$, the momentum is rotated by angle $-\theta$, \vec{k}[-\theta]=\Big{[}k_{x}Cos[\theta]+k_{y}Sin[\theta],k_{y}Cos[\theta]-k_{x}Sin[\theta]\Big{]}

[TABLE]

$g_{x}[\theta]$ and $g_{y}[\theta]$ are the new BZ with two lattice vectors $L_{1}$,$L_{2}$.

At special angles $g_{x}[\theta=\theta_{magic}]=g^{Moire}_{x}$, $g_{y}[\theta=\theta_{magic}]=g^{Moire}_{y}$ given by $Cos[\theta(n)]=\frac{3n^{2}+3n+\frac{1}{2}}{3n^{2}+3n+1}$ n=1,2…Shallcross ; Lopes , we obtain the Moire comensurate rotations such that $g_{x}[\theta_{magic}=\theta(n)]L_{1}=2\pi$ ,$g_{y}[\theta_{magic}=\theta(n)]L_{2}=2\pi$.

Using the periodicity in the BZ with respect to the $Moire$ reciprocal lattice $a_{2}(p_{x},p_{y})=a_{2}(p_{x}+g_{x}[\theta],p_{y}+g_{y}[\theta])$ , $b_{2}(p_{x},p_{y})=b_{2}(p_{x}+g_{x}[\theta],p_{y}+g_{y}[\theta])$, we find that at the magic angle $[\theta_{magic}]$ it is commensurate with the hexagonal lattice $a_{2}(p_{x},p_{y})=a_{2}(p_{x}+g_{x}[\theta=0],p_{y}+g_{y}[\theta]=0)$ , $b_{2}(p_{x},p_{y})=b_{2}(p_{x}+g_{x}[\theta=0],p_{y}+g_{y}[\theta=0])$ .As a result of the periodicity and commensuration we obtain:

[TABLE]

We replace $\vec{p}\cdot\vec{\delta^{\prime}}_{r}(\theta)$ with $\vec{p}[-\theta]\cdot\vec{\delta}_{r}$ and compute $\varphi_{2}(p)$:

[TABLE]

The Dirac points in layer $2$ are at momentum \vec{K}_{2}=\frac{4\pi}{3a}\Big{[}Sin[\theta],\frac{1}{\sqrt{3}}Cos[\theta]\Big{]} and \vec{K^{\prime}}_{2}=-\frac{4\pi}{3a}\Big{[}Sin[\theta],\frac{1}{\sqrt{3}}Cos[\theta]\Big{]} .

The tunneling Hamiltonian between layer $1$ and $2$ is given by Lopes :

[TABLE]

where \vec{p}[-\theta]=\Big{[}p_{x}Cos[\theta]+p_{y}Sin[\theta],p_{y}Cos[\theta]-p_{x}Sin[\theta]\Big{]}

For small angle rotations, we replace $H_{\perp,3}$ with the $Moire$ BZ:

[TABLE]

Here, the tunneling coupling constant is given by $\gamma_{0}=2.8ev$,$\gamma_{3}=0.3ev$ Castro .

IV-Computation of the eigenvalues

We will compute the eigenvalues for small angles which correspond to a $Moire$ commensurate lattice. The Hamiltonian $H_{1}$ and $H_{2}$ are diagonalyzed using the following representation : for layer $i=1$ the eigenvalues are $\pm|\varphi_{1}(\vec{p})|$, with the two eigenvectors u_{1}(\vec{p})=\frac{1}{\sqrt{2}}\Big{[}1,-e^{-i\alpha_{1}(\vec{p})}\Big{]}^{T}=\frac{1}{\sqrt{2}}\Big{[}1,-\frac{|\phi_{1}(\vec{p}|}{\phi_{1}(\vec{p}}\Big{]}^{T} and for negative eigenvalues the eigenvector is v_{1}(\vec{p})=\frac{1}{\sqrt{2}}\Big{[}1,e^{-i\alpha_{1}(\vec{p})}\Big{]}^{T}

For layer $i=2$ we have the eigenvalue $\pm|\varphi_{2}(p)|$ The eigenvector for layer $i=2$ is u_{2}(\vec{p})=\frac{1}{\sqrt{2}}\Big{[}1,-e^{-i\alpha_{2}(\vec{p})}\Big{]}^{T}=\frac{1}{\sqrt{2}}\Big{[}1,-\frac{\varphi^{*}_{2}(p)}{|\varphi_{2}(p)|}\Big{]}^{T} for positive eigenvalues, while for the negative eigenvalues v_{2}(\vec{p})=\frac{1}{\sqrt{2}}\Big{[}1,e^{-i\alpha_{2}(\vec{p})}\Big{]}^{T}

[TABLE]

$C^{\dagger}_{2}(\vec{p})$,$C_{2}(\vec{p})$,$C^{\dagger}_{1}(\vec{p})$,$C_{1}(\vec{p})$ are the particle operators and $D^{\dagger}_{2}(\vec{p})$,$D_{2}(\vec{p})$,$D^{\dagger}_{1}(\vec{p})$,$D_{1}(\vec{p})$ are the anti-particles operators.

We neglect the anti -particles and rewrite the Hamiltonian in terms of the particle operators only. We consider the small angles such that the $Moire$ lattice is commensurate with the hexagonal lattice.

[TABLE]

The lowest eigenvalue is given by:

[TABLE]

We include the chemical potential $\mu$ and observe that the band is quasi- flat .

In order to obtain a better description of the bands,we will expand the model around the Dirac Cones and we will observe the Dirac dispersion.

IV-The continuum model

In order to see how the interactions are affected by the rotations we will use a continuum model. The continuum model will show the Dirac dispersion around the Dirac cones. For each layer we introduce a linear model around the position of the two Dirac nodes. For layer $1$, we replace $\vec{p}=\vec{K^{\prime}}_{1}+\vec{q}$ for the left valley and $\vec{p}=\vec{K}_{1}+\vec{q}$ for the right valley. The two valley are represented by the Pauli matrix $\vec{\tau}$.

[TABLE]

where

[TABLE]

where $\mu[q_{x}+K_{x,1}]$ is the step function which obeys $\mu[-q_{x}-K^{\prime}_{x,1}]+\mu[q_{x}+K_{x,1}]=1$. From equation $(15)$ we obtain the eigen spinors and represent $\Phi_{1,L}(\vec{q})$, $\Phi_{1R}(\vec{q})$ in terms of the valley operators $C_{1,L}(\vec{q})$ , $C_{1,R}(\vec{q})$

[TABLE]

For layer $i=2$ with the rotated nodes at momentum \vec{K}_{2}=\frac{4\pi}{3a}\Big{[}Sin[\theta],\frac{1}{\sqrt{3}}Cos[\theta]\Big{]} and \vec{K^{\prime}}_{2}=-\frac{4\pi}{3a}\Big{[}Sin[\theta],\frac{1}{\sqrt{3}}Cos[\theta]\Big{]} , eq.$(8)$ gives the linearized form:

[TABLE]

The field in layer $i=2$ has the representation:

[TABLE]

with the representation : $\Phi_{2,L}(\vec{q})$, $\Phi_{2,R}(\vec{q})$ in terms of the operators $C_{2,L}(\vec{q}$ , $C_{2,R}(\vec{q})$

[TABLE]

The rotated layer $i=2$ ,$\Psi_{2}(\vec{r^{\prime}})$ is represented in terms of the rotated Dirac nodes , \vec{K^{\prime}}_{2}=-\frac{2\pi}{3a}\Big{[}Sin[\theta],\frac{1}{\sqrt{3}}Cos[\theta]\Big{]}, \vec{K}_{2}=\frac{2\pi}{3a}\Big{[}Sin[\theta],\frac{1}{\sqrt{3}}Cos[\theta]\Big{]} .

The eigenvalues for the two layers around the two Dirac points for particles with respect to the momentum $\vec{q}$ are : $\epsilon_{1,L}(\vec{q})=\epsilon_{1,R}(\vec{q})=v_{F}\sqrt{(q_{x})^{2}+(q_{y})^{2}}$ and $\epsilon_{2,L}(\vec{q})=\epsilon_{2,R}(\vec{q})=v_{F}\sqrt{(q_{x})^{2}+(q_{y})^{2}}$.

For the tunneling Hamiltonian, we obtain:

[TABLE]

where

[TABLE]

[TABLE]

Using the periodicity with respect to the $Moire$ reciprocal lattice which at magical angles is commensurate with the honeycomb lattice, at small angles we obtain in the $Moire$ BZ the representation:

[TABLE]

[TABLE]

where the effective tunneling coefficients are given by:

[TABLE]

In order to address the question of flat bands, we will solve the model under an approximations which neglects the higher order couplings .

We diagonalize the left $H_{L}$ Hamiltonian and find two eigenvalues:

[TABLE]

We find the two operators, $C^{(-)}_{L}$ and $C^{(+)}_{L}$ using the approximation, $E^{(+)}_{L}>>E^{(-)}_{L}$. The approximations is based on projecting out the states $E^{(+)}_{L}$ using the constraint $C^{(+)}_{L}\approx 0$ (which is justified when $E^{(+)}_{L}>>E^{(-)}_{L}$ ).Thus we obtain:

[TABLE]

We diagonalize the $H_{R}$ Hamiltonian using the approximation based on projecting out the state $E^{(+)}_{L}$ :

[TABLE]

Performing the projection in the right valley gives:

[TABLE]

The effective tunneling Hamiltonian which includes the coupling between the two valleys is:

[TABLE]

The lowest eigenvalue of the effective Hamiltonian will give the band $E_{0}$:

[TABLE]

The lowest energy band $E_{0}$ is flat $E_{flat}=E_{0}$ and justifies the name when $\theta\approx\theta(n)_{magic}$

V-Superconductivity induced by the rotated layer $i=2$

Here we will use the spinor representation to demonstrate that the rotation by angle $\theta$ affects the electron-electron interactions. For simplicity we consider a repulsive Hubbard interaction. Due to the spinor rotation the Hubbard interaction becomes attractive in the $y$ direction and periodic in the $x$ direction .Effectively, this is described as a set of one dimensional superconducting wires separated by metallic regions. This result is additive to the attractive interactions mediated by the phonons.

The interaction in layer $i=2$ is controlled by two fields , $\Psi_{2}(\vec{R^{\prime}})$ and $\Psi_{2}(\vec{R^{\prime}}+\delta_{\vec{r^{\prime}}})$:

[TABLE]

where the spinors are given by: U_{2,L}(\vec{p}-\vec{K^{\prime}}_{2})\approx U_{2,L}(-\vec{K^{\prime}}_{2})=\frac{1}{\sqrt{2}}\Big{[}1,ie^{i\theta}e^{-i\vec{K^{\prime}}_{2}}\Big{]},

U_{2,R}(\vec{p}-\vec{K}_{2})\approx U_{2,R}(-\vec{K}_{2})=\frac{1}{\sqrt{2}}\Big{[}1,ie^{i\theta}e^{-\vec{K}_{2}}\Big{]}.

We observe that the spinor depends on the a rotated angle $\theta$.

The Hubbard interaction in layer $i=2$ is:

[TABLE]

We will use the summation over the twisted vector $\vec{\delta}_{r^{\prime}}$ introduced in equation $(8)$ which relates the rotated momentum to the $Moire$ reciprocal lattice (see Eq.$(6)$ ) :

[TABLE]

Next, in the Hubbard interaction we substitute the fields $\Psi_{2,\sigma}(\vec{R^{\prime}})$ and $\Psi_{2,\sigma}(\vec{R^{\prime}}+\delta_{\vec{r^{\prime}}})$ given in Eq.$(33)$ including the spin dependence. We notice that the intervalleys depend on the rotated angles . If we use a long range Coulomb potential we can use the same strategy as used for the Hubbard model , by identifying directions where we have an attractive potential.Those terms are given by $\tilde{C}^{\dagger}_{2,L,\uparrow}(\vec{p}_{1})\tilde{C}^{\dagger}_{2,L,\downarrow}(\vec{p}_{2})\tilde{C}_{2,R,\downarrow}(\vec{p}_{3})\tilde{C}_{2,R,\uparrow}(\vec{p}_{4})$ and $\tilde{C}^{\dagger}_{2,R,\uparrow}(\vec{p}_{1})\tilde{C}^{\dagger}_{2,R,\downarrow}(\vec{p}_{2})\tilde{C}_{2,L,\downarrow}(\vec{p}_{3})\tilde{C}_{2,L,\uparrow}(\vec{p}_{4})$. We introduce the short notation :

[TABLE]

Using the periodicity with respect to the $Moire$ reciprocal lattice $\tilde{C}_{2,L}(\vec{p}+\vec{g}[\theta])=\tilde{C}_{2,L}(\vec{p})$, $\tilde{C}_{2,R}(\vec{p}+\vec{g}[\theta])=\tilde{C}_{2,R}(\vec{p})$ we obtain the following form of the Hubbard interaction:

[TABLE]

At certain angles, the imaginary part of the effective Hubbard potential vanishes and the real part is negative:

[TABLE]

We define :

U[\bar{\theta}]=\frac{U}{2}\Big{(}Cos[4\theta]+Cos[4\theta+g_{x}[\theta]+g_{y}[\theta]]+2Cos[(4\theta-\frac{1}{2}g_{x}[\theta]-\frac{1}{2}g_{y}[\theta])Cos[\frac{\sqrt{3}}{2}(g_{y}[\theta]-g_{x}[\theta])]\Big{)}. We observe in Figure $1$ that the interaction term $U[\bar{\theta}]$ is attractive for certain angles $\theta=s$.

[TABLE]

This allows us to write a one dimensional pairing Hamiltonian at a fixed linear momentum $p_{x}=0$ ( this corresponds to $\vec{K^{\prime}}_{2,x}=\vec{K}_{2,x}$) :

[TABLE]

Following Weinberg we use the Lagrangian representation and perform a saddle point computation:

[TABLE]

We use the Hubbard Stratonovici fields $\Delta$,$\Delta^{*}$, $D$ and $D^{*}$. This is done by replacing:

\tilde{C}^{\dagger}_{2,R,\uparrow}(p_{x}=0,y)\tilde{C}^{\dagger}_{2,R,\downarrow}(p_{x}=0,y)\tilde{C}_{2,L,\downarrow}(p_{x}=0,y)\tilde{C}_{2,L,\uparrow}(p_{x}=0,y)=\frac{1}{4}\Big{(}\tilde{C}^{\dagger}_{2,R,\uparrow}(p_{x}=0,y)\tilde{C}^{\dagger}_{2,R,\downarrow}(p_{x}=0,y)+\tilde{C}_{2,L,\downarrow}(p_{x}=0,y)\tilde{C}_{2,L,\uparrow}(p_{x}=0,y)\Big{)}^{2}-\frac{1}{4}\Big{(}\tilde{C}^{\dagger}_{2,R,\uparrow}(p_{x}=0,y)\tilde{C}^{\dagger}_{2,R,\downarrow}(p_{x}=0,y)-\tilde{C}_{2,L,\uparrow}(p_{x}=0,y)\tilde{C}_{2,L,\downarrow}(p_{x}=0,y)^{2}

and further decouplingthe two body interaction by a Gaussian integration Weinberg .

[TABLE]

The variation with respect to the fields $\Delta$,$\Delta^{*}$, $D$ and $D^{*}$ gives the equations :

[TABLE]

From these equations we obtain :

\langle\tilde{C}^{\dagger}_{2,L,\uparrow}(p_{x}=0,y;t)\tilde{C}^{\dagger}_{2,L,\downarrow}(p_{x}=0,y;t)\rangle=\Big{(}\langle\tilde{C}_{2,L,\uparrow}(p_{x}=0,y;t)\tilde{C}_{2,L,\downarrow}(p_{x}=0,y;t)\rangle\Big{)}^{*}. This allows us to reduce the four fields to only two:$D=\Delta=\chi(p_{x}=0,y)$, $\Delta^{*}=D^{*}=\chi(p_{x}=0,y)^{*}$.Using this saddle point, we simplify the interaction term:

[TABLE]

This gives a one-dimensional superconducting order parameter periodic in $x$ with periodicity $K_{1,x}$.

[TABLE]

To compute the critical temperature $T=T_{c}$, we perform a variation with respect to $\chi$ . Performing the computation in the Euclidean form, we obtain:

[TABLE]

We obtain the equation:

[TABLE]

with $E(\vec{p})=\epsilon(\vec{p})-\mu<\Lambda$

The elctronic dispersion is not known.The flat band allows us to replace $E(p)=\epsilon(\vec{p})_{max}-\mu\approx 0$ and superconductivity is destroyed . The critical temperature for limit $\mu\approx 0$ gives $T_{c}\approx\frac{L}{\epsilon(\vec{p})_{max}}|U[\bar{\theta}]|$.

VI-Conclusions

To conclude, we have shown that in graphene lower rotation induces an attractive interaction for some directions and certain valley components. This results in a one-dimensional superconducting parameter periodic in $x$ with periodicity $K_{1,x}$ .We want to mention that alternative explanations have been proposed .These explanations are complementary to the model proposed here and can give rise to two dimensional superconductivity.

These results have been obtained under the following conditions:

a)A superlattice with basis vectors $\vec{t}_{1}=i\vec{a}_{1}+(i+1)\vec{a}_{2}$; $\vec{t}_{2}=-(i+1)\vec{a}_{1}+(2i+1)\vec{a}_{2}$ is formed at magic angles $cos[\theta_{i}]=\frac{3i^{2}+3i+0.5}{3i^{2}+3i+1}$,i=0,1,2…

b)A uniform rotation in real space by an angle $\theta$ is equivalent to a rotation $-\theta$ of the momentum space.

c) The discrete summation with respect to the lattice points relates the momentum to the reciprocal lattice vectors $(G^{(1)}_{x}+G^{(2)}_{x})Cos[\theta]$ and $(G^{(1)}_{x}+G^{(2)}_{x})Sin[\theta]$ which at the magic angles become the $Moire$ reciprocal lattice .