Graphene

Latex如果看著費勁照例可以去我的leanote： Graphene

# Outline

- Bonding and lattice

- Tight-binding model

- Spectrum

- Relativistic Dirac fermions

- DOS

- Next-nearest neighbor

- Quantum Hall Effect

- Other facts

# Bonding and lattice

- $^{12}C$ has nuclear spin $I=0$ and configuration $2s^12s^22p^2$, where the $2p$ orbitals $(2p_x,2p_y,2p_z)$ are $~4eV$ higher than the $2s$ orbital.

- In the excited state, it is favourabe to excite one electron from the $2s$ to the third $2p$ orbital (in order to form covalent bonds, for instance). See fig1.

- In the excited state one therefore has four equivalent quantum states, $|2srangle$, $|2p_xrangle$, $|2p_yrangle$, and $|2p_zrangle$. A superposition of the state $|2srangle$ with $n$ $|2p_jrangle$ states is called $sp^n$ hybridization.

- $sp^1$ hybridization happens for exampe in $H-cequiv C-H$, where overlaping $sp^1$ orbitals of the two carbon atoms form a strong $sigma$ bond. The remaining unhybridized $2p$ state are furthermore involved in the formation of two weaker $pi$ bonds. See fig2.

- For $sp^2$ hybridization, the three states are given by $|psi_1rangle=frac{1}{sqrt{3}}left( |2srangle-sqrt{2}|2p_yrangleright)$, $|psi_2rangle=frac{1}{sqrt{3}}left(|2srangle+sqrt{frac{3}{2}}|2p_xrangle+sqrt{frac{1}{2}}|2p_yrangleright)$, $|psi_3rangle=frac{1}{sqrt{3}}left(-|2srangle-sqrt{frac{3}{2}}|2p_xrangle+sqrt{frac{1}{2}}|2p_yrangleright)$.

- These three orbitals are oriented in the $xy$-plane and have mutual 120 degrees angle. The remaining unhybridized $2p_z$ orbital is perpendicular to the plane. Famous example is the benzene molecule, see fig3: in addition to the six covalent $sigma$ bonds between the carbon atoms, there are three $psi$ bonds.

- $sp^3$ hybridization example: dimond. The orbitals form a tetrahedron.

- The carbon atoms in graphene form a honeycomb lattice due to their $sp^2$ hybridization. The distance between nearest neighbor carbon atoms is 0.142nm, which is the average of the single (C-C) and double (C=C) covalent bonds as in benzene. All $sp^2$ orbitals form $sigma$-bonds with the $sp^2$ orbitals of the neighboring atom. The bonding orbital associated with each $sigma$-bond is occupied by two electrons (spin up and down).

- There is one electron per carbon atom left in the $2p_z$ orbital. Thus each primitive cell contributes two $2p_z$-orbitals that participate in bonding. The $2p_z$ orbital stick out of the plane of the chain and form $pi$-bonds with neighboring $2p_z$ orbitals. The $pi$-electrons happen to be those responsible for the electronic properties at low energies, whereas the $sigma$ electrons form energy bands far away from the Fermi energy.

- Now we set the notation. We choose our Bravais lattice to have primitive lattice vectors given by $mathbf{a}_1=frac{a}{2}(3,sqrt{3})$ and $mathbf{a}_2=frac{a}{2}(3,-sqrt{3})$, where $a$ is the above-mentioned nearest neighbor lattice spacing 0.142nm. See fig4.

- For convenience, we denote for an A-sublattice atom, the three nearest-neighbor vectors in real space are given by $mathbf{delta}_1=frac{a}{2}(1,sqrt{3})$, $mathbf{delta}_2=frac{a}{2}(1,-sqrt{3})$ and $mathbf{delta_3}=-a(1,0)$. While those for the B-sublattice are the negatives of these.

- The reciprocal lattice vectors are defined through the relation $mathbf{a}_icdotmathbf{b}_j=2pidelta_{ij}$ , $mathbf{b}_1=frac{2pi}{3a}(1,sqrt{3})$ and $mathbf{b}_2=frac{2pi}{3a}(1,-sqrt{3})$.

- We define the first Brillouin zone as bounded by the planes bisecting the vectors to the nearest reciprocal lattice points, see fig5. The six points at the corners of the BZ fall into two groups of three which are equivalent, so we need only to consider two equivalent corners that we label by $mathbf{K}=frac{2pi}{3a}(1,frac{1}{sqrt{3}})$ and $mathbf{K}=frac{2pi}{3a}(1,-frac{1}{sqrt{3}})$. We stress that $mathbf{K}$ and $mathbf{K}$ are not connected by a reciprocal lattice vector.

# Tight-binding model

- Introduce the atomic orbitals $phi_m(mathbf{r})$, which are the eigenfunctions of the Hamiltonian of a single isolated atom. Tight-binding assumes the overlap between these atomic wave functions on adjacent sites is small.

- A solution to the time-independent single electron Schroedinger equation is then approximated as a linear combination of atomic orbital: $psi_m(mathbf{r})=sum_{mathbf{R})_n}b_m(mathbf{R}_n)phi_m(mathbf{r}-mathbf{R}_n)$, where $m$ refers to the $m$-th atomic energy level and $mathbf{R}_n$ locates an atomic site in the lattice.

- The Bloch theorem states that the wave function in crystal can change under translation only by a phase factor: $psi(mathbf{r}+mathbf{R}_l)=e^{imathbf{k}cdotmathbf{R}_l}psi(mathbf{r})$, where $k$ is the wave vector. Plugging in the expression for $psi$ above and simplifying, we have $b_m(mathbf{R}_l)=e^{imathbf{k}cdotmathbf{R}_l}b_m(mathbf{0})$.

- After normalization, one obtains $b_m(0)approx frac{1}{sqrt{N}}$ and $psi_m(mathbf{r})approx frac{1}{sqrt{N}}sum_{mathbf{R}_n}e^{imathbf{k}cdotmathbf{R}_n}phi_m(mathbf{r}-mathbf{R}_n)$.

- By tight-binding, the Hamiltonian $H(mathbf{r})=sum_{mathbf{R}_n}H_{at}(mathbf{r}-mathbf{R}_n)+Delta U(mathbf{r})$, where the second term describes the small correction, or the small overlap beetween neighboring atomic wave functions $phi$.

- Assuming only the $m$-th atomic energy level is important for the $m$-th energy band of the whole Hamiltonian, the Bloch energies $varepsilon_m$ are of the form $varepsilon_m=int d^3r~psi^*(mathbf{r})H(mathbf{r})psi(mathbf{r})approx E_m+b^*(0)sum_{mathbf{R}_n}e^{-imathbf{k}cdotmathbf{R}_n}int d^3rphi^*(mathbf{r}-mathbf{R}_n)Delta U(mathbf{r})psi(mathbf{r})$, where $E_m$ is the energy of the $m$-th atomic level.

- Plugging in the expression of $psi$ above, one recognizes three terms that appear in the above expression: $beta_m=-intphi^*_m(mathbf{r})Delta U(mathbf{r})phi_m(mathbf{r})d^3r$, $t_{m,l}(mathbf{R}_n)=-intphi^*_m(mathbf{r})Delta U(mathbf{r})phi_l(mathbf{r}-mathbf{R}_n)d^3r$, and $alpha_{m,l}(mathbf{R}_n)=intphi^*_m(mathbf{r})phi_l(mathbf{r}-mathbf{R}_n)d^3r$. Here $beta$ describes tha tomic energy shift due to the potential on neighboring atoms, which is relatively small in most cases. $t$ is the hopping matrix element. $alpha$ is the overlap integral between the atomic orbitals $m$, and $l$ on adjacent atoms.

# Spectrum

- Now we model graphene using tight-binding theory. The relevant atomic orbital is the single $pi$ orbital which is left unfilled by the bonding electrons, and which is normal to the plane of the lattice. If we denote the the orbital on atom $i$ with spin $sigma$ as $(i,sigma)$, and the creation operator for an atom in the A(B) sublattice as $a^dagger$($b^dagger$), then in the second quantization regime the nearest-neighbor tight-binding Hamiltonian is $H=-tsum_{(ij),sigma}(a_{(i,sigma)}^dagger b_{(j,sigma)} + h.c.)$. For graphene, $t$ is believed to be around 2.8eV.

- We right the tight-binding eigenfunctions in the form of a spinor, whose components correspond to the amplitudes on the A and B atoms repsectively within the unit cell labeled by a reference point $mathbf{R}_i^0$. (Thats why we call the A/B psuedospins.)

- By convention, we choose B to be separated from A by $mathbf{delta}_1$ and $mathbf{R}_i^0$ as the position of A, see fig6.

- Then the tight-binding eigenfunctions have the form (apart from the spin index) $left(begin{array}{c} alpha_mathbf{k} beta_mathbf{k}end{array}right)=sum_i e^{imathbf{k}cdot mathbf{R}_i^0} left(begin{array}{c} a_i^dagger e^{-imathbf{k}cdotmathbf{delta}_1/2} b_i^dagger e^{imathbf{k}cdotmathbf{delta}_1/2}end{array}right)$. The factor $e^{pm imathbf{k}cdotmathbf{delta}_1/2}$ is inserted for subsequent convenience.

- The resulting Hamiltonian in the $k$-representation is purely off-diagonal: $H_mathbf{k}=left(begin{array}{cc} 0 & Delta_{mathbf{k}} Delta_{mathbf{k}}^* & 0 end{array}right)$, with $Delta_{mathbf{k}}equiv -tsum_l e^{imathbf{k}cdotmathbf{delta}_l}$.

- From the explicit form of $mathbf{delta}_l$, we obtain the eigenvalues of $H$ as $varepsilon_{mathbf{k}}=pm |Delta_{mathbf{k}}=pm t sqrt{1+4cos frac{3k_xa}{2}cosfrac{sqrt{3}k_ya}{2}+4cos^2frac{sqrt{3}k_ya}{2}}$.

- $varepsilon_{mathbf{k}}=0$ happens at the following pints: $frac{3k_xa}{2}=2npi$, $cosfrac{sqrt{3}k_ya}{2}=-frac{1}{2}$; $frac{3k_xa}{2}=(2n+1)pi$, $cosfrac{sqrt{3}k_ya}{2}=frac{1}{2}$. The second set of solution at $n=0$ lies within the first Brillouin zone and is satisfied exactly at points $mathbf{K}$ and $mathbf{K}$. These are the Dirac points.

- Since the energy band is exactly symmetric about the point $varepsilon_{mathbf{k}}=0$ and this condition is met only at the two Dirac points (not met over some complete surface), it follows that for exactly half-filling of the band, the density of states at the Fermi level is exactly zero.

- But graphene has exactly one electron per spin per atom, so the band is indeed exactly filled. Thus undoped graphene is a perfect semimetal.

# Relativistic Dirac fermions

- Now we look at what happens near the Dirac points.

- Since the two sublattices are physically equivalent, then the eigenfunctions must be either symmetric or antisymmetric with respect to the exchange of A and B (apart from trivial phase factors involved in the definition of $a_i$ and $b_i$).

- Do expansion with respect to $mathbf{q}=mathbf{k}-mathbf{K}$. Then $Delta(mathbf{q})approx-ifrac{3ta}{2}e^{-iK_xa}(q_x+iq_y)$.

- Extract an overall constant factor which doesnt affect physical results. Then $Delta(mathbf{q})=hbar v_F(q_x+iq_y)$ with $v_F=frac{3ta}{2hbar}=10^6 m/s$.

- Had we done the expansion around $mathbf{K}$, we would have obtained $Delta(mathbf{q})=hbar v_F(q_x-iq_y)$.

- The Hamiltonian then becomes, $H=hbar v_Fleft(begin{array}{cc} 0 & q_x+iq_y q_x-iq_y & 0 end{array}right)=hbar v_F mathbf{sigma}cdot mathbf{q}$ and the bands are $varepsilon(q)=pm v_F |q|$.

- This Hamiltonian is exactly that of an ultra-relativistic particle of spin-1/2. with velocity of light $c$ replaced by the Fermi velocity $v_F$, which is $~300$ times smaller.

- The eigenfunctions near $K$ point are $psi^pm(mathbf{q})=frac{1}{sqrt{2}}left(begin{array}{c} e^{itheta_q /2} pm e^{-itheta_q/2} end{array}right)$, with $theta_q=tan^{-1}frac{q_x}{q_y}$. (Remark: when $mathbf{q}$ rotates once around the Dirac point, phase of this state changes by $pi$, as expected for spin-1/2.)

- One can check these two eigenfunctions at $mathbf{K}$ and $mathbf{K}$ have different helicities defined as $mathbf{h}=mathbf{sigma}cdotmathbf{p}/p$.

# DOS

- Density of states $rho(varepsilon)$ counts the number of quantum states in the vicinity of a fixed energy $varepsilon$. It will be useful in the discussion of transport. It is defined as $rho(varepsilon)=frac{1}{A}frac{partial N^+}{partial varepsilon}$, where $N^+=gsum_{k, varepsilon_k^+leq varepsilon}=Aint_0^varepsilon dvarepsilonrho(varepsilon)$ is the total number of states below the energy $varepsilon$ and $A$ is the total surface. The factor $g=4$ counts the spin and valley degeneracy.

- The calculation of DOS may be involved because one needs to calculate a 2D integral over the wave vector as a function of energy by inverting the energy dispersion. But near the Dirac point, energy dispersion is isotropic and one can identify $sum_{k,varepsilon_k^+leqvarepsilon}simeq frac{Ag}{2pi}int_0^{q(varepsilon)}dq~q$. (By isotropy we mean the bands only dipend on the magnitude of $mathbf{q}$.)

- One thus obtains near the Dirac point, $rho(varepsilon)=frac{g}{2pi}frac{q(varepsilon)}{partialvarepsilon/partial q}=frac{g|varepsilon|}{2pihbar^2v_F^2}$.

- As expected, the DOS vanishes linearly at the Dirac point at zero energy. This particular situation needs to be contrasted to the conventional case of electrons in 2D metals, with $varepsilon=hbar^2q^2/2m^*$ and $rho(varepsilon)=gm^*/2pihbar^2$ is a constant.

- One can define an effective mass $m^*=hbar q_F/v_F$.

- The mass measured by a cyclotron resonance experiment is in the semiclassical limit $m_c=frac{1}{2pi}frac{partial A}{partial varepsilon}$, which in our case is identical to $m^*$. That is something one can measure and verify.

# Next nearest neighbor hopping

- Now we discuss the correction due to nex nearest neighbor hopping, described by some parameter $t$: $H=-tsum_{((i,j)),sigma}left(a_{isigma}^dagger a_{jsigma}+b_{isigma}^dagger b_{jsigma}+h.c.right)$.

- One observes that this is symmetric between A and B sublattices. Such a term would destroy the perfect symmetry of the band around $varepsilon=0$, since some function $-tf(mathbf{k})$ is added to both the upper and lower branches.

- Bascially the whole spectrum shifts but the degeneracy between $mathbf{K}$ and $mathbf{K}$ will not be spoiled. It will only be spoiled if one introduces some kind of asymmetry between the A and B sublattices.

- More explicitly, in the spinor representation, this term gives a contribution $-tf(mathbf{k})$, with $f(mathbf{k})=sum_{((i,j))}cosmathbf{k}cdotmathbf{R}_{ij}$, where $mathbf{R}_{ij}$ is $mathbf{a}_i-mathbf{a}_j$ the difference between the primitive vectors.

- The value of $f(mathbf{k})$ at the Dirac points is $3t$ and its gradients there are zero. Namely, the $mathbf{k}$ position of Dirac points are not shifted at all.

# Quantum Hall Effect

- Quantum hall effect can be observed in graphene even at room temperature. But there is one nontrivial difference: the ladder in Hall conductivity is shifted with respect to the standard IQHE sequence by 1/2, see fig7. $sigma_{xy}=pm frac{4e^2}{h}(n+frac{1}{2})$. (The factor of four is due to the valley and spin degeneracies.)

- This is actually just due to the Dirac points. The existence of a quantized level at zero energy, which is shared by electrons and holes, is essentially everything one needs to know to explain the anomalous QHE sequence. An alternative explanation is to invoke the coupling between pseudospin and orbital motion, which gives rise to a geometrical phase of $pi$ accumulated along cyclotron trajectories and often is referred to as Berrys phase. The additional phase leads to a $pi$-shift in the phase of quantum oscillations and in the QHE limit to a half-step shift.

- Lets give an intuitive but actually incorrect derivation[5] using Laughlins explanation of IQHE (I wrote a note on Ch9 of Atland&Simons before which included this flux insertion picture). Each Landau level contributes with one state times its degeneracy four. Hence we expect that when flux changes by one flux quantum, the change in energy would be $delta E=pm 4 n eV_H$. Hense we conclude that $I_{inc}=pm 4e^2/hV_H$ and hense $sigma_{xy,inc}=pm 4ne^2/h$. But when the chemical potential is excatly at half-filling (Dirac points), it would predict a Hall plateau at $n=0$ with $sigma_{xy,inc}=0$, which is not possible since there is an $n=0$ Landau level, with extended states at this energy. The solution for this paradox: because of the presense of zero mode that is shared by the two Dirac points, there are excatly $2times(2n+1)$ occupied states that are transferred from one edge to another. Hence $delta E=pm 2(2n+1)eV_H$ upon insertion of $deltaPsi=hc/2$. Thus $sigma_{xy}=frac{I}{V_H}=frac{c}{V_H}frac{delta E}{delta Phi}=pm 2(2N+1) frac{e^2}{h}$ without any Hall plateau at $n=0$.

- Bilayer graphene exhibits an equally anomalous QHE too. One measures the standard $sigma_{xy}=pm 4ne^2/h$ but the very first plateau at $n=0$ is missing, which also implies that bilayer graphene remains metallic at the neutrality point. We refer to other references for this topic.

# Other facts

- Graphenes zero-field conductivity $sigma$ does not disappear in the limit of vanishing $n$ but instead exhibits values close to the conductivity quantum $e^2/h$ per carrier type, see fig8.

- At low temperatures, all metallic systems with high resistivity should inevitably exhibit large quantum interference (localization) magnetoresistance, eventually leading to the metal-insulator transition at $sigmasim e^2/h$. Such behavior is usually universal but found missing in graphene.

# References

[1] J. N. Fuchs and M. O. Goerbig, Introduction to the physical properties of graphene.

[2] F. Rana, Energy bands in graphene: tight binding and the nearly free electron approach.

[3] A. J. Leggett, Graphene: electronic bands tructure and Dirac fermions.

[4] A. K. Geim and K. S. Novoselov, The rise of gaphene.

[5] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov and A. K. Geim, The electronic properties of graphene.

[7] Wikipedia