User:Jatosado

Welcome

Over the past three years I have been working on various projects which have led to corrections, additions and expansions of Wikipedia. See below for any current projects. I encourage you, the reader, to give me comments on any of the material you see here since it is at this time when corrections to the material are easiest to make.

Current Projects

• Graphene: The purpose is to give a clear and basic explanation of graphene's band structure and possibly other characteristics

Graphene

Carbon has four valence electrons. From Valence bond theory three out of the four half filled valent electron orbitals $sp^{2}$ hybridize to create carbon-carbon $\sigma$ bonds which form the graphene crystal. The fourth $p_{z}$ orbital overlaps with neighboring $p_{z}$ orbitals creating out of plane $\pi$ bonds. In the context of Molecular orbital theory the bonding of the carbon atoms has the effect of removing the energy degeneracy possessed by each atomic orbital in isolation serving to create bonding and antibonding states. For $\pi$ bonds, bonding and antibonding states are refered to as $\pi$ and $\pi ^{*}$ states, respectively. The bonding state has a spatially symmetric wave function with a lower energy than the antibonding one. The $\sigma$ bonds account for graphene's structural properties while the $\pi$ bonds account for graphene's electronic properties at low energies. The analysis below is concerned with the band structure which originates from the overlap of the $\pi$ electrons .

The Graphene Lattice

Graphene has a 2-dimensional honeycomb structure which is described by a hexagonal lattice with two atoms at each lattice point and thus two atoms within each Wigner–Seitz cell. Alternatively, it can be viewed as a bipartite lattice composed of two interpenetrating hexagonal sub-lattices. Since the real space lattice is hexagonal so too is the corresponding reciprocal lattice rotated by 90 degrees. Let $\mathbf {\Gamma }$ be the center of the Brillouin zone and $\mathbf {\Gamma } _{i}$ be the center of the $i^{th}$ Brillouin zone, i.e. the position of the respective lattice points in reciprocal space. The reciprocal lattice can be described by $n_{1}\Gamma _{1}+n_{2}\Gamma _{2}$ such that,

$\mathbf {\Gamma _{1}} =2\pi {\frac {({\hat {x}}\otimes {\hat {y}}-{\hat {y}}\otimes {\hat {x}})\mathbf {r_{2}} }{\mathbf {r_{1}} \cdot ({\hat {x}}\otimes {\hat {y}}-{\hat {y}}\otimes {\hat {x}})\mathbf {r_{2}} }}$

$\mathbf {\Gamma _{2}} =2\pi {\frac {({\hat {y}}\otimes {\hat {x}}-{\hat {x}}\otimes {\hat {y}})\mathbf {r_{1}} }{\mathbf {r_{2}} \cdot ({\hat {y}}\otimes {\hat {x}}-{\hat {x}}\otimes {\hat {y}})\mathbf {r_{1}} }}$

The corners of the Brillouin zone denoted by $\mathbf {K} _{i}$ are thus,

$\mathbf {K} _{1}={\frac {1}{3}}(2\mathbf {\Gamma } _{2}+\mathbf {\Gamma } _{1})$,

$\mathbf {K} _{2}={\frac {1}{3}}(\mathbf {\Gamma } _{2}-\mathbf {\Gamma } _{1})$,

$\mathbf {K} _{3}=-\mathbf {K} _{2}+\mathbf {\Gamma } _{1}$,

$\mathbf {K} _{4}=-\mathbf {K} _{1}$,

$\mathbf {K} _{5}=-\mathbf {K} _{2}$,

$\mathbf {K} _{6}=\mathbf {K} _{2}-\mathbf {\Gamma } _{1}$.

Notice that there are two distinct corners or "K points", $\mathbf {K} _{1}$ and $\mathbf {K} _{2}$, from which all the rest are derivable.

Graphene's lattice can be classified as the plane group p6mm. In this sense all translations commute with reflections in the plane of the lattice. This implies that all electron (and phonon) eigenstates are either even or odd under reflection. The segregation of the electron states into $\sigma$ and $\pi$ bonds accentuate this idea. The even states lie in the nodal plane of the crystal and are symmetrical with respect to rotation about the bond axis. These states compose the $\sigma$ bonds. The odd states lie outside of the nodal plane but are cylindrically symmetric within it. These half-filled states lie near the Fermi level, are electrically active (in the low energy limit) and thus compose $\pi$ bonds. For this reason, the $\pi$ states are the easiest to access by experimental probing.

The Hamiltonian of Graphene using the Tight-Binding Model

To understand the basic electronic behavior of graphene it is necessary to describe the behavior of its $\pi$ electrons. Using the tight-binding model for their description assumes that each $\pi$ electron should be tightly bound to its originating carbon atom and should have limited interactions with the states and potentials of neighboring atoms in the crystal. The degree of limitation will be conceptualized by the overlap integral matrix ${\hat {S}}$.

Each atom in the lattice is located at sublattice points "A" and "B" corresponding to vectors $\mathbf {R} _{a}=n_{1}\mathbf {r} _{1}+n_{2}\mathbf {r} _{2}$ and $\mathbf {R} _{b}=\mathbf {R} _{a}+\Delta \mathbf {r}$ with $\Delta \mathbf {r} =-{\frac {1}{3}}(\mathbf {r} _{1}+\mathbf {r} _{2})$ such that $\mathbf {r} _{1}={\sqrt {3}}a_{o}{\hat {x}}$ and $\mathbf {r} _{2}={\frac {1}{2}}{\sqrt {3}}a_{o}{\hat {x}}+{\frac {3}{2}}a_{o}{\hat {y}}$. If now $\chi (\mathbf {r} )$ is the normalized $2p_{z}$ atomic orbital wave function of an isolated carbon atom then let $|\mathbf {R} \rangle$ be the corresponding carbon atom orbital wave function positioned at lattice point $\mathbf {R}$. Since there are two atoms in each Wigner–Seitz cell (at site A and B), one can expect the $\pi$ electron wave function to have a 2-dimensional basis such that each basis function is formed from the isolated carbon atom wave function at the respective lattice sites. Translational symmetry from Bloch's theorem and normalization of these basis functions in the context of the tight binding model of graphene yield two basis wave functions which are Bloch wave functions, i.e.,

$|\phi _{1}\rangle ={\frac {1}{\sqrt {N}}}\sum _{a}e^{i\mathbf {k} \cdot \mathbf {R} _{a}}|\mathbf {R} _{a}\rangle ,$

$|\phi _{2}\rangle ={\frac {1}{\sqrt {N}}}\sum _{b}e^{i\mathbf {k} \cdot \mathbf {R} _{b}}|\mathbf {R} _{b}\rangle .$

In the low energy limit (near the Fermi energy) it is safe to assume that no other orbitals can mix with with the $p_{z}$ orbitals. Therefore, from the basis wave functions the eigenstates can be written as,

$|\psi \rangle =\alpha |\phi _{1}\rangle +\beta |\phi _{2}\rangle .$

These eigenstates must of course satisfy the Schrödinger equation, i.e.,

${\hat {H}}|\psi \rangle =E|\psi \rangle$

where ${\hat {H}}$ is the graphene Hamiltonian and $E$ is the energy of the $\pi$ electron. The components of the Hamiltonian can now be described in terms of the basis states of the crystal, i.e., the inner product of the Schrödinger equation with either basis function yields,

$\langle \phi _{1}|{\hat {H}}|\psi \rangle =E\langle \phi _{1}|\psi \rangle$ and

$\langle \phi _{2}|{\hat {H}}|\psi \rangle =E\langle \phi _{2}|\psi \rangle .$

This implies that,

${\begin{bmatrix}H_{11}&H_{12}\\H_{21}&H_{22}\end{bmatrix}}{\begin{bmatrix}\alpha \\\beta \end{bmatrix}}=E{\begin{bmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{bmatrix}}{\begin{bmatrix}\alpha \\\beta \end{bmatrix}}$

where $H_{ij}=\langle \phi _{i}|{\hat {H}}|\phi _{j}\rangle$ and $S_{ij}=\langle \phi _{i}|\phi _{j}\rangle$ correspond to the elements of ${\hat {H}}$ and ${\hat {S}}$, respectively. While there are two distinct lattice sites each carbon atom is identical to its neighbor. With this in mind, the energy of a $p_{z}$ electron of an isolated carbon atom is then

$E_{p_{z}}=\langle \phi _{1}|{\hat {H}}|\phi _{1}\rangle =\langle \phi _{2}|{\hat {H}}|\phi _{2}\rangle .$

The off diagonal elements are related due to the Hamiltonian being Hermitian, i.e.,

$H_{12}=H_{21}^{*}$

These elements correspond to the energy needed for a $\pi$ electron to "hop" form one lattice site to another. If only hops to nearest neighbors are considered then these off-diagonal elements take a fairly simple expression. To formulate this, first consider an A lattice site at the origin, namely, $\mathbf {R} _{a}=\mathbf {0}$. Next, consider the three B site nearest neighbors to this A site, namely, $\mathbf {R} _{b}=\mathbf {R} _{1},\mathbf {R} _{2}$ & $\mathbf {R} _{3}$ where

$\mathbf {R} _{1}=\mathbf {0} +\Delta \mathbf {r} ,$

$\mathbf {R} _{2}=\mathbf {r} _{1}+\Delta \mathbf {r} ,$

$\mathbf {R} _{3}=\mathbf {r} _{2}+\Delta \mathbf {r} .$

The off-diagonal elements then take the form,

$H_{12}=\langle \phi _{1}|{\hat {H}}|\phi _{2}\rangle$

$={\frac {1}{\sqrt {N}}}\sum _{a}e^{-i\mathbf {k} \cdot \mathbf {R} _{a}}\langle \mathbf {R} _{a}|{\hat {H}}({\frac {1}{\sqrt {N}}}\sum _{b}e^{i\mathbf {k} \cdot \mathbf {R} _{b}}|\mathbf {R} _{b}\rangle )$

$={\frac {1}{N}}\langle \mathbf {0} |{\hat {H}}|\mathbf {R} _{N.N.}\rangle (e^{i\mathbf {k} \cdot \mathbf {R} _{1}}+e^{i\mathbf {k} \cdot \mathbf {R} _{2}}+e^{i\mathbf {k} \cdot \mathbf {R} _{3}})$

where $\langle \mathbf {0} |{\hat {H}}|\mathbf {R} _{N.N.}\rangle =\langle \mathbf {0} |{\hat {H}}|\mathbf {R} _{1}\rangle =\langle \mathbf {0} |{\hat {H}}|\mathbf {R} _{2}\rangle =\langle \mathbf {0} |{\hat {H}}|\mathbf {R} _{3}\rangle$. The quantity ${\frac {1}{N}}\langle \mathbf {0} |{\hat {H}}|\mathbf {R} _{N.N.}\rangle =t\approx 2.75$ eV is the "hopping integral" which represents the kinetic energy of electrons hopping between atoms. The value of $t$ is chosen to match first principle calculations of graphene's band structure around the corners of the Brillouin zone to experimental observation.

As for the overlap integral matrix ${\hat {S}}$, its elements can be formulated similarly. The overlap integral can be visualized as a measure of the mutual resemblance of the wave functions of two basis states. In this case, $S_{11}=S_{22}=\langle \phi _{1}|\phi _{1}\rangle =1$ (i.e., a basis wave function resembles itself 100%) and

$S_{12}=S_{21}^{*}=\langle \phi _{1}|\phi _{2}\rangle ={\frac {1}{N}}\langle \mathbf {0} |\mathbf {R} _{N.N.}\rangle (e^{i\mathbf {k} \cdot \mathbf {R} _{1}}+e^{i\mathbf {k} \cdot \mathbf {R} _{2}}+e^{i\mathbf {k} \cdot \mathbf {R} _{3}})$

Here, the quantity ${\frac {1}{N}}\langle \mathbf {0} |\mathbf {R} _{N.N.}\rangle =s$ is also experimentally determined. For the purpose of this article $s\approx 0$ which simplifies the Schrödinger equation to yield the secular equation

$0=det(\mathbf {\hat {H}} -E\mathbf {1} )$

$0=det{\begin{bmatrix}E_{p_{z}}-E&t(e^{i\mathbf {k} \cdot \mathbf {R} _{1}}+e^{i\mathbf {k} \cdot \mathbf {R} _{2}}+e^{i\mathbf {k} \cdot \mathbf {R} _{3}})\\t(e^{-i\mathbf {k} \cdot \mathbf {R} _{1}}+e^{-i\mathbf {k} \cdot \mathbf {R} _{2}}+e^{-i\mathbf {k} \cdot \mathbf {R} _{3}})&E_{p_{z}}-E\end{bmatrix}}$

$0=E_{p_{z}}^{2}+E^{2}-2E_{p_{z}}E-t^{2}(e^{-i\mathbf {k} \cdot \mathbf {R} _{1}}+e^{-i\mathbf {k} \cdot \mathbf {R} _{2}}+e^{-i\mathbf {k} \cdot \mathbf {R} _{3}})(e^{i\mathbf {k} \cdot \mathbf {R} _{1}}+e^{i\mathbf {k} \cdot \mathbf {R} _{2}}+e^{i\mathbf {k} \cdot \mathbf {R} _{3}})$

$0=E^{2}-2EE_{p_{z}}+E_{p_{z}}^{2}-t^{2}(3+e^{i\mathbf {k} \cdot \mathbf {r} _{1}}+e^{i\mathbf {k} \cdot \mathbf {r} _{2}}+e^{-i\mathbf {k} \cdot \mathbf {r} _{1}}+e^{i\mathbf {k} \cdot \mathbf {r} _{2}-i\mathbf {k} \cdot \mathbf {r} _{1}}+e^{-i\mathbf {k} \cdot \mathbf {r} _{2}}+e^{i\mathbf {k} \cdot \mathbf {r} _{1}-i\mathbf {k} \cdot \mathbf {r} _{2}})$

substituting for $r_{1}$ and $r_{2}$ gives,

$0=E^{2}-2EE_{p_{z}}+E_{p_{z}}^{2}-t^{2}(3+e^{{\sqrt {3}}ik_{x}a_{o}}+e^{-{\sqrt {3}}ik_{x}a_{o}}+e^{{\frac {1}{2}}{\sqrt {3}}ik_{x}a_{o}+{\frac {3}{2}}ik_{y}a_{o}}+e^{-{\frac {1}{2}}{\sqrt {3}}ik_{x}a_{o}-{\frac {3}{2}}ik_{y}a_{o}}+e^{-{\frac {1}{2}}{\sqrt {3}}ik_{x}a_{o}+{\frac {3}{2}}ik_{y}a_{o}}+e^{{\frac {1}{2}}{\sqrt {3}}ik_{x}a_{o}-{\frac {3}{2}}ik_{y}a_{o}})$

$0=E^{2}-2EE_{p_{z}}+E_{p_{z}}^{2}-t^{2}[3+e^{{\sqrt {3}}ik_{x}a_{o}}+e^{-{\sqrt {3}}ik_{x}a_{o}}+(e^{{\frac {1}{2}}{\sqrt {3}}ik_{x}a_{o}}+e^{-{\frac {1}{2}}{\sqrt {3}}ik_{x}a_{o}})(e^{{\frac {3}{2}}ik_{y}a_{o}}+e^{-{\frac {3}{2}}ik_{y}a_{o}})]$

$0=E^{2}-2EE_{p_{z}}+E_{p_{z}}^{2}-t^{2}[3+2cos({\sqrt {3}}k_{x}a_{o})+4cos({\frac {1}{2}}{\sqrt {3}}k_{x}a_{o})cos({\frac {3}{2}}k_{y}a_{o})]$

which implies that

$E=E_{p_{z}}\pm |H_{12}|=E_{p_{z}}\pm t{\sqrt {3+2cos({\sqrt {3}}k_{x}a_{o})+4cos({\frac {1}{2}}{\sqrt {3}}k_{x}a_{o})cos({\frac {3}{2}}k_{y}a_{o})}}$

(see Wallace, (1947)for comparison).

It is common to use this energy corresponding to $s\approx 0$ where the "plus" case is energy of the antibonding state and the "minus" is the energy of the bonding states. Notice that these two energies are degenerate at just the $\mathbf {K}$ points of the Brillouin zone. For this reason, graphene is a zero band gap semiconductor.

The Low-Energy Limit

Low energy excitations of $\pi$ electrons into the conducting $\pi ^{*}$ state are more likely to occur near the $\mathbf {K}$ points. A description of situations where only these excitations are likely gives reason to Taylor expanded graphene's Hamiltonian about these points. Consider a circle about the $K_{1}$ point, i.e., some $\mathbf {k} =\mathbf {K} _{1}+\Delta \mathbf {k}$.

${\hat {H}}={\hat {H}}_{o}-{\frac {\sqrt {3}}{2}}a_{o}t{\frac {1}{\hbar }}{\begin{bmatrix}0&\hbar (\Delta k_{x}-i\Delta k_{y})\\\hbar (\Delta k_{x}+i\Delta k_{y})&0\end{bmatrix}}={\hat {H}}_{o}-{\frac {\sqrt {3}}{2}}a_{o}t{\frac {1}{\hbar }}(\hbar \Delta k_{x}\sigma _{x}+\hbar \Delta k_{y}\sigma _{y})={\hat {H}}_{o}-v_{F}\mathbf {\sigma } \cdot \mathbf {p}$

where the Fermi velocity is $v_{F}={\frac {\sqrt {3}}{2}}a_{o}t\approx {\frac {1}{300}}c$. A massless Dirac fermion refers to the linearly increasing energy state, i.e., from the form of the Hamiltonian above $E=\pm v_{F}\hbar {\sqrt {\Delta \mathbf {k} _{x}^{2}+\Delta \mathbf {k} _{y}^{2}}}$.

Moving adiabatically in k-space around this "k-point", modifies the wave function by a Berry phase, i.e., $e^{i\pi }=-1$ per revolution.

Misc

Two Dimensional Reciprocal Lattice

$\mathbf {\Gamma _{1}} =2\pi {\frac {({\hat {x}}\otimes {\hat {y}}-{\hat {y}}\otimes {\hat {x}})\mathbf {r_{2}} }{\mathbf {r_{1}} \cdot ({\hat {x}}\otimes {\hat {y}}-{\hat {y}}\otimes {\hat {x}})\mathbf {r_{2}} }}$

$\mathbf {\Gamma _{2}} =2\pi {\frac {({\hat {y}}\otimes {\hat {x}}-{\hat {x}}\otimes {\hat {y}})\mathbf {r_{1}} }{\mathbf {r_{2}} \cdot ({\hat {y}}\otimes {\hat {x}}-{\hat {x}}\otimes {\hat {y}})\mathbf {r_{1}} }}$

Berry Phase Concepts

$(d\mathbf {V} /d\lambda )=0$

$\mathbf {U} \cdot \nabla \mathbf {V} =0$

$(\mathbf {U} \cdot \nabla \mathbf {V} )$

$\phi =\oint _{C}\!d\lambda (\mathbf {U} \cdot \nabla \mathbf {U} )$

$\phi =\oint _{C}\!d\lambda \langle \mathbf {U} |\nabla |\mathbf {U} \rangle$

Photoelectrons in a Metal

$I(E)=\lambda _{tot}(E)I_{1}(E)+\int _{E'>E}D(E',E)G(E')dE'$

$I=A\int _{Emin}^{Emax}\!D(E-E')G(E'-E_{o})dE'$

$D(E')=cos[{\frac {1}{2}}\pi a+(1-a)atan(-[E'-E_{o}]\Delta E_{L}^{-1})][(E'-E_{o})^{2}+\Delta E_{L}^{2}]^{{\frac {1}{2}}(a-1)}$

$G(E,E')=exp[-{\frac {1}{2}}(E-E')^{2}\Delta E_{G}^{-2}]$

$I=A\sum _{i=1}^{N}\!D(E_{i}-E_{o})G(E-E_{i})$

$I(E)=\lambda _{tot}(E)I_{1}(E)+\int _{E'>\omega }g(E',E)I_{2}(E')dE'$

$I_{1}(E)={\frac {1}{2\pi }}\int _{\pm \infty }I_{o}(t)e^{i{\frac {E}{\hbar }}t}dt$

$I_{o}(t)=e^{-{\frac {t}{\tau }}}$

$I_{o}(t)=e^{-{\frac {\Delta E_{L}t}{\hbar }}}{\frac {e^{-i\epsilon _{o}t}}{(iDt)^{\alpha }}}$

$I_{2}(E)=B_{o}\int _{E}^{E_{k,max}}(I(E')-I_{E_{k,max}})dE'$

$G_{A}=(K+CE')^{-1}exp[-{\frac {1}{2}}(E-E')^{2}(\Delta E_{G}+CE')^{-2}]$

Method of Least Squares

Consider the data set $Y$ such that $y_{i}$ ∈ $Y$ and $i$ ∈ {1,2,3,... N}. If then $z$ is a function which best describes $Y$ over the domain $X$ then there exists an element $z_{ij}(x_{i},\beta _{j})$ such that $x_{i}$ ∈ $X$ and $j$ ∈ {1,2,3... M} when there M parameters $\beta$. The error between the description of $Y$ and itself can then be defined, i.e., let

$\epsilon _{ij}\equiv y_{i}-z_{ij}(x_{i},\beta _{j})$

Assuming that the error is not systematic, then for a given j the error is expected to be normally distributed about zero. The best description of $Y$ is then one where the mean absolute error is minimized, i.e., where the RMS error is a minimum. The RMS error, though, does not necessarily have a global minimum whereas the square of the Euclidean norm does.

Squared Euclidean Norm

$E_{j}^{2}\equiv \sum _{i=1}^{N}\epsilon _{ij}^{2}$

Root Mean Square Error

$\epsilon _{RMS}={\sqrt {\frac {E_{j}^{2}}{N}}}$

The squared Euclidean norm is globally parabolic in the space formed by the union of the error and parameter spaces. Hence, the minimum of $E^{2}$ occurs when,

$0=(\partial E_{j}^{2}/\partial \beta _{j})$

$0=\sum _{i=1}^{N}\epsilon _{ij}(\partial \epsilon _{ij}/\partial \beta _{j})$

Since $y_{i}$ is an independent function, only the derivative of $z_{i}$ is needed. In general, $z_{i}$ in unknown, however if one assumes that the function is smooth then one can describe it with a Taylor series, i.e.,

$z_{ij}=z_{ij}|_{\beta =\beta _{kj}}+(\partial z_{ij}/\partial \beta _{j})|_{\beta =\beta _{kj}}(\beta _{j}-\beta _{kj})+...$

Here the function is defined in the neighborhood of a given set of parameters $\beta _{k}$. That neighborhood can be expressed as,

$\Delta \beta _{j}\equiv (\beta _{j}-\beta _{kj})$

$z_{ij}\approx z_{ij}|_{\beta =\beta _{kj}}+(\partial z_{ij}/\partial \beta _{j})|_{\beta =\beta _{kj}}\Delta \beta _{j}$

and so

$0\approx -2\sum _{i=1}^{N}[y_{i}-z_{ij}|_{\beta =\beta _{kj}}-\Delta \beta _{j}(\partial z_{ij}/\partial \beta _{j})|_{\beta =\beta _{kj}}](\partial z_{i}/\partial \beta _{t})$

The local error within this neighborhood can also be defined as,

$\Delta y_{i}\equiv y_{i}-z_{i}|_{\beta =\beta _{kj}}$

such that the partial derivative defines the Jacobian, i.e.,

$J_{ij}\equiv (\partial z_{i}/\partial \beta _{j})$

These definitions allow one to write,

$0=2\sum _{i=1}^{N}[\sum _{j=1}^{M}J_{it}J_{ij}\Delta \beta _{j}-J{it}\Delta y_{i}]$.

Using the notion of tensor products, the indices can be rearranged, i.e.,

$0=\sum _{i=1}^{N}[\sum _{j=1}^{M}J_{ti}J_{ij}\Delta \beta _{j}-J_{ti}\Delta y_{i}]$

and thus in matrix notation one can write,

$0=J^{T}J\Delta \mathbf {\beta } -J^{T}\Delta \mathbf {y}$

$\Delta \mathbf {\beta } =(J^{T}J)^{-1}J^{T}\Delta \mathbf {y}$

The calculation of $\Delta \mathbf {\beta }$ is that change to the vector $\mathbf {\beta _{k}}$ within its neighborhood for which $(\partial E_{j}^{2}/\partial \beta _{j})$ is nearer to zero. Therefore, if $\mathbf {\beta _{k}}$ is initially far from zero then the global minimum can only be reached after some number of iterations.