Welcome

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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

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

Graphene

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Carbon has four valence electrons. From Valence bond theory three out of the four half filled valent electron orbitals   hybridize to create carbon-carbon   bonds which form the graphene crystal. The fourth   orbital overlaps with neighboring   orbitals creating out of plane   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   bonds, bonding and antibonding states are refered to as   and   states, respectively. The bonding state has a spatially symmetric wave function with a lower energy than the antibonding one. The   bonds account for graphene's structural properties while the   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   electrons .

The Graphene Lattice

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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   be the center of the Brillouin zone and   be the center of the   Brillouin zone, i.e. the position of the respective lattice points in reciprocal space. The reciprocal lattice can be described by   such that,

 
 

The corners of the Brillouin zone denoted by   are thus,

 ,
 ,
 ,
 ,
 ,
 .

Notice that there are two distinct corners or "K points",   and  , 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   and   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   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   bonds[1][2]. For this reason, the   states are the easiest to access by experimental probing.

The Hamiltonian of Graphene using the Tight-Binding Model

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To understand the basic electronic behavior of graphene it is necessary to describe the behavior of its   electrons. Using the tight-binding model for their description assumes that each   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  .

Each atom in the lattice is located at sublattice points "A" and "B" corresponding to vectors   and   with   such that   and  . If now   is the normalized   atomic orbital wave function of an isolated carbon atom then let   be the corresponding carbon atom orbital wave function positioned at lattice point  . Since there are two atoms in each Wigner–Seitz cell (at site A and B), one can expect the   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.,

 
 

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

 

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

 

where   is the graphene Hamiltonian and   is the energy of the   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,

  and
 

This implies that,

 

where   and   correspond to the elements of   and  , respectively. While there are two distinct lattice sites each carbon atom is identical to its neighbor. With this in mind, the energy of a   electron of an isolated carbon atom is then

 

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

 

These elements correspond to the energy needed for a   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,  . Next, consider the three B site nearest neighbors to this A site, namely,   &   where

 
 
 

The off-diagonal elements then take the form,

 
 
 

where  . The quantity   eV is the "hopping integral" which represents the kinetic energy of electrons hopping between atoms. The value of   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  , 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[3]. In this case,   (i.e., a basis wave function resembles itself 100%) and

 

Here, the quantity   is also experimentally determined. For the purpose of this article   which simplifies the Schrödinger equation to yield the secular equation

 

 

 

 

substituting for   and   gives,

 

 

 

which implies that

 

(see Wallace, (1947)for comparison).[4]

It is common to use this energy corresponding to   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   points of the Brillouin zone. For this reason, graphene is a zero band gap semiconductor.

The Low-Energy Limit

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Low energy excitations of   electrons into the conducting   state are more likely to occur near the   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   point, i.e., some  .

 

where the Fermi velocity is  . A massless Dirac fermion refers to the linearly increasing energy state, i.e., from the form of the Hamiltonian above  .

Moving adiabatically in k-space around this "k-point", modifies the wave function by a Berry phase, i.e.,   per revolution.

Phonons in Graphene

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References

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  1. ^ Jorio, A., et al., "Raman Spectroscopy in Graphene Related Systems", Wiley, Germany (2011)
  2. ^ Delhaès, P., "Graphite and precursors", Gordon and Breach Science Publishers (2001), p.27-31
  3. ^ Turro, N.J., et al., "Principles of molecular photochemistry: an introduction", University Science Books (2009), p.63-6
  4. ^ Wallace, P.R., "The Band Theory of Graphite", Phys. Rev. 71, 9, p.622-34

Misc

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Two Dimensional Reciprocal Lattice

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Berry Phase Concepts

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Photoelectrons from a Metal

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Photoelectrons in a Metal

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Method of Least Squares

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Consider the data set   such that    and  {1,2,3,... N}. If then   is a function which best describes   over the domain   then there exists an element   such that    and  {1,2,3... M} when there M parameters  . The error between the description of   and itself can then be defined, i.e., let

 

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   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

 

Root Mean Square Error

 


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

 


 


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

 

Here the function is defined in the neighborhood of a given set of parameters  . That neighborhood can be expressed as,

 


 

and so

 

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

 

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

 


These definitions allow one to write,

 .

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

 


and thus in matrix notation one can write,

 


 


The calculation of   is that change to the vector   within its neighborhood for which   is nearer to zero. Therefore, if   is initially far from zero then the global minimum can only be reached after some number of iterations.