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- Graphene: The purpose is to give a clear and basic explanation of graphene's band structure and possibly other characteristics
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
[edit]
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,
![{\displaystyle \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}} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4a88a44a77e9dfc0c72571ae0def04cb0ca8ef)
![{\displaystyle \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}} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3aac1174570cb8018302ed832fbd377e5d9e6066)
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
[edit]
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.,
![{\displaystyle |\phi _{1}\rangle ={\frac {1}{\sqrt {N}}}\sum _{a}e^{i\mathbf {k} \cdot \mathbf {R} _{a}}|\mathbf {R} _{a}\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abd0d319932729bee3480b0659356a74d47995f6)
![{\displaystyle |\phi _{2}\rangle ={\frac {1}{\sqrt {N}}}\sum _{b}e^{i\mathbf {k} \cdot \mathbf {R} _{b}}|\mathbf {R} _{b}\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49a78d3d44303832bf0d708fbbcdc49718bdec3f)
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,
![{\displaystyle |\psi \rangle =\alpha |\phi _{1}\rangle +\beta |\phi _{2}\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b46d87b9e3b716ed1808ab640345c9acd293042)
These eigenstates must of course satisfy the Schrödinger equation, i.e.,
![{\displaystyle {\hat {H}}|\psi \rangle =E|\psi \rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/194cc6fdb038cc890c7b223c9bc6dc89aa516141)
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
![{\displaystyle \langle \phi _{2}|{\hat {H}}|\psi \rangle =E\langle \phi _{2}|\psi \rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68b58f4412b9acc47b15a1078102ce83767e6e90)
This implies that,
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f35d8e1f8104a887b764a1aff5f17a3bb15cae3c)
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
![{\displaystyle E_{p_{z}}=\langle \phi _{1}|{\hat {H}}|\phi _{1}\rangle =\langle \phi _{2}|{\hat {H}}|\phi _{2}\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ba471bc6e5d3e511260ef6e90994bac829374d)
The off diagonal elements are related due to the Hamiltonian being Hermitian, i.e.,
![{\displaystyle H_{12}=H_{21}^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca76416693bd73850707debf7aaf3c0e78f472c)
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
![{\displaystyle \mathbf {R} _{1}=\mathbf {0} +\Delta \mathbf {r} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dbfcb5cb53b903bc61e68630251801c73daaa14)
![{\displaystyle \mathbf {R} _{2}=\mathbf {r} _{1}+\Delta \mathbf {r} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e74997bebaf2a59a3cb976ea26cc318919bf03a)
![{\displaystyle \mathbf {R} _{3}=\mathbf {r} _{2}+\Delta \mathbf {r} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47d8e903fad7dc1fdfc1e31ce333a35069749f34)
The off-diagonal elements then take the form,
![{\displaystyle ={\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/112d9540200d56502516ffe11faf9e220660297c)
![{\displaystyle ={\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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a26a8d5bd81097ac2c547dbfdeacc0c3951d27)
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
![{\displaystyle 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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45bbf36d09334fbdb0f5055f83035e0fff9f0a1e)
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
[edit]
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
.
![{\displaystyle {\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe5a066b909ef6eb56719e493067901ce97709b)
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
[edit]
- ^ Jorio, A., et al., "Raman Spectroscopy in Graphene Related Systems", Wiley, Germany (2011)
- ^ Delhaès, P., "Graphite and precursors", Gordon and Breach Science Publishers (2001), p.27-31
- ^ Turro, N.J., et al., "Principles of molecular photochemistry: an introduction", University Science Books (2009), p.63-6
- ^ Wallace, P.R., "The Band Theory of Graphite", Phys. Rev. 71, 9, p.622-34
Two Dimensional Reciprocal Lattice
[edit]
![{\displaystyle \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}} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4a88a44a77e9dfc0c72571ae0def04cb0ca8ef)
![{\displaystyle \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}} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3aac1174570cb8018302ed832fbd377e5d9e6066)
Berry Phase Concepts
[edit]
![{\displaystyle G_{2}=\int _{Emin}^{Emax}G_{1}(E'-E_{i})G_{2}(E-E')dE'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cdf3a4349e8cc6c8e2c26ed06fb4ff586666637)
![{\displaystyle I_{1}(E)={\frac {1}{2\pi }}\int _{\pm \infty }I_{o}(t)e^{i{\frac {E}{\hbar }}t}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458f6363e6113c76e450573a1705d390ffd27bd0)
![{\displaystyle I_{o}(t)=e^{-{\frac {t}{\tau }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd732123e2c316698589519904b17be11ac3254)
![{\displaystyle I_{o}(t)=e^{-{\frac {\Delta E_{L}t}{\hbar }}}{\frac {e^{-i\epsilon _{o}t}}{(iDt)^{\alpha }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc28cd3de6fb3cf848d3acd6f762d2388a0929e0)
Method of Least Squares
[edit]
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,
![{\displaystyle \Delta \beta _{j}\equiv (\beta _{j}-\beta _{kj})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5667bb862915db133eab07196ad13b071d532f2)
and so
The local error within this neighborhood can also be defined as,
![{\displaystyle \Delta y_{i}\equiv y_{i}-z_{i}|_{\beta =\beta _{kj}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/091ec7344013c029b1dc4a2547e2330b548ded8b)
such that the partial derivative defines the Jacobian, i.e.,
![{\displaystyle J_{ij}\equiv (\partial z_{i}/\partial \beta _{j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75e780f089667eda62e5b574f435e4cad3e61d49)
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.