Homework Set 3: Difference between revisions

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==Problem 1: Shubnikov-de Haas oscillations in 2DEG in high magnetic field==
==Problem 1: Energy-momentum dispersion in graphene==
The energy momentum-dispersion of graphene valence and conduction bands is given by the formula derived in class:


[[Image:sdh.png||thumb|left|400px|Shubnikov-de Haas oscillations in the longitudinal voltage.]]
<math> E(k_x,k_y) = \pm t\sqrt{1 + 4 \cos \left( \frac{\sqrt{3}}{2} k_x a \right) \cos \left( \frac{k_y a}{2} \right) + 4 \cos^2 \left( \frac{k_y a}{2} \right )} </math>


Physical properties of a 2DEG can be measured by embedding it into a multiterminal Hall bridge, as shown in the Figure above. The energy spectrum of 2D electrons in magnetic field <math> \mathbf{B}=B\hat{z} </math> orthogonal to such bridge consists of discrete Landau levels:
which assumes tight-binding Hamiltonian projected to a basis set of single <math> \pi </math> orbital per carbon atom and hopping allowed only between the nearest neighbor carbon atoms. The honeycomb lattice spacing is <math> a </math>.


<math> E_n=\hbar \omega \left(n + \frac{1}{2} \right), \ \omega=eB/m^* </math>,
:'''(a)''' Using MATLAB or Mathematica plot this function as a surface in 3D k-space within the range of <math> (k_x,k_y) </math> vector values defined by the first Brillouin zone.


which are well-separated for large enough B. Thus, the step-like density of states associated with 2DEG <math> g(E) = \frac{m^*}{\pi \hbar^2} \theta(E-E_b) </math> (<math> E_b </math> is the lowest level in the triangular potential that confines 2DEG) is then broken up into a series of peaks spaced by <math> \hbar \omega </math>:
:'''(b)''' Near one of the two inequivalent K points [the so-called Dirac points at which <math> E(k_x,k_y) \equiv 0 </math>] in the corners of the  first Brillouin zone, where the conduction (top surface) and valence (bottom surface) bands touch, the dispersion has a conical shape characteristics of massless relativistic particles (such as photons or neutrinos). Up to what energy away from the Dirac point <math> E=0 </math> (which is the Fermi energy of ideal undoped graphene) can the real dispersion shown in your figure be approximated by linear dispersion of the "Dirac cones"? You can subtract <math> E(k_x,k_y) </math> surface around one of the K points from the conical surface <math> E(q_x, q_y) = \pm v_F \hbar \sqrt{q_x^2+q_y^2} </math> (corrections are <math> O[(q/K)]^2</math>; here <math> \mathbf{q} = \mathbf{K} + \mathbf{k} </math> and the Fermi velocity is <math> v_F = \frac{\sqrt{3}}{2} t a/\hbar \approx 10^6 m/s </math>) and find at which energy <math> \pm E_0 </math> their difference starts to be more than 5%.


<math> g(E,B) = \frac{2eB}{h} \sum_{n=0}^{\infty} \delta \left[ E- E_b - \hbar \omega \left(n + \frac{1}{2} \right)\right] </math>. (1)
==Problem 2: Density of state of graphene==


Although the spikes are ideally delta functions, scattering processes spread them out in energy. As one changes the magnetic field B, longitudinal voltage (from which one computes the longitudinal resisitivity <math> \rho_{xx}=V_x W/ I L </math>) in the Figure on the left goes through one cycle of oscillations as the Fermi energy moves from the center of one Landau level to the center of the next Landau level. These oscillations are refered to as ''Shubnikov-de Haas (SdH)'' oscillations.
The honeycomb lattice of graphene is topologically equivalent to a brick lattice shown in the Figure below:


[[Image:Graphene_brick.png||thumb|center|400px|Honeycomb lattice of graphene (top left), its equivalent brick lattice (top right) and submatrices of the matrix representation of the tight-binding Hamiltonian defined on either of these two lattices.]]


'''(a)''' Devise a procedure that will allow you to extract electron density <math> n </math> within 2DEG from the values of magnetic field <math> B </math> at which <math> V_x </math> (or the corresponding <math> \rho_{xx} </math>)  reaches maximum in the course of SdH oscillations.


Thus, the matrix representation of the Hamiltonian defined on this lattice can be obtained in the same fashion as for the square lattice (used in Problem 4. of Homework Set 2), i.e., by loading its submatrices <math> \mathbf{H}_0 </math> describing graphene supercells coupled to adjacent supercells via  the hopping submatrices <math> \mathbf{H}_1 </math>. The carbon atoms within the supercells as well as the structure of these submatrices are shown in the Figure. Here it is assumed that electron can hop between only the nearest neighbor carbon atoms with probability <math> t </math>.


'''(b)''' What is <math> n </math> for the 2DEG in the Figure above?
:'''(a)''' Using the honeycomb (or equivalent brick) lattice of <math> 59 \times 30 </math> carbon atoms (for example, the honeycomb lattice in the top inset of the Figure above is <math> 5 \times 4 </math> in this terminology), construct the full Hamiltonian matrix <math> \mathbf{H} </math> of finite size graphene sheet and compute its density of states (DOS) via the Green function method used in Homework Set 2 in the energy range <math> [-3.5t, 3.5t]</math>.


:'''(b)''' Introduce periodic boundary into <math> \mathbf{H} </math> (this requires to input two additional submatrices <math> \mathbf{H}_1 </math> into <math> \mathbf{H} </math>, as well as to introduce extra hoppings <math> t </math> into <math> \mathbf{H}_0 </math>) and compute its DOS via the same method as in (a).


'''HINT:''' From Eq. (1) we see that the number of occupied Landau levels is <math> n/(2eB/h) </math>, if electron density <math> n </math> is already known.
:'''(c)''' Does any of your numerically obtained DOS for finite-size graphene sheet resembles analytical DOS shown in Fig. 5 (page 114) of A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, ''The electronic properties of graphene'', Rev. Mod. Phys. '''81''', 109 (2009). [http://link.aps.org/doi/10.1103/RevModPhys.81.109 [PDF]]?


==Problem 2: Sharvin formula for classical point contact conductance==
'''HINT:''' Test if your Hamiltonian is properly set up with the MATLAB function '''visual_graphene_H.m''' posted on the [[Computing]] page of PHYS 824 wiki. The function  will plot real space position of the carbon atoms and the hoppings between them, based on the matrix content and lattice size you supply as an input.


[[Image:cpc.png||thumb|left|400px|Model of a classical point contact between two massive metallic electrodes.]]
==Problem 3: Massive Dirac particles in graphene==


The Sharvin formula for the electrical conductance of an extremely short contact area <math> A </math> between two pieces of metal is given by
Consider a tight-binding model on a honeycomb lattice with on-site potential1 different on the sublattices A and B, <math> V_A = - V_B = \lambda </math>.


<math> G_\mathrm{Sharvin}=\frac{2e^2}{h}\frac{k_F^2 A}{4\pi} </math>
: '''(a)''' Develop a general solution of this problem in terms of plane wave states, expressing the spectrum of excitations in terms of the nearest neighbor hopping amplitude <math> t </math> and the asymmetry parameter <math> \Delta  = \frac{1}{2} \left(V_A - V_B \right) </math>.


where <math> k_F </math> is the Fermi wavelength. Derive the Sharvin formula by considering the total current flowing through a hole of area <math> A </math> in a thin insulating barrier separating two free electron gases with different Fermi energies, as shown in the Figure above. The gas on the left has Fermi energy <math> E_F + eV </math>, while the gas on the right has Fermi energy <math> \varepsilon_F^0 </math>, where <math> V </math> is the applied voltage bias.
: '''(b)''' Linearize the solution found in part a) near the points <math> K </math> and <math> K^\prime </math> treating the asymmetry  as a small perturbation, and find how the massless Dirac picture (valid for <math> V_A = V_B = 0 </math>) of low energy states of graphene  is altered. That is, show that <math> \Delta </math> generates a finite mass of Dirac quasiparticles whose Hamiltonian is given by:


Use purely (semi)classical arguments:


'''(1)''' The total current through the contact is
<math> \hat{H} = v_F \hat{\mathbf{\sigma}} \cdot \hat{\mathbf{p}}  + \lambda \hat{\sigma}_z </math>,


<math> I = j_z A </math>


where current density along the z-axis is (carried by electrons within the bias window <math> E_F </math> to <math> E_F+eV </math>):
where <math> \lambda = V_A = - V_B </math>.


<math> j_z = e\int_{E_F}^{E_F+eV} dE\, \int d\theta v_z \frac{\partial^2 n}{\partial E \partial \theta} </math>.
==Problem 4: Bilayer graphene==


The graphene bilayers are expected to play an important role in nanoelectronic device fabrication since one can manipulate their electronic structure and band gaps (see, e.g., [http://www.nature.com/nmat/journal/v7/n2/abs/nmat2082.html Nature Materials 7, 151 (2007)]] Suppose that two graphene layers are stacked atop one another according to the "Bernal stacking" (the stacking fashion of graphite). This means that the B sublattice sites of the upper layer (layer 1) are directly above the A sublattice sites of the lower layer (layer 2). The A sublattice sites of the upper layer and the B sublattice sites of the lower layer have no "partner" atoms below/above them, as shown in the Figure below.


'''(2)''' In a free electron gas, the number of electrons with energies between <math> E </math> and <math> E+dE </math> traveling at an angle between <math>\theta</math> and <math>\theta + d\theta</math> with respect to a given axis is
[[Image:bilayer.jpg||thumb|center|400px|Lattice of bilayer graphene.]]


<math> \frac{\partial^2 n}{\partial E \partial \theta} dE d\theta = \frac{g(E)}{2} \sin \theta dE d\theta </math>,
:'''(a)''' Formulate and solve the tight-binding model for this system consisting of the usual hopping <math> t </math> between nearest-neighbor sites in each layer, and an additional smaller hopping <math> t_b </math>  between the B1-A2 sites in different layers. Choose the zero of energy to equal that  of an isolated atom. You should find 4 bands.


where <math> g(E) </math> is the density of states in three dimensions.
:'''(b)''' Two of the bands found above touch at the Fermi energy <math> E_F= 0 </math> at the Brillouin zone boundary points ±K. Find the effective mass around these points.


== Problem 3: Finite temperature effects on conductance quantization in quantum point contacts==
:'''(c)''' Calculate the Berry phase acquired by an electron encircling one of the zone boundary points in each of the two touching bands. Use your result to argue that the bands cannot split if a small perturbation is applied provided inversion and time-reversal symmetry are preserved. Where is the inversion center for the bilayer? REFERENCE: [http://link.aip.org/link/?LTPHEG/34/794/1 The Berry phase in graphene and graphite multilayers].


[[Image:qpc_exp.png||thumb|left|400px|Exerimental discovery of conductance quantization in quantum point contacts within 2DEG and temperature effects.]]
:'''(d)''' ''Check this conclusion in a simple case:'' Because the B1 and A2 sites have more atoms close by than the B2 and A1 sites, there will generally be some difference of the site energies for these orbitals. Add an energy <math> +U_0 </math> to the electrons on the B1, A2 sites and <math> -U_0 </math> to the electrons on the B2, A1 sites. Show that for small enough <math> U_0 </math>, the two bands still touch.


When the size of the point contact from Problem 2 becomes comparable to the Fermi wavelength <math> \lambda_F = 2\pi/k_F </math>, the contact enters the quantum regime where its conductance becomes quantized, as explained by the Landauer formula:
'''REFERENCE:''' B. Partoens and F. M. Peeters, ''From graphene to graphite: Electronic structure around the K point'', Phys. Rev. B '''74''', 075404 (2006). [http://link.aps.org/doi/10.1103/PhysRevB.74.075404 [PDF]]


<math> G(E_F) = \frac{2e^2}{h} \sum_{n=1}^{N_{\rm open}} \int\limits_{-\infty}^{+\infty} dE\, \left(-\frac{\partial f }{\partial E} \right) T_n(E) </math>
== Problem 5: Chiral dynamics and Klein tunneling  of electrons in graphene==


and observed experimentally in the Figure on the left. Here <math> N_\mathrm{open} </math> is the number of open "conducting channels", whose transmission is  perfect <math> T_n(E) = \theta (E-\varepsilon_n) </math> [<math> \theta(x) </math> is the step function] since in ballistic wires a channel is either open (when the Fermi energy is above the bottom of the corresponding subband) or closed (otherwise). One can also view the Sharvin formula from Problem 3 as a limiting case of such Landauer formula where the number of channels <math> N_{\mathrm{classical}} = \frac{k_F^2 A}{4 \pi} </math> (in three dimensions) is very large, so that <math> N_{\mathrm{classical}}</math> is a continuous function of <math> k_F </math>  and temperature is set to zero, <math> \left(-\frac{\partial f }{\partial E} \right) = \delta(E-E_F) </math>.
Consider motion of massless Dirac fermions in an external potential, as described by the Hamiltonian:


Suppose that a quantum point contact (QPC) can be modeled by a 2D strip with hard-wall boundary conditions in the direction transverse to the current flow. The width of the strip is <math> a=100 </math> nm and the effective mass of electrons is <math> m^* = 0.067m_0 </math> of the bare electron mass <math> m_0 </math> (which is the case for GaAs heterostructures).
<math> \hat{H}_\mathrm{Weyl} = v_F \hat{\mathbf{\sigma}} \cdot \mathbf{p} + U(x) </math>


a) Plot the conductance of the quantum point contact as function of the Fermi energy for a number of temperatures <math> T < E_F </math>. For the plot <math> G(E_F) </math> you can select an interval of Fermi energies encompassing up to 5 subband bottoms <math> \varepsilon_6 > E_F \ge 0 </math>  determined by the hard-wall boundary conditions as <math> \varepsilon_n = \frac{\hbar^2 n^2 \pi^2}{ 2m^*a^2} </math>. Express all relevant energies in your plot (<math> E_F </math> along the x-axis, <math> k_B T </math> as parameter, and <math> \varepsilon_n </math> as discrete energies along the x-axis at conductance jumps at zero temperature in steps of <math> 2e^2/h </math>) in units of eV.
:'''(a)''' For a rectangular potential barrier, <math> U(x)=V_0, \, 0<x<D </math> and <math> U(x)=0</math> otherwise, find transmission as a function of the incidence angle.


b) Using your plot or analytical considerations, determine characteristic temperature(s) at which features of the quantization plateau begin to disappear. In the latter case, for simplicity, you can consider only the first two subbands (i.e., conducting channels), <math> n=1,2 </math> to  obtain explicit analytical expression for <math> G(E_F,T) </math>. How does your conclusion compare to experimental observation, shown in the Figure on the left, of washing out of quantized conductance steps with increasing temperature?
:'''(b)''' For a step-like potential <math> U(x) = U_0 \mathrm{sign}(x) </math> find transmission as a function of the incidence
angle. For simplicity, consider the case of particle at zero Fermi energy (zero doping).


== Problem 4: Magnetotransport in two quantum point contacts in series==
:'''(c)''' What happens at normal incidence <math> k_y = 0 </math> in both (a) and (b)?


[[Image:two_qpcs.png||thumb|left|200px|Two narrow point contacts in series connecting terminals 1 and 3 (modeled as semi-infinite leads).]]
'''REFERENCES'''


In semiclassical electronic transport theory, the conductance of two QPCs in series, illustrated in Figure on the left, is half of that of a single quantum point, so that according to Problem 3  <math> G_\mathrm{2QPC} = 0.5 \times 2e^2/h \times N_\mathrm{open} </math> assuming zero temperature. However, if two QPCs are sufficiently close together than the electrons emerging from one QPC do not have the chance to spread out before they reach the second QPC. Consequently, the conductance is the same as that of a single QPC, <math> G_\mathrm{2QPC} =  2e^2/h \times N_\mathrm{open} </math>.  
===Theory===
*M. I. Katsnelson, K. S. Novoselov,  and A. K. Geim, ''Chiral tunnelling and the Klein paradox in graphene'', Nature Physics '''2''', 620 (2006). [http://www.condmat.physics.manchester.ac.uk/pdf/mesoscopic/publications/graphene/Naturephys_2006_Klein.pdf [PDF]]


But if we turn on an external magnetic field, then the electrons get deflected and conductance is reduced, see [https://doi.org/10.1103/PhysRevB.39.10445 Phys. Rev. B '''39''', 10445(R) (1989)] and [https://doi.org/10.1103/PhysRevB.41.8461 Phys. Rev. B '''41''', 8461 (1990)]. Using the Büttiker formula for multiterminal phase-coherent nanostructures, show that conductance is given by:
===Experiment===
 
* A. F. Young and P. Kim, ''Quantum interference and Klein tunnelling in graphene heterojunctions'', Nature Phys. '''5''', 222 (2009). [http://www.nature.com/nphys/journal/v5/n3/abs/nphys1198.html [PDF]]
<math> G = \frac{e^2}{h} \left[ N_\mathrm{open} + T_F + \frac{(T_R - T_L)^2}{2T_F^'+T_R+T_L} \right] </math>
 
where the transmission functions are
 
<math> T_F = T_{13} = T_{31} </math>,
 
<math> T_F^' = T_{42} = T_{24} </math>,
 
<math> T_R = T_{21} = T_{32} = T_{43} = T_{14}</math>,
 
<math> T_L = T_{41} = T_{12} = T_{23} = T_{34} </math>,
 
where <math> T_{pq} \equiv T_{p \leftarrow q} = \mathrm{Tr}\, [\mathbf{t_{pq} t_{pq}^\dagger}] </math> is the transmission functions from lead <math> q </math> to lead <math> p </math>.

Latest revision as of 14:35, 18 October 2018

Problem 1: Energy-momentum dispersion in graphene

The energy momentum-dispersion of graphene valence and conduction bands is given by the formula derived in class:

which assumes tight-binding Hamiltonian projected to a basis set of single orbital per carbon atom and hopping allowed only between the nearest neighbor carbon atoms. The honeycomb lattice spacing is .

(a) Using MATLAB or Mathematica plot this function as a surface in 3D k-space within the range of vector values defined by the first Brillouin zone.
(b) Near one of the two inequivalent K points [the so-called Dirac points at which ] in the corners of the first Brillouin zone, where the conduction (top surface) and valence (bottom surface) bands touch, the dispersion has a conical shape characteristics of massless relativistic particles (such as photons or neutrinos). Up to what energy away from the Dirac point (which is the Fermi energy of ideal undoped graphene) can the real dispersion shown in your figure be approximated by linear dispersion of the "Dirac cones"? You can subtract surface around one of the K points from the conical surface (corrections are ; here and the Fermi velocity is ) and find at which energy their difference starts to be more than 5%.

Problem 2: Density of state of graphene

The honeycomb lattice of graphene is topologically equivalent to a brick lattice shown in the Figure below:

Honeycomb lattice of graphene (top left), its equivalent brick lattice (top right) and submatrices of the matrix representation of the tight-binding Hamiltonian defined on either of these two lattices.


Thus, the matrix representation of the Hamiltonian defined on this lattice can be obtained in the same fashion as for the square lattice (used in Problem 4. of Homework Set 2), i.e., by loading its submatrices describing graphene supercells coupled to adjacent supercells via the hopping submatrices . The carbon atoms within the supercells as well as the structure of these submatrices are shown in the Figure. Here it is assumed that electron can hop between only the nearest neighbor carbon atoms with probability .

(a) Using the honeycomb (or equivalent brick) lattice of carbon atoms (for example, the honeycomb lattice in the top inset of the Figure above is in this terminology), construct the full Hamiltonian matrix of finite size graphene sheet and compute its density of states (DOS) via the Green function method used in Homework Set 2 in the energy range .
(b) Introduce periodic boundary into (this requires to input two additional submatrices into , as well as to introduce extra hoppings into ) and compute its DOS via the same method as in (a).
(c) Does any of your numerically obtained DOS for finite-size graphene sheet resembles analytical DOS shown in Fig. 5 (page 114) of A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009). [PDF]?

HINT: Test if your Hamiltonian is properly set up with the MATLAB function visual_graphene_H.m posted on the Computing page of PHYS 824 wiki. The function will plot real space position of the carbon atoms and the hoppings between them, based on the matrix content and lattice size you supply as an input.

Problem 3: Massive Dirac particles in graphene

Consider a tight-binding model on a honeycomb lattice with on-site potential1 different on the sublattices A and B, .

(a) Develop a general solution of this problem in terms of plane wave states, expressing the spectrum of excitations in terms of the nearest neighbor hopping amplitude and the asymmetry parameter .
(b) Linearize the solution found in part a) near the points and treating the asymmetry as a small perturbation, and find how the massless Dirac picture (valid for ) of low energy states of graphene is altered. That is, show that generates a finite mass of Dirac quasiparticles whose Hamiltonian is given by:


,


where .

Problem 4: Bilayer graphene

The graphene bilayers are expected to play an important role in nanoelectronic device fabrication since one can manipulate their electronic structure and band gaps (see, e.g., Nature Materials 7, 151 (2007)] Suppose that two graphene layers are stacked atop one another according to the "Bernal stacking" (the stacking fashion of graphite). This means that the B sublattice sites of the upper layer (layer 1) are directly above the A sublattice sites of the lower layer (layer 2). The A sublattice sites of the upper layer and the B sublattice sites of the lower layer have no "partner" atoms below/above them, as shown in the Figure below.

Lattice of bilayer graphene.
(a) Formulate and solve the tight-binding model for this system consisting of the usual hopping between nearest-neighbor sites in each layer, and an additional smaller hopping Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_b } between the B1-A2 sites in different layers. Choose the zero of energy to equal that of an isolated atom. You should find 4 bands.
(b) Two of the bands found above touch at the Fermi energy Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E_F= 0 } at the Brillouin zone boundary points ±K. Find the effective mass around these points.
(c) Calculate the Berry phase acquired by an electron encircling one of the zone boundary points in each of the two touching bands. Use your result to argue that the bands cannot split if a small perturbation is applied provided inversion and time-reversal symmetry are preserved. Where is the inversion center for the bilayer? REFERENCE: The Berry phase in graphene and graphite multilayers.
(d) Check this conclusion in a simple case: Because the B1 and A2 sites have more atoms close by than the B2 and A1 sites, there will generally be some difference of the site energies for these orbitals. Add an energy Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle +U_0 } to the electrons on the B1, A2 sites and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -U_0 } to the electrons on the B2, A1 sites. Show that for small enough Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_0 } , the two bands still touch.

REFERENCE: B. Partoens and F. M. Peeters, From graphene to graphite: Electronic structure around the K point, Phys. Rev. B 74, 075404 (2006). [PDF]

Problem 5: Chiral dynamics and Klein tunneling of electrons in graphene

Consider motion of massless Dirac fermions in an external potential, as described by the Hamiltonian:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{H}_\mathrm{Weyl} = v_F \hat{\mathbf{\sigma}} \cdot \mathbf{p} + U(x) }

(a) For a rectangular potential barrier, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U(x)=V_0, \, 0<x<D } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U(x)=0} otherwise, find transmission as a function of the incidence angle.
(b) For a step-like potential Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U(x) = U_0 \mathrm{sign}(x) } find transmission as a function of the incidence

angle. For simplicity, consider the case of particle at zero Fermi energy (zero doping).

(c) What happens at normal incidence Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_y = 0 } in both (a) and (b)?

REFERENCES

Theory

  • M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Chiral tunnelling and the Klein paradox in graphene, Nature Physics 2, 620 (2006). [PDF]

Experiment

  • A. F. Young and P. Kim, Quantum interference and Klein tunnelling in graphene heterojunctions, Nature Phys. 5, 222 (2009). [PDF]
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