Diffusive Conductivity In Magnetically Modulated Gapped Graphene.

1Heading OneHeading One.OneHeading One.TwoHeading One.Two.OneHeading One.Two.One.OneElectrical Transport in Monolayer Gapped GrapheneBand Structure of Single Layer GrapheneMonolayer graphene is described by a honeycomb arrangement of carbon atoms. A sketch of a lattice section is shown in Fig.2.1 (a), where the circles indicate the position of the carbon nuclei. The primitive cell of this system is identified to be a rhombus containing the basis of two carbon atoms (here denoted by A1 and B1) and it is spanned by the vectors a1 and a2. Hence, the hexagonal lattice can be seen as the result of the interpenetration of two sublattices A1 and B1.

Each one of the C atoms is sp2 hybridized with its three neighbors via strong covalent bonds. The corresponding ?-bands (bonding and anti-bonding) are split to high absolute energy values and can therefore be neglected, when investigating the low energy band structure. The remaining valence electron in the pz orbital leads to the formation of another pair of bands, the so-called ?-bands. Carrying out a tight-binding calculation to obtain the dispersion relation of the graphene lattice as a function of the wave vector q, yield ADDIN EN.CITE <EndNote><Cite><Author>Wallace</Author><Year>1947</Year><RecNum>1</RecNum><DisplayText>1</DisplayText><record><rec-number>1</rec-number><foreign-keys><key app=”EN” db-id=”frv9s9rxnfrtxxe2vdkxdxrhpex2pzvfpsrw” timestamp=”1520854959″>1</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Wallace, Philip Richard</author></authors></contributors><titles><title>The band theory of graphite</title><secondary-title>Physical Review</secondary-title></titles><periodical><full-title>Physical Review</full-title></periodical><pages>622</pages><volume>71</volume><number>9</number><dates><year>1947</year></dates><urls></urls></record></Cite></EndNote>1.

?±(q)= ±?01 +4cos2qxa2+ 4cosqxa2cos3qya2 (2. SEQ Eq * MERGEFORMAT 1)

Here, only the nearest neighbor was included, assuming the hopping parameter ?0 between adjacent sites (see Fig. 2.1 (a)). Further, the lattice constant a = 2.46 ?A enters this equation. The plus and minus sign describe the ??-band and the ?-band, respectively. In Fig. 2.1(b), the resulting band structure is plotted for the first Brillouin zone (BZ). Whereas the bands exhibit a large energy separation of 20 eV in the center of the BZ, points of intersection at s = 0 are obtained at the six corners of the hexagonal reciprocal unit cell. For the Fermi surface to be located at these special points, two valence electrons per primitive cell are necessary to fill the valence band completely. The points denoted by K and K’ are the basis of the reciprocal primitive cell and constitute the two inequivalent valleys, reflecting the presence of two sublattices Q and B.

Figure STYLEREF 1 s 2. SEQ Figure * ARABIC s 1 1: (a) Sketch of single layer lattice with sublattice atoms for A1 and B1. The unit cell with the corresponding vectors is indicated in red. Hopping between sites requires the hopping energy ?0. (Adapted from Ref. HYPERLINK l “_bookmark150” 15) (b) Band structure according to Eq. (2.1) for the first Brillouin zone. (c) Zoom into (b) at the K point where the linear dispersion occurs. Blue and red arrows indicate the direction of motion for electrons and holes, respectively. Grey and black double arrows show the direction of the pseudo spin for the two parts of the spectrum. The energy axis is normalized to the interlayer hopping energy ?1 to enable a direct comparison with Fig.2.2 (b) and (c).

The experimentally reachable range is restricted to low energies and therefore, we focus on the dispersion around the K-points in the following. We replace the wave vector q by k = Kq, describing the wave vector k with respect to the next K point. Expanding Eq. (2.1) around the corners of the Brillouin zone results in the linear dispersion relation for small k

?±k= ±?3?0a2?k= ±?vk(2. SEQ Eq * MERGEFORMAT 2)

with the Fermi velocity vF in graphene being determined by the intralayer hop- ping energy ?0. The corresponding energy eigenvalues are plotted Fig.2.1(c) and reveal the gapless dispersion of graphene as well as the electron-hole symmetry. One implication of the linear dispersion relation is the fact that the density of states in4886960165735F

00F

creases linearly with increasing?, namely D(?) = 2| ? |/(? ? 2vF2 ) and has zero states at ? = 0.

Besides the linear dispersion, another peculiarity of graphene is comprised in the description of its quasiparticles. Equation (2.1) gives the energy eigenvalues of the Dirac equation for massless fermions in two dimensions, which reads

?vFk??= ? ?(2. SEQ Eq * MERGEFORMAT 3)

with the Pauli matrices ? = (?x, ?y) around K and ?? = (?x, ??y) around K’ and the speed of light c replaced by the Fermi velocity vF. The two component state vector ? takes the role of the spin in neutrino physics and is therefore called pseudo spin. The eigenvector is given ADDIN EN.CITE <EndNote><Cite><Author>Haldane</Author><Year>1988</Year><RecNum>9</RecNum><DisplayText>2</DisplayText><record><rec-number>9</rec-number><foreign-keys><key app=”EN” db-id=”frv9s9rxnfrtxxe2vdkxdxrhpex2pzvfpsrw” timestamp=”1520856948″>9</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Haldane, F Duncan M</author></authors></contributors><titles><title>Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the" parity anomaly"</title><secondary-title>Physical Review Letters</secondary-title></titles><periodical><full-title>Physical Review Letters</full-title></periodical><pages>2015</pages><volume>61</volume><number>18</number><dates><year>1988</year></dates><urls></urls></record></Cite></EndNote>2.

?±,K(k)= 12e- i?2±ei?2(2. SEQ Eq * MERGEFORMAT 4)

It denotes the relative wave function amplitude on the two sublattices and includes the angle in momentum space, ? = arctan (kx/ky). Therefore, the direction of the pseudo spin is coupled to the direction of k and chirality can be attributed to the quasiparticles. As indicated in Fig.2.1(c), for electrons, the pseudo spin points in the same direction as the momentum, whereas for holes they are aligned anti-parallel. According to the formal definition of the chirality being the projection of the pseudo spin onto the direction of motion, electron-like states have positive chirality whereas holes exhibit negative chirality.

This property has an important implication for transport. Since the direction of the pseudo-spin is maintained in scattering processes in the absence of a short range scattering potential, the only allowed transitions are from e.g. a right-moving electron to a right-moving electron or to a left-moving hole (see Fig.2.1(c)). The latter process is known from neutrino physic and is called Klein tunneling and poses a challenge for the confinement of charge carriers in graphene ADDIN EN.CITE <EndNote><Cite><Author>Young</Author><Year>2009</Year><RecNum>3</RecNum><DisplayText>3</DisplayText><record><rec-number>3</rec-number><foreign-keys><key app=”EN” db-id=”frv9s9rxnfrtxxe2vdkxdxrhpex2pzvfpsrw” timestamp=”1520855904″>3</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Young, Andrea F</author><author>Kim, Philip</author></authors></contributors><titles><title>Quantum interference and Klein tunnelling in graphene heterojunctions</title><secondary-title>Nature Physics</secondary-title></titles><periodical><full-title>Nature Physics</full-title></periodical><pages>222</pages><volume>5</volume><number>3</number><dates><year>2009</year></dates><isbn>1745-2481</isbn><urls></urls></record></Cite></EndNote>3. Nevertheless, the suppression of back scattering due to the fact that e.g. a right-moving electron cannot be scattered into a left-moving electron, is expected to allow for charge carrier mobilities of up to 200000 cm2/Vs even at room temperature ADDIN EN.CITE <EndNote><Cite><Author>Chen</Author><Year>2008</Year><RecNum>2</RecNum><DisplayText>4</DisplayText><record><rec-number>2</rec-number><foreign-keys><key app=”EN” db-id=”frv9s9rxnfrtxxe2vdkxdxrhpex2pzvfpsrw” timestamp=”1520855815″>2</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Chen, Jian-Hao</author><author>Jang, Chaun</author><author>Xiao, Shudong</author><author>Ishigami, Masa</author><author>Fuhrer, Michael S</author></authors></contributors><titles><title>Intrinsic and extrinsic performance limits of graphene devices on SiO 2</title><secondary-title>Nature nanotechnology</secondary-title></titles><periodical><full-title>Nature nanotechnology</full-title></periodical><pages>206</pages><volume>3</volume><number>4</number><dates><year>2008</year></dates><isbn>1748-3395</isbn><urls></urls></record></Cite></EndNote>4.

Graphene in a perpendicular magnetic field

The trajectory of a charged particle in a perpendicular magnetic field follows an orbit due to the Lorentz force acting on it. Since the wave character of electrons leads to interference effects along such orbits, only discrete energies exist for which constructive interference occurs. These are referred to as Landau levels and allow for the observation of the quantum Hall effect ADDIN EN.CITE <EndNote><Cite><Author>Klitzing</Author><Year>1980</Year><RecNum>4</RecNum><DisplayText>5</DisplayText><record><rec-number>4</rec-number><foreign-keys><key app=”EN” db-id=”frv9s9rxnfrtxxe2vdkxdxrhpex2pzvfpsrw” timestamp=”1520856100″>4</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Klitzing, K v</author><author>Dorda, Gerhard</author><author>Pepper, Michael</author></authors></contributors><titles><title>New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance</title><secondary-title>Physical Review Letters</secondary-title></titles><periodical><full-title>Physical Review Letters</full-title></periodical><pages>494</pages><volume>45</volume><number>6</number><dates><year>1980</year></dates><urls></urls></record></Cite></EndNote>5.

The density of states in conventional semiconductor 2DEG condenses into a sequence of Landau levels at energies EN = ± k?c(N+1), where N is an integer numbering the Landau levels. Each filled Landau level contributes with ge /h to the

Hall conductivity, resulting in steps at ?xy = i ge2/h, where g is the LL-degeneracy and i an integer. In Fig.2.3 (a), the position of the LLs as well as the ladder for the Hall conductivity are shown as a ction of charge carrier density. The factor eB/h describes the Landau level degeneracy nL of the system.

Figure STYLEREF 1 s 2. SEQ Figure * ARABIC s 1 2: QHE for different materials. The position of the LLs is marked by the peaks and the dependence of ?xy on the charge carrier density is plotted as the red trace. (a) Conventional 2DEG, (b) single layer graphene and (c) bilayer graphene. (Figure reprinted from Ref.10)

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We now compare the spectra of graphene shown in Fig.2.3(b) and (c) to the standard 2DEG displayed in Fig.2.3(a). An obvious difference between the conventional semiconductor and the case of graphene is the position of the plateaus in ?xy. For single layer graphene the plateaus are located at ?xy=±ge2/h (i-12 ) meaning that they are shifted by 12ge2 nevertheless the step height is given by ge2h, with g=4 accounting for both spin and valley degeneracy.

A way of understanding the observed shift in the conductivity spectrum is provided by the pseudo spin of graphene, which leads to the accumulation of a Berry phase. As a quasiparticle adiabatically rotates its pseudo spin by ? = 2?, meaning it moves between the two sublattices, it picks up a phase shift of j? (j = 1 for SL and j = 2 for BL) and its wave functions change sign (see Eq. (2.4)). This Berry phase contributes to the total phase, which a particle acquires while encircling a cyclotron orbit. In single layer graphene it induces a phase shift of ? to the Shubnikov- de Haas oscillations and in turn an offset of 2e2/h to the Hall conductivity.

Single layer graphene shows a linear density of states, the number of states in bilayer is constant with energy. Since the states have to condense into the discrete Landau levels, the spacing between energy levels is expected to differ for single and bilayer. Indeed, the sequences are given by ADDIN EN.CITE <EndNote><Cite><Author>McCann</Author><Year>2006</Year><RecNum>7</RecNum><DisplayText>2, 6</DisplayText><record><rec-number>7</rec-number><foreign-keys><key app=”EN” db-id=”frv9s9rxnfrtxxe2vdkxdxrhpex2pzvfpsrw” timestamp=”1520856594″>7</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>McCann, Edward</author><author>Fal’ko, Vladimir I;/author;;/authors;;/contributors;;titles;;title;Landau-level degeneracy and quantum Hall effect in a graphite bilayer;/title;;secondary-title;Physical Review Letters;/secondary-title;;/titles;;periodical;;full-title;Physical Review Letters;/full-title;;/periodical;;pages;086805;/pages;;volume;96;/volume;;number;8;/number;;dates;;year;2006;/year;;/dates;;urls;;/urls;;/record;;/Cite;;Cite;;Author;Haldane;/Author;;Year;1988;/Year;;RecNum;8;/RecNum;;record;;rec-number;8;/rec-number;;foreign-keys;;key app=”EN” db-id=”frv9s9rxnfrtxxe2vdkxdxrhpex2pzvfpsrw” timestamp=”1520856613″;8;/key;;/foreign-keys;;ref-type name=”Journal Article”;17;/ref-type;;contributors;;authors;;author;Haldane, F Duncan M;/author;;/authors;;/contributors;;titles;;title;Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the;quot; parity anomaly;quot;;/title;;secondary-title;Physical Review Letters;/secondary-title;;/titles;;periodical;;full-title;Physical Review Letters;/full-title;;/periodical;;pages;2015;/pages;;volume;61;/volume;;number;18;/number;;dates;;year;1988;/year;;/dates;;urls;;/urls;;/record;;/Cite;;/EndNote;2, 6.

E±N=2e?vFBN (SL)and E±N=2e?vFBN

where B is the magnetic field and ?c is the cyclotron frequency. The bilayer spectrum equals the one for conventional semiconductors except for an additional level at zero energy, which contains both the N = 0 and the N = 1 LL. Instead of the four-fold degeneracy due to spins and valleys, it is hence eight-fold degenerate and leads to a doubled step height in ?xy. Single layer graphene exhibits an N = 0 Landau level at s = 0 as well, which is only four-fold degenerate, however. It is shared by electrons and holes and hence can accommodate only 2 nL instead of 4 nL of each carrier type. Since ?xy features a plateau if the energy is lying in-between two LLs, the N = 0 LL induces a shift of the conductivity ladder by 2e2/h.

Electric field effect in Single layer grapheneThe back gate characteristics obtained for the single layer device is displayed in Fig.3.1(b). The measured resistanceR and the conductivity ? are related via the device length L and width W by ? = 1/R · L/W. Since the width of the flakes is not well defined here, we take the largest value (Wmax in Fig.3.1(a)) to obtain the lower bound for the charge carrier mobility µ. Whereas the upper x-axis displays the measurement parameter VBG, the charge carrier density n is given by the bottom axis.

Clearly, a minimum in the conductivity is visible as the BG is tuned from negative to positive voltages. This is expected for single layer graphene since the density of states decreases linearly as the absolute value of the Fermi energy EF is reduced. Ideally, the graphene sheet exhibits no density of states and hence zero conductivity at the charge neutrality point (CNP). In experimental data, a finite value of the order of 4e2/h is obtained, however. Whether this value is universal and the origin of it are still under dispute 16, HYPERLINK “file:///C:\Users\Muhammad%20Jawad\Downloads\eth-5731-02.docx” l “_bookmark166” 32–37. As an explanation, the formation of electron- hole puddles due to potential fluctuations is put forward. Within these puddles, a finite density of states is present and due to Klein tunneling, charge carriers can be transferred through the graphene sheet, leading to an increased conductance. As an enhanced disorder amplitude is induced by impurities residing on the graphene surface, scattering of charge carriers will reduce the conductivity on the other hand. It is indeed observed that clean samples (e.g. supported by BN or suspended) exhibit an increased minimum conductivity 31, HYPERLINK “file:///C:\Users\Muhammad%20Jawad\Downloads\eth-5731-02.docx” l “_bookmark168” 38,39 as compared to devices with a high charge carrier inhomogeneity.

The shift of the charge neutrality point away from zero gate voltage indicates the doping asymmetry. In the back gate dependence shown in Fig.3.1(b), the minimum of ? is located at a slightly negative voltage, indicating that an excess of positive dopants is present. When determining the density n, this shift was compensated for.

Since the position of the charge neutrality point only provides information about

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the relative doping of positive and negative impurities, a closer look at the curvature of the conductivity trace at its minimum is instructive 39. In the double logarith- mic plot of Fig.3.1(b) two regimes are observed for the conductivity. Below a threshold density (indicated by the vertical line), the conductivity is independent of the number of charge carriers in the system. Here, the transport is dominated by potential fluctuations and the Drude model is not valid anymore. We can extract a saturation density of nsat 9.5 1010 cm?2 for the single layer device. Following Ref. 39, we can determine the Fermi energy at which the saturation sets in to be EF,sat = kvF?nsat ? 35 meV. Such a low value is achieved only rarely in grapheme flakes supported by SiO2 40 and indicates the high quality of the present flake. By fits to the linear part of the conductivity curve, we can estimate the charge carrier mobility. The curve is symmetric indicating that electrons and holes exhibit the same carrier mobility. As a lower limit, assuming the whole width of the flake contributing to transport, we obtain µmin ? 14 400 cm2/Vs. Supposed that only the inner part, defined by the distance Wmin between the contacts, carries current, the upper limit for the mobility µmax ? 45 200 cm2/Vs is found. In agreement with the extraordinarily low disorder density nsat, the mobility values are among the highest reported so far for non-suspended graphene on SiO2. This finding demonstrates, that the fabrication process does not necessarily affect the transport characteristics.

From these values, the m?inimum mean free path for n = 2·1012 cm?2 is determined to be lmfp = (kµmin/e) ?n ? 238 nm. A comparison to the system size hence justifies the application of the Drude model for diffusive transport a posteriori.

Magnetotransport in grapheneThe smoking gun of a single layer of graphene is its quantum Hall effect since the chirality of the massless fermions present in graphene leads to conductance quantization at half-integer multiples of 4e2/h in a finite magnetic field. For bilayer graphene, the coupling between the two graphene sheets imposes a finite mass to the charge carriers. Nevertheless, they remain chiral particles and accumulate a Berry’s phase of 2? along cyclotron orbits (instead of ? for single layer graphene). The resulting conductance ladder exhibits plateaus at integer multiples of 4e2/h except for the one at zero, which is absent due to the presence of a combined Landau level (LL) consisting of the lowest electron and hole LL.

Quantum Electrodynamics In CarbonGraphene consists of a two-dimensional hexagonal lattice of carbon atoms as shown in figure 2.4(a). The two sublattices consisting of atoms A and B can be distinguished. Figure 2.4(b) displays the band structure of graphene calculated by a nearest-neighbor tight- binding approach, which takes only the pz orbitals that are responsible for the delocalized ?-electron system into account. The hexagonal Brillouin zone possesses the high symmetry points ?, M , K and K’, where K and K’ are in equivalent since they correspond to the two sublattices A and B. The dispersion relation is given by 33

Ekx,ky=±?o1+4cos3kxa2coskya2+4cos2kya212Here, ?0 ? 3.2 eV is the nearest neighbor interaction. The carbon-carbon bond length of a’=1.42 A? determines the magnitude of the lattice vector a=3a’. Since the valence and conduction bands form conically shaped valleys that meet at the K and K’ points, graphene is a semi-metal and exhibits an ambipolar field effect. If the Fermi level lies below the Dirac point, the unoccupied states in the valence band can be described as positively charged holes that contribute to conduction. undoped graphene, the Fermi level lies E=0 with a completely filled valence band and empty conduction band. The charge carrier concentration thus equals zero. Upon raising the Fermi energy, electrons in the conduction band become available for conduction (see gure2.4(c)). In the vicinity of the K and K’ points (displacement from the K and K points in reciprocal space: ??k), the dispersion relation can be approximated as 44

E?k=??F?kwith the Fermi velocity ?F=3 ?0a2? ? 106 ms. The Fermi energy depends on the electron density n according to EF= ?n??F. The K and K’ are also called Dirac points since in the vicinity of these points, at low energies, the dispersion is linear rendering charge carriers mass-less. Due to the linear dispersion, all charge carriers below the Fermi energy have the same constant velocity in contrast to conventional semiconductors, where the parabolic dispersion causes a rapid carrier velocity decrease away from the band edge. The mass-less carriers in graphene are to be described by the 2D-Dirac equation instead of the Schrödinger equation.

Figure STYLEREF 1 s 2. SEQ Figure * ARABIC s 1 3: Electronic structure of monolayer graphene. (a) Hexagonal lattice with atoms A and B and lattice constant a. (b) Band structure derived from tight binding model with valleys K and Kj (adapted from 44). (b) Ambipolar conduction by electrons or holes, depending on the Fermi level position

In the 2D-Dirac equation 32, HYPERLINK l “_bookmark194” 45.

±?F0px-ipypx+ipy0?A(r)?B(r)=E?A(r)?B(r)p is the momentum, while ?A(r) and ?B(r) represent the two di erent carrier wavefunc- tions on the sublattices A and B. Thus, instead of using two separate Schrödinger equations for the two charge carrier types, as common practice for 2DEGs, the Dirac equa- tion is used for the charge carriers with an additional degree of freedom. In particular, additionally to the spin, the carriers also posses a so-called valley or pseudo-spin degree of freedom (associated with the sublattices A and B). Within one band, corresponding to4730115260350?

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one of the sublattices, the direction of motion for electrons (positive energy E) is oppo- site to the direction of motion for holes (negative energy E). Since the pseudo-spin is the same for both carrier types in the band, it is parallel to the electron momentum and antiparallel to the hole momentum in the band. In the other band the opposite applies. De ning chirality 17 as the projection of the pseudo-spin onto the direction of motion of the charge carrier leads to electrons with positive and holes with negative chirality in the rest energy band and vice versa in the other band.

Diffusive Conductivity

To calculate the electrical conductivity in the presence of weak magnetic modulation, we use the Kubo formula to calculate the linear response to an applied external field. In a magnetic field, the main contribution to Weiss oscillations comes from the scattering induced migration of the Larmor circle center. This is diffusive conductivity and we shall determine it. Where in it was shown that the diagonal component of ?yy can be calculated by the following expression in the case of quasielastic scattering of electrons:

?yy=?e2LxLy?fE?1-fE??E?(?y?)where Lx and Ly are the dimensions of the layer, ?=1kBT is the inverse temperature with kB as the Boltzmann constant, f(E) is the Fermi-Dirac distribution function, ?(E) is the electron relaxation time, and ? denotes the quantum numbers of the electron eigenstate. The diagonal component of the conductivity ?yy is due to the modulation induced broadening of the Landau bands and, hence, it carries the effects of modulation, in which we are primarily interested in this work. ?xx does with the result not contribute, as the component of velocity here in the x direction is zero. The collisional contribution due to impurities is not taken into account in this work.

Gapped GrapheneWhen a graphene sample is placed on a boron-nitride substrate a staggered potential is formed in the plane of graphene. In practice, this means that the on-site energies of the carbon atoms belonging to the A and B sublattices have different values, denoted by O and O, respectively. As a consequence, an energy gap between the conduction and valence bands appears. A new situation brought about by the staggered potential can still be described by a Dirac-like Hamiltonian5 but with additional diagonal terms.

The novel substrate induces a finite gap and brings about non-equivalence of the two sub lattices that make up the honeycomb crystal structure of graphene. The gap is of the order of several tens of meV at the Dirac point and leads to finite masses for the Dirac fermions. Theoretical value determined from ab-initio calculations. An alternative way to introduce a gap in graphene monolayer using h-BN is to synthesize atomically thin films composed of hybridized h-BN and graphene domains.

The one electron Hamiltonian for gaped graphene monolayer under magnetic field is given by:

41243253032260042462457493000

The corresponding eigenvalue

whereResult and Discussion, Formulation

Formulation

H=vF?.p+ ?z.?(3. SEQ Eq * MERGEFORMAT 5)

Where p is momentum operator, written as

p=pxi+ pyj+pzk = (px,py,pz) (3. SEQ Eq * MERGEFORMAT 6)

? is the and pauli matrix and vF is the electron velocity in graphene having value of 106 ms-1

We can written pauli matrix as

?=?xi+ ?yj+?zk(3. SEQ Eq * MERGEFORMAT 7)

Mathematically the value of pauli matrices are:

?x=0110, ?y=0-ii0, ?z=100-1(3. SEQ Eq * MERGEFORMAT 8)

as our system in 2D so we just use ?x and ?y initially, we have the unmodulated graphene case for which V(x,y) = 0, and suitable landau gauge we use is as

A=(0,Bx,0)(3. SEQ Eq * MERGEFORMAT 9)

In the case of low energy quasiparticles, the tight binding approximation boots up the Hamiltonian of graphene. We use minimal approximation ( p ? p + eA ) when the graphene sheet is placed under the magnetic field B, now the one electron Hamiltonian will become

H0=vF?.p +eA+ ? .?z(3. SEQ Eq * MERGEFORMAT 10)

H0=vF(?xpx+?ypy +e?yBx)+ ??z(3. SEQ Eq * MERGEFORMAT 11)

H0=vF?xpx+vF?ypy +evF?yBx+ ??z(3. SEQ Eq * MERGEFORMAT 12)

H0=vFpx0110+vFpy 0-ii0+eBvF0-ii0+ ?100-1(3. SEQ Eq * MERGEFORMAT 13)

Magnetic length is denoted as l so eB= l2?, =vF0pxpx0+0-ipy ipy 0+0-il2?il2?0+?00-?(3. SEQ Eq * MERGEFORMAT 14)

H0=?vF(px-ipy -il2?)vx(px+ipy +il2?)-?(2. SEQ Eq * MERGEFORMAT 15)

Making the substitution ?±=px±ipy±il2?xH0=?vx?-vx?+-?(3. SEQ Eq * MERGEFORMAT 16)

H0-E?= 0(3. SEQ Eq * MERGEFORMAT 17)

?n,ky= eikyyLy-iasn?n-1x+x0lsbsn?nx+x0l(3. SEQ Eq * MERGEFORMAT 18)

Where x0=l2kyand ?= x+x0l

The index n (n=0, 1, 2,…..) labels the discrete landau levels while s labels the conduction (s = +1) and valance bands (s = -1). With Esn being the Eigen value the coefficient asn and bsn are given by

asn = Esn + ?2Esn ; bsn = Esn + ?2Esn?0,ky= eikyyLy0?0(3. SEQ Eq * MERGEFORMAT 19)

Magnetically modulated gapped grapheneH0=vF?.p +eA+ ? .?z(3. SEQ Eq * MERGEFORMAT 20)

H0=vF?.p +eA?+ ?z?(3. SEQ Eq * MERGEFORMAT 21)

B= B0+Bmcoskx ; K=2?aA=(0, B0x+ BmKsinkx0)(3. SEQ Eq * MERGEFORMAT 22)

H=vF?.p+vFe?.A?+?z?(3. SEQ Eq * MERGEFORMAT 23)

H=vF?xpx+vF?ypy +evF?y(B0x+ BmKsinkx0)+ ??z(3. SEQ Eq * MERGEFORMAT 24)

H=vF?xpx+?y(vFpy +evFB0x)+ ??z+ evFBmK?ysinkx0(3. SEQ Eq * MERGEFORMAT 25)

H0=vF?xpx+?y(vFpy +evFB0x)+ ??z ; H’=u?ysinKx0 ; u=evFBmKH=H0+H'(3. SEQ Eq * MERGEFORMAT 26)

?En,ky= -??dx0Lydy?†n,ky(r) H’?n,ky(r)(3. SEQ Eq * MERGEFORMAT 27)

?n,ky= eikyyLy-iCn?n-1x+x0lsDn?nx+x0l(3. SEQ Eq * MERGEFORMAT 28)

=-eikyyLy-iCn?*n-1(x,)sDn?*nx,0-iumsinkxiumsinkx0eikyyLy-iCn?n-1(x,)sDn?n(x,)(3. SEQ Eq * MERGEFORMAT 29)

=1LylisDn?*n(x,)umsinkxCn?*n-1(x,)umsinkx-iCn?n-1(x,)sDn?nlx,(3. SEQ Eq * MERGEFORMAT 30)

=1LylsCnDnumsinkx?*n(x,)?n-1(x,)sDnCnumsinkx?*n-1(x,)?n(x,)(3. SEQ Eq * MERGEFORMAT 31)

H*n(x)=Hn(x)(3. SEQ Eq * MERGEFORMAT 32)

?n=e-x ‘222nn!? Hn(x)(3. SEQ Eq * MERGEFORMAT 33)

1LysCnDnumsinkxe-x ‘222nn!? Hn(x,)e-x ‘222n(n-1)!? Hn-1(x,)sDnCnumsinkxe-x ‘222nn!? Hn(x,)e-x ‘222n(n-1)!? Hn-1(x,) (3. SEQ Eq * MERGEFORMAT 34)

( STYLEREF 1 s 3. SEQ Equation * ARABIC s 1 1)

=2LylsCnDnum12nn!?2n(n-1)!?sinkxe-x ‘2Hn(x,) Hn-1(x,)(3. SEQ Eq * MERGEFORMAT 35)

Let? = sCnDnum12nn!?2n(n-1)!?

?En,ky= 2?Lyl-??dx0Lydysinkxe-x ‘2Hn(x,) Hn-1(x,)s (3. SEQ Eq * MERGEFORMAT 36)

The y-integral yields Ly and we lift with

?En,ky= 2?l-??dxsinkxe-x ‘2Hn(x,) Hn-1(x,)(3. SEQ Eq * MERGEFORMAT 37)

x’=x+ x0l , x = lx’- x0 , dx = ldx’

?En,ky= 2?l-??ldx’sink(lx’- x0)e-x ‘2Hn(x,) Hn-1(x,)(3. SEQ Eq * MERGEFORMAT 38)

?En,ky= 2?-??dx’sinklx’coskx0-cosklx’sinkx0)e-x ‘2Hn(x,) Hn-1(x,)(3. SEQ Eq * MERGEFORMAT 39)

?En,ky= 2?-??dx’sinklx’coskx0e-x ‘2Hn(x,) Hn-1(x,)-0(3. SEQ Eq * MERGEFORMAT 40)

?En,ky= 4?coskx00?dx’sinklx’e-x ‘2Hn(x,) Hn-1(x,)(3. SEQ Eq * MERGEFORMAT 41)

Now we compare iit with 7.388.6 in Gradshleyn and Ryzhik page 8+6

0?sinklx’e-x ‘2Hn(x,) Hn+2m+1(x,)dx= 2n(-1)m?2 n!b2m+1e-b24Ln2m+1b22(3. SEQ Eq * MERGEFORMAT 42)

In equation 2.37 b= kl , n+2m-1=n-1, m= -1

?En,ky= 4?coskx02n(-1)-1?2 n!(kl)-1e-(kl)24Ln-1(kl)22(3. SEQ Eq * MERGEFORMAT 43)

?En,ky= – ?2n+2?2 n!coskx0kle-(kl)24Ln-1(kl)22(3. SEQ Eq * MERGEFORMAT 44)

Now we use Gredsteyn and rizhikk 8.971.5

Ln?-1(x)= Ln?(x)- Ln-1?(x)(3. SEQ Eq * MERGEFORMAT 45)

In our case ?=0 and Ln0(x)=Ln(x)Ln-1x=Lnx-Ln-1(x) (3. SEQ Eq * MERGEFORMAT 46)

Hence equation 2.40 can be written as

?En,ky= – ?2n+2?2 n!coskx0kle-(kl)24Ln(kl)22-Ln-1(kl)22 (3. SEQ Eq * MERGEFORMAT 47)

?En,ky= ?2n+2?2 n!coskx0kle-(kl)24Ln-1(kl)22-Ln(kl)22(3. SEQ Eq * MERGEFORMAT 48)

Where K=2?a , l=?eB and x0=l2ky

Let u=(kl)22 , then we can express equation 2.44 as follows

?En,ky= ?2n+2?2 n!coskx0kle-u2Ln-1u-Lnu(3. SEQ Eq * MERGEFORMAT 49)

Now ? = SCnDnevFBmK12nn!?2n(n-1)!?? = SCnDnevFBmK1(2n)2n!(n-1)!?(3. SEQ Eq * MERGEFORMAT 50)

? = SCnDnevFBmK1(2n)2n(n-1)!(n-1)!?(3. SEQ Eq * MERGEFORMAT 51)

? = SCnDnevFBmK2?2.n((n-1)!2n+1(3. SEQ Eq * MERGEFORMAT 52)

By putting the value of ? in equation 2.45

?En,ky= SCnDnevFBmK2.2n+2?2.n((n-1)!2n+1?2 n!coskx0kle-u2Ln-1u-Lnu(3. SEQ Eq * MERGEFORMAT 53)

?En,ky= 4n sCnDnevFBmK2l e-u2Ln-1u-LnucosKx0(3. SEQ Eq * MERGEFORMAT 54)

?En,ky= 4SCnDnevFBm?K2l ? n e-u2Ln-1u-LnucosKx0(3. SEQ Eq * MERGEFORMAT 55)

?En,ky= SCnDn? evFBmk l ?4n e-u2klLn-1u-LnucosKx0(3. SEQ Eq * MERGEFORMAT 56)

We put ?m= evFBmk l ? in eq. 2.52 then

?En,ky= SCnDn? ?m4n e-u2klLn-1u-LnucosKx0(3. SEQ Eq * MERGEFORMAT 57)

We put, Gn,B=? ?m2n e-u2klLn-1u-LnucosKx0 in eq. 2.53

?En,ky= 2SCnDn Gn,BcosKx0(3. SEQ Eq * MERGEFORMAT 58)

vy?= 1? ??ky ?En,ky(3. SEQ Eq * MERGEFORMAT 59)

vy? = 1?(SCnDn? ?m)4n e-u2klLn-1u-LnusinKx0(-kl2)(3. SEQ Eq * MERGEFORMAT 60)

vy? = (-4SCnDn?mn le-u2)Ln-1u-LnusinKx0(3. SEQ Eq * MERGEFORMAT 61)

vy?2 = 16S2C2nD2n?2mn l2e-u)Ln-1u-Lnusin2Kx0(3. SEQ Eq * MERGEFORMAT 62)

?yydiff= ?e2LxLy?f?1 – f??(E?)(vy?2)(3. SEQ Eq * MERGEFORMAT 63)

?yydiff= ?e2LxLy?n=0?gEng(En+1)216S2C2nD2n?2mn l2e-u)Ln-1u-LnuLy4?0Lxl2sin2Kx0 dky(3. SEQ Eq * MERGEFORMAT 64)

?yydiff= ?e2LxLy?n=0?gEng(En+1)216S2C2nD2n?2mn l2e-u)Ln-1u-Lnu LyLx2?l212012sin22?xdx (3. SEQ Eq * MERGEFORMAT 65)

?yydiff= ?e2LxLy?n=0?gEng(En+1)216S2C2nD2n?2mn l2e-u) Ln-1u-Lnu LyLx2?l212 (1) (3. SEQ Eq * MERGEFORMAT 66)

?yydiff= ?e2LxLy?n=0?gEng(En+1)216S2C2nD2n?2mn l2e-u Ln-1u-Lnu LyLx4?l2(3. SEQ Eq * MERGEFORMAT 67)

?yydiff= 4e2??n=0??gEng(En+1)216S2C2nD2n?2mn e-u Ln-1u-Lnu (3. SEQ Eq * MERGEFORMAT 68)

?yydiff= 4?2e2S2C2nD2n?2m??2?n=0?n e-u?f(E)?EE=EnLn-1u-Lnu (3. SEQ Eq * MERGEFORMAT 69)

?yydiff= 4?2e2S2C2nD2n?2m?(h2?)??n=0?n e-u?f(E)?EE=EnLn-1u-Lnu (3. SEQ Eq * MERGEFORMAT 70)

?yydiff= 8?2e2S2C2nD2n?2me-uh.??n=0?n ?f(E)?EE=EnLn-1u-Lnu (3. SEQ Eq * MERGEFORMAT 71)

Multiply by 2, due to spin degeneracy.

?yydiff= 16e2h??SCnDn? ?m2e-un=0? n?f(E)?EE=EnLn-1u-Lnu (3. SEQ Eq * MERGEFORMAT 72)

Refrences

ADDIN EN.REFLIST 1.Wallace, P.R., The band theory of graphite. Physical Review, 1947. 71(9): p. 622.

2.Haldane, F.D.M., Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the” parity anomaly”. Physical Review Letters, 1988. 61(18): p. 2015.

3.Young, A.F. and P. Kim, Quantum interference and Klein tunnelling in graphene heterojunctions. Nature Physics, 2009. 5(3): p. 222.

4.Chen, J.-H., et al., Intrinsic and extrinsic performance limits of graphene devices on SiO 2. Nature nanotechnology, 2008. 3(4): p. 206.

5.Klitzing, K.v., G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Physical Review Letters, 1980. 45(6): p. 494.

6.McCann, E. and V.I. Fal’ko, Landau-level degeneracy and quantum Hall effect in a graphite bilayer. Physical Review Letters, 2006. 96(8): p. 086805.