Free Essay

Submitted to the Department of Physics on 25 June 2012, in partial ful…llment of the requirements for the degree of Doctor of Philosophy

Abstract

Recent experimental as well as theoretical works have shown that it is possible to create periodic, electric as well as magnetic, potentials in graphene. The e¤ects of these potentials on charge carriers in graphene leads to novel physical e¤ects with important consequences for transport. Whereas a strong periodic potential can lead to new Dirac points in the band structure of graphene, a weak periodic potential along with a perpendicular magnetic …eld B introduces a new length scale, period of modulation, in the system in addition to the cyclotron diameter at the Fermi energy. Commensurability of these two length scales leads to new observable physical e¤ects. These e¤ects were observed earlier in transport studies in conventional 2DEG systems realized in semiconductor heterostructures. Our aim is to study these e¤ects in a graphene monolayer in order to highlight the similarities and di¤erences in the two systems, conventional 2DEG and graphene. Therefore, in this thesis we have carried out a detailed investigation of the electrical magnetotransport properties of a one-dimensional weakly modulated graphene monolayer. It is found that the periodic modulation broadens the sharp Landau Levels into bands and they start oscillating with B. The electronic conduction in this system can take place through either di¤usive scattering or collisional scattering o¤ impurities. Both these contributions to the electronic transport are taken into account in this work. In addition to the appearance of commensurability oscillations in both the collisional and di¤usive contributions, we …nd that Hall resistance also exhibits commensurability oscillations. Furthermore, the period and amplitude of these commensurability oscillations in the transport parameters and the e¤ect of temperature on them are also analyzed in this work. In addition, we also study a graphene monolayer which is subjected to electric as well as magnetic modulations. Theoretical study of thermodynamic properties of a monolayered graphene, subjected to a magnetic …eld B and one-dimensional weak periodic modulation, is also performed. In this regard, we have investigated the e¤ects of modulation on the thermodynamic properties of graphene and comparison is made with the results of standard two-dimensional electron gas (2DEG) systems. The Landau levels degeneracy is lifted, due to electric modulation. The periodic potential converts the Landau levels in to bands such that their bandwidth oscillates with B. We …nd commensurability (Weiss) oscillations for small values of B whereas de Haas– van Alphen (dHvA)-type oscillations appear for larger values of B. We …nd that the modulation-induced e¤ects on the thermodynamic properties are enhanced and less damped with temperature in graphene compared with conventional 2DEG systems. Furthermore, we have derived analytic asymptotic expressions which allow us to determine the critical tempera9

ture and critical magnetic …eld for the damping of magnetic oscillations in the thermodynamic quantities considered here. Lastly, we draw motivation from recent experimental studies and present a comprehensive study of magnetothermoelectric transport in a graphene monolayer within the linear response regime. We employ the modi…ed Kubo formalism developed for thermal transport in a magnetic …eld. Thermopower as well as thermal conductivity as a function of the gate voltage of a graphene monolayer in the presence of a magnetic …eld perpendicular to the graphene plane are determined for low magnetic …elds (~1 Tesla) as well as high …elds (~8 Tesla). We include the e¤ects of screened charged impurities on thermal transport. We …nd good agreement with recent experimental work on the subject. In addition, in order to analyze the e¤ects of modulation, which can be induced by various means, on the thermal transport in graphene, we evaluate the thermal transport coe¢ cients for a graphene monolayer subjected to a periodic electric modulation in a magnetic …eld. The results are presented versus the gate voltage and the magnetic …eld respectively.

10

Chapter 1

Introduction

1.1 Graphene

Graphene is a monolayer of carbon atoms that are tightly packed into a two-dimensional (2D) honeycomb lattice structure. It is the fundamental building block of all other graphitic materials. Graphene can be rolled up into one dimensional nanotubes and it can be stacked as three-dimensional graphite [1, 2]. More than seventy years ago, Landau and Peierls pointed out that two-dimensional crystals were thermodynamically unstable and could not exist [3]. Their theory showed that a divergent contribution of thermal ‡ uctuations in low-dimensional crystal lattices should lead to such displacements of atoms that they become comparable to interatomic distances at any …nite temperature [3, 4]. Therefore, the scienti…c community was surprised by the recent discovery that graphene can be experimentally extracted from bulk graphite and it can be deposited on the surfaces of any bulk material (crystalline or amorphous) with which it can have only van der Waals interactions [5]. The graphene crystal can also be partially suspended with the edges attached to a substrate [6, 7, 8]. Weak graphene substrate interaction makes it possible to retain some intrinsic features of graphene such as its band structure, its mechanical strength, etc. The original approach to a graphene monolayer was to isolate a single layer by micromechanical cleavage or exfoliation technique and it is based on using adhesive tape to peel o¤ ‡ akes from three-dimensional graphite [5]. In these ‡ akes, a search is carried out for single layers. As the 11

‡ akes are deposited on a silicon substrate, the contrast allows the search for single layers by an optical microscope. Andre’ Geim and Kostantin Novoselov won the 2010 Nobel Prize in Physics for experiments they conducted to isolate a single layer of graphene and for studying its properties. Graphene can now be produced in sizes up to 100 m using this technique. Other methods include use of chemical exfoliation to produce graphene. It is based on growing graphene epitaxially by chemical vapor deposition of hydrocarbons on metal substrates or by thermal decomposition of silicon carbide [9, 10]. It has been shown that graphene is the strongest two-dimensional material which is stretchable [11]. It is an extremely good electrical and thermal conductor. It is almost impermeable and a completely transparent material. Graphene is a zero-gap semiconductor with high crystal purity. It exhibits a dramatic decrease in resistance with the application of gate voltage and high charge mobility [2]. At room temperature, the mobilities are of the order of 105 cm2 =V s. Such mobilities are possible in the case of high charge carrier density ( > 1012 cm

2

). The

conductivity even in the limit of very low charge carrier density does not go to zero but shows a minimum. Such minimum conductivity is found by some groups [12] close to 4e2 =h which is a universal value for massless Dirac fermions. Moreover, in graphene the integer quantum Hall e¤ect has been observed even at room temperature ( 300K) due to high cyclotron energies of the charge carriers in it [13]. The soxy

called "half-integer" quantum Hall e¤ect is found where Hall conductivity

exhibits plateaus

at half integers of 4e2 =h as a function of charge carrier density. The anomalous Hall e¤ects are rooted in graphene’ chiral nature [10]. s One very important feature of graphene is that its charge carriers behave like relativistic particles and are described by the Dirac equation [14, 15, 16, 17, 18, 19, 20, 21, 22]. The interaction of conduction electrons with the periodic potential of graphene’ honeycomb lattice s gives rise to new quasiparticles that at low energies E obey the (2+1) dimensional Dirac equation with e¤ective speed of 106 m=s. These quasiparticle are called massless Dirac fermions. One can imagine them as electrons that have lost their rest mass mo or as neutrinos that have acquired electron charge e. The relativistic description of electron waves on a honeycomb lattices have now provided a way to study quantum electrodynamics (QED) e¤ects by studying electronic properties of graphene [1].

12

1.1.1

Band Structure of Graphene

Graphene has two sublattices in a single layer of carbon atoms arranged in a honeycomb lattice. Each carbon atom has four valence atomic orbitals. The 2s, 2px and 2py orbitals hybridize to form three planar sp2 orbitals whereas 2pz orbital perpendicular to the surface of the graphene plane remains unhybridized. Therefore, graphene has six two bands formed by the 2pz orbital. The bands formed by sp2 orbitals and

bands are closer to the Fermi-surface when

compared to the at low energies.

bands [1, 9, 23, 24, 25, 26, 27] and thus determine the transport in graphene

In a graphene monolayer the carbon atoms are arranged at the corners of regular hexagons forming the honeycomb lattice. The lattice and its Brillouin zone are shown in Figure (1-1) and Figure (1-2). The lattice is made out of two interpenetrating triangular lattices having a1 and a2 as unit vectors. The lattice vectors are de…ned as: a1 = a2 = where a a p (3; 3) 2 p a (3; 3) 2 (1.1) (1.2)

1:42 Ao is the interatomic distance. The reciprocal lattice of graphene is also a

hexagonal lattice with vectors shown in Figure (1-2). They are given as b1 = p 2 (1; 3) 3a p 2 (1; 3): 3a (1.3)

b2 =

(1.4)

The graphene Brillouin zone shown in Figure (1-2) has two points K and K0 at the corners. These points are called Dirac points and their positions in reciprocal space are: K= 2 2 ; p 3a 3 3a 2 ; 3a 2 p 3 3a ; (1.5)

K0 =

:

(1.6)

13

Figure 1-1: Lattice Structure of graphene.

14

Figure 1-2: The …rst Brillouin zone of graphene.

Moreover, the three nearest-neighbor atoms in real space are located at = a p (1; 3); 2 (1.7) (1.8) (1.9)

1

2

=

3

p a (1; 3); 2 a = (1; 0): 2

1.1.2

Dirac Electrons in the Low Energy Limit

Using the tight binding approximation, the energy dispersion for graphene has been derived in this section. We start with the wave function of graphene, which is taken as a linear

combination of Bloch functions n. Then for both sublattices A and B; it becomes: = X cn; (k) (k; r); (1.10)

n;

r; =A;B

15

where, n; 1 X (k; r) = p e N k (r

ik:R

n;

(r

R):

(1.11) is used

In this expression, R indicates the Bravais vectors, N is the number of unit cells and to identify sublattices. Also, the n; R) is the atomic orbital centered at R in one of the

sublattices. Here, the values of n runs over all the four orbitals of a carbon atom. With this ansatz, we solve the Schrödinger equation:

(Hat +

U)

="

(1.12) U is arising due to interaction

where, Hat represents the atomic Hamiltonian. The potential

of atoms in the crystal. Solving the above equation, eight bands are obtained. These bands corresponding to eight orbitals in a unit cell. However, we focus on the two 2pz orbitals which are decoupled from the in-plane orbitals. The above Schrödinger equation has been multiplied with h h pz;A (r)j

and h R)j

pz;B (r)j,

and only nearest neighbor hopping through h 6= 10 A@

0;

pz;

(r R)j U j

pz;

0

(r

R)i = t is included. Moreover, the next nearest neighbor (and all other) overlap integrals i.e. pz; (r

pz;

0

(r

R)i = 0 if R 6= R0 8 0

are neglected. Then, we obtain equations

two equations for the coe¢ cients in the following form: "o t g(k) ik:a2 )

where g(k) = t(1 + e

ik:a1

@

t g(k) "o

cA cB

1

+e

and "o = h

pz;

(r)jHat j

A = "@ pz; 0

cA cB

0

1 A

(1.13)

(r)i. For more illustration, let

us convert this system of Eqs. (1.13) in the form of Hamiltonian H, applied to a wave function = (cA ; cB )T such that H = " . Then using the determinant given in the Eq. (1.13), we

obtain the dispersion relation after taking "o = 0 as [28, 29]: p

" (k) = where, f (k) = 2 cos p

t

3 + f (k) p ! 3 ky a cos 2 p ! 3 kx a : 2

(1.14)

3ky a + 4 cos

(1.15)

16

Here, the plus sign applies to the upper (

) and the minus sign to the lower ( ) band. It is clear

from Eq. (1.15) that the spectrum is symmetric around zero energy. The full band structure of graphene is shown in Figure (1-3). We have obtained two bands which are degenerate at the dirac points K and K0 . In the vicinity of these degenerate points, also termed as valley K and valley K0 respectively, the energy dispersion is characterized by double cones. The transport properties of graphene are mostly determined by the nature of the spectrum around these two points. The Fermi level of intrinsic graphene is situated at these dirac points. The two electrons from the two atoms per unit cell …ll the lower band and the upper band is empty. Three of the six bands, corresponding to in-plane orbitals, lie below the pz bands while the other three above. At the Fermi energy the density of states is zero and hence the electrical conductivity of intrinsic graphene is quite low. In the above derivation leading up to Eq. (1.14), we have neglected next-nearest neighbor’ overlap. If that is included, an asymmetry between electrons s and holes appears in the system, that is the two bands are not symmetrical any more. The low-energy properties, related to the electronic states near the Fermi energy, can be described by expanding the energy dispersion relation around the K and K0 points. De…ning graphene wave vector for valley K as q = K + k with k 0:5T are observed, as shown

in Figure (3-10). However oscillations in Hall resistivity of in-phase E&M modulations are observed only if the derivative of xy with respect to B is taken, as shown in Figure (3-11).

These oscillations are greater in amplitude than the ones for electric or magnetic modulations alone. From the asymptotic expressions for the di¤usive conductivity for in-phase and out-of-phase modulations, given by Eq. (3.37) and Eq. (3.48) respectively, the damping temperature for SdH oscillations is found to be the same as that of the system under either electric or magnetic modulations. The same is true for the damping temperature of Weiss oscillations. The period

74

0.010

(a)

Conductivity(X10− 3Ω )

0.008 0.006 0.004 0.002 0.000 0.0

−1

In phase modulation Ve=1meV, Bm = 0.01645 T a = 382 nm, ne = 3.16x1015 m− 2

σyy σxx

0.2

σyx/1000

0.4

0.6

0.8

1.0

B(Tesla)

0.010 In phase modulation Ve=1meV, Bm = 0.01645 T a = 382 nm, ne = 3.16x1015 m− 2

∆ConductivityX(10− 3Ω−1)

(b)

0.008 0.006 0.004 0.002 0.000

∆σyy

-0.002 0.0 0.2

∆σyx/5

0.4 0.6

∆σxx

0.8 1.0

B(Tesla)

Figure 3-7: The components of (a) the conductivity tensor of graphene and (b) the change in conductivity due to the 1D modulation as a function of the magnetic …eld at temperatures T = 2K (solid curve) and T = 6K (broken curve) for in-phase E&M modulations of equal strength respectively.

75

11 10 9 8 7 6 5 4 3 2 1 0 -1

In phase modulation Ve=1meV, Bm = 0.01645 T a = 382 nm, ne = 3.16x1015 m− 2

Resistivity(Ω)

(a)

ρxy/100 ρxx ρyy

0.0

0.2

0.4

B(Tesla)

0.6

0.8

1.0

6 5

∆Resistivity(Ω)

4 3 2 1 0 -1 -2 0.0

In phase modulation Ve=1meV, Bm = 0.01645 T a = 382 nm, ne = 3.16x1015 m− 2

(b)

∆ρxx

∆ρxy

0.2 0.4

∆ρyy

0.6 0.8 1.0

B(Tesla)

Figure 3-8: The components of (a) the resistivity tensor of graphene and (b) the change in resistivity due to the 1D modulation as a function of the magnetic …eld at temperatures T = 2K (solid curve) and T = 6K (broken curve) for in-phase E&M modulations of equal strength respectively.

76

0.010 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000

(a) σyx/1000 σyy

Conductivity(X10− 3Ω )

Out of phase modulation Ve=1meV, Bm = 0.01645 T a = 382 nm, ne = 3.16x1015 m− 2

−1

σxx

0.0 0.2 0.4 0.6 0.8 1.0

B(Tesla)

0.012

∆ConductivityX(10− 3Ω−1)

(b)

0.008 0.004 0.000

∆σyy ∆σxx ∆σyx/5

0.0 0.2 0.4 Out of phase modulation Ve=1meV, Bm = 0.01645 T a = 382 nm, ne = 3.16x1015 m− 2

-0.004 -0.008

B(Tesla)

0.6

0.8

1.0

Figure 3-9: The components of (a) the conductivity tensor of graphene and (b) the change in conductivity due to the 1D modulation as a function of the magnetic …eld at temperatures T = 2K (solid curve) and T = 6K (broken curve) for out of phase E&M modulations of equal strength respectively.

77

11 10 9 8 7 6 5 4 3 2 1 0

Resistivity(Ω)

Out of phase modulation Ve=1meV, Bm = 0.01645 T a = 382 nm, ne = 3.16x1015 m− 2

(a)

ρxy/100 ρxx

ρyy

0.0 0.2 0.4 0.6 0.8 1.0

B(Tesla)

6 5 4 3 2 1 0 -1 -2 -3 -4 0.0 0.2 0.4 Out of phase modulation Ve=1meV, Bm = 0.01645 T a = 382 nm, ne = 3.16x1015 m− 2

∆Resistivity(Ω)

(b)

∆ρxx

∆ρyy ∆ρxy/5 B(Tesla)

0.6 0.8 1.0

Figure 3-10: The components of (a) the resistivity tensor of graphene and (b) the change in resistivity due to the 1D modulation as a function of the magnetic …eld at temperatures T = 2K (solid curve) and T = 6K (broken curve) for out-of-phase E&M modulations of equal strength respectively.

78

1.40

3 dρxy/dB(10Ω/Τ )

1.18 1.16

1.35 1.30 1.25

Ve=1meV, Bm = 0.01645 T a = 382 nm, ne = 3.16x1015 m− 2

1.14 1.12 1.10 1.08 1.06 1.04 1.02 1.00 0.98 0.96 0.1

Ve = 1 meV, a = 382 nm ne = 3.16 x 1015 m-2

ele(T=2K) ele(T=6K)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

dρxy/dB(103Ω/Τ)

1.20 1.15 1.10 1.05 1.00 0.95 0.90 0.0 0.2 0.4

B(Tesla)

in phase ele+mag(T = 2K) in phase ele+mag(T = 6K) mag(T = 2K) mag(T = 6K) B(Tesla)

0.6 0.8 1.0

Figure 3-11: The derivative (d xy =dB) as a function of the magnetic …eld in the cases of 1D weak magnetic modulation (for T = 2K; solid curve and T = 6K; broken curve) and in-phase electric and magnetic modulation (for T = 2K; dash-dot curve and T = 6K; dot curve). The inset represents the derivative in case of 1D weak electric modulation as a function of the magnetic …eld.

79

of oscillations for both electric & magnetic (in-phase and out-of-phase) modulations is 2:056T

1.

This period of oscillation is the same as when only electric/magnetic modulation is applied.

Thus, it is found from this work that characteristic damping temperatures of Weiss and SdH oscillations as well as the period of oscillations are independent of the type of modulation applied to the system. It is also evident from the results presented that complete suppression of Weiss oscillations can occur if the graphene monolayer is subjected to out-of-phase electric & magnetic modulations.

3.4

Conclusions

We have determined the transport properties in this chapter, by calculating the magnetoconductivity tensor for a graphene monolayer with a weak periodic magnetic modulation. The results have being compared with those of a graphene under weak electric modulation. We …nd that the period of commensurability oscillations in the transport coe¢ cients is the same for both the electrically and the magnetically modulated graphene. Furthermore, for equal modulation strengths, the oscillations are greater in amplitude for magnetically modulated graphene and =2 out of phase with the electrically modulated graphene. We have also determined the collisional, di¤usive and Hall conductivity when E&M modulations are simultaneously applied to the graphene monolayer. We have considered the situations: when modulations are in-phase as well as out-of-phase to each other. When in-phase modulations being applied, the extrema of oscillations in the transport coe¢ cients is dependent on the comparative strength of these modulations. Moreover, study of out-of-phase modulations reveals that the oscillations occur at the same position as in the system under only magnetic modulation and the Weiss oscillations get completely suppressed by choosing particular values of the relative strengths of the two modulations.

80

Chapter 4

Thermodynamic Properties of Periodically Modulated Graphene

4.1 Introduction

As stated in the introduction, the charge carriers in graphene obey the two-dimensional Dirac equation and the emergent quasiparticles are Dirac Fermions. In this chapter, we investigate the thermodynamic properties of Dirac Fermions in the presence of a perpendicular magnetic …eld and a weak periodic electric potential. We were motivated by studies performed on Two Dimensional Electron Gas (2DEG) systems found in semiconductor heterostructures [58, 59]. It has to be realized that electrons in 2DEG behave as free particles and follow a parabolic dispersion relation. Therefore, we perform a detailed comparison between the thermodynamic properties of the two systems to highlight the di¤erences. In this chapter we show systematically that in all thermodynamic quantities of the graphene system, commensurability oscillations (Weiss oscillations) and dHvA (de Haas-van Alphen) type oscillations are observed. Particularly, Weiss and dHvA type oscillations menifest themselves in the chemical potential, Helmholtz free energy, electronic speci…c heat, orbital magnetization and orbital magnetic susceptibility of weakly modulated graphene systems. It is also shown in this work that these e¤ects are more pronounced in graphene systems. Furthermore, in the limits of weak magnetic …elds and small temperatures, analytic expression of Helmholtz free energy is determined which enables us to calculate the critical temperature and critical magnetic 81

…eld for damping of magnetic oscillations in the thermodynamic properties of an electrically modulated graphene monolayer. This chapter is arranged as follows. In section 4.2, we give the formulation of the problem. Calculation of the thermodynamic quantities is given in section 4.3 and numerical results are discussed in section 4.4. Numerical results are supported by analytical results derived in the asymptotic limit, which are then compared with 2DEG systems and presented in section 4.5.

4.2

Formulation

We consider a graphene monolayer in the xy plane with magnetic …eld B being applied along the z direction. In the Landau gauge, the unperturbed single particle Dirac- like Hamiltonian may be written as [21, 39, 67] Ho = vF : ( i}r + eA) : Here, =f x; yg

(4.1)

are the Pauli matrices and vF = 106 m=s characterizes the electron velocity

with A = (0; Bx; 0) the vector potential. The normalized eigenfunctions of the Hamiltonian given in Eq. (4.1) are n;ky 2

where

n

=2) p = exp(n x p Hn (x), Hn (x) are the Hermite Polynomials, Ly is the normalization length 2 n!

eiky y =p 2Ly l

i

+ xo )=l] ; n [(x + xo )=l]

n 1 [(x

(4.2)

in the y direction; n is an integer corresponding to the Landau Level index and xo = ky l2 is p the center of the cyclotron orbit. The corresponding eigenvalue are En = }! g n, where p ! g = vF 2eB=}. In order to investigate the e¤ects of modulation, we express the Hamiltonian in the presence of modulation as H = Ho + V (x): (4.3)

Here, V (x) is the one-dimensional weak periodic modulation potential applied along the x axis and is given by V (x) = Ve cos Kx (4.4)

82

with K =

2 a

such that a represents the modulation period and Ve is the constant modulation EF . Thus …rst order energy correction, using

strength. To account for weak modulation Ve perturbation theory is:

En;xo = En + jVn j cos Kxo Here, jVn j =

Ve 2

(4.5)

1 (u)

exp(

u 2 )[Ln (u)

+ Ln

1 (u)],

u=

K 2 l2 2

and, Ln (u) and Ln

represents the

Laguerre polynomials. As mentioned in Chapter 2 that we have taken the Fermi level in this system to be upshifted from the Dirac point, which indicates that this model relates to n-doped graphene. However to carry out the calculations performed below to p-type graphene, where the Fermi energy (EF ) is downshifted from the Dirac point, all we have to do is substitute E by E, ! g by ! g and the Fermi Dirac distribution function of holes, 1 f (E), for that of

electrons, f (E), in the expressions given below. The density of states D(E) expressions turn out to be the same as that of the n-type system. Same is true for the Helmholtz free energy F . Therefore, our results are valid for p-doped graphene as well. Although similar features in the energy spectrum have also been found in the 2DEG system there are substantial di¤erences between the two spectra. The Landau level spectrum of Dirac electrons is proportional to the square root of B and n as compared to the linear dependence on B and n in the case of standard electrons in 2DEG. The Eq. (4.5) shows that the energy eigenvalues of a periodically modulated graphene have a term which is a sum of two successive Laguerre polynomials, whereas standard electrons in 2DEG obey a relation containing a one Laguerre polynomial having index n. We …nd that the modulation potential lifts the degeneracy of the Landau levels and broadens the formerly sharp levels into electric Landau bands. Further, broadening of the energy spectrum induced by electric modulation is nonuniform. The Landau band width Vn oscillates with n since Laguerre polynomials are an oscillatory function of the n. Landau bands vanishes for those values of B for which amplitude of modulation becomes zero. By putting Vn = 0; one can obtain the ‡ band condition: at exp u [Ln (u) + Ln 2

1 (u)]

= 0:

(4.6)

83

Applying the asymptotic expression [63]: exp p 1 u Ln (u) ' p p cos(2 nu 2 nu ) (4.7)

4

with Ln (u) = Ln

1 (u);

one obtains from Eqs. (4.6) and (4.7) the following condition 2Rc = a(i 1=4); i = 1; 2; 3; :::::::::: (4.8)

where Rc = kF l2 is the radius of classical cyclotron orbit. It can be observed from Eqs. (4.5) and (4.7) that in the large n limit, width of the Landau levels oscillates sinusoidally, and is periodic in 1=B; for …xed values of n and a. When n is small, the bandwidth still keep on oscillating, but the condition (Eq. (4.8)) does not holds anymore because Eq. (4.7) and Ln (u) ' Ln condition, for low values of B, when many Landau levels are …lled. [58, 59]. It is well known that for an unmodulation system, the density of states (DOS) are made up of delta functions. However, the presence of electric modulation modi…es these delta functions leading to density of state broadening. The density of states D(E) are given by [35]: D(E) = A X (E l2 n;x o 1 (u)

are not valid here. Interestingly both the graphene and 2DEG systems have the same ‡ band at

En;xo )

(4.9)

where, the sum over n runs over all occupied states. By substituting the energy eigenvalues given in Eq. (4.5), D(E) can be expressed as: Z A X dxo D(E) = 2 (E l n a a 0

En

Vn cos Kxo ) :

(4.10)

The density of states can also be expressed in terms of the self energy as: (E)

2 l2 2 o o

D(E) = Im where

(4.11) is the collisional broadening pa-

(E) is the self energy given in Appendix-A and

rameter of Landau bands. In the limiting case, when impurity induced broadening of Landau

84

bands

o

is smaller than the modulation induced broadening Eq. (4.10) and Eq. (4.11) reduces A X (jVn j q 2 l2 n jVn j2 jE (E En j) En )

to [78, 79] D(E) =

(4.12)

2

where (x) is the Heaviside unit step function. From Eq. (4.12) it can be seen that, the onedimensional van Hove singularities appears at the energy edges of the broadened Landau bands. These singularities are of the inverse square-root type and form a double peak like structures in the DOS. The numerical solution DOS of an electrically modulated graphene verses energy at a …xed magnetic …eld of B = 0:055T which corresponds to }! g = 8:5meV is shown in Figure

2 (4-1). The density of states has been scaled using Do = E=( }2 vF ) where Do is the DOS of a

graphene monolayer with no magnetic …eld, such that y axis is dimensionless. The strength of modulation is taken as Ve = 1meV . In the absence of the modulation the DOS consists of functions where transport occurs through hopping of electrons between Landau states. However in the presence of modulation these function DOS is broadened into a band with van Hove

singularities appearing at the edge of the LLs re‡ ecting the 1D nature of the electron motion in these bands. The peaks of DOS of di¤erent LLs are well separated. The wide band corresponds to lower peaks of DOS and the narrow band to the higher peaks because each LL contains the same number of states.

4.3

Equilibrium Thermodynamic Quantities

We determine the electronic contribution to the graphene’ equilibrium thermodynamic propers ties with magnetic …eld and weak electric modulation. To facilitate comparison with results for a 2DEG, we compute the following thermodynamic quantities: chemical potential, Helmholtz free energy, electronic speci…c heat, orbital magnetization and orbital magnetic susceptibility. The chemical potential the following relation N= (B; T ) of a graphene monolayer can be determined by inverting

1 Z 0

D(E)f (E)dE: i

1

(4.13)

h Here, the Fermi Dirac distribution function f (E) = exp 85

E kB T

+1

, kB is the Boltzmann

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 40 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 40 42 44 46 48 50 52 54 56 58 60 41 42 43 44 45 46 47 48

D(E)/Do

E(meV)

D(E)/Do

E(meV)

Figure 4-1: The D(E)=Do in a periodically modulated graphene versus energy for B = 0:055T . 2 The modulation strength is taken as Ve = 1meV; a = 382nm and Do = E=( }2 vF ):

86

constant and N is the total number of electrons. The total internal energy is de…ned as:

1 Z 0

U=

ED(E)f (E)dE:

(4.14)

We observe that (B; T ) is a¤ected by changes in the D(E). Substituting Eq. (4.9) into Eq. (4.13), we obtain A X N= 22 l Here x = jE En j jVn j ; n 1

n=0 1

Z1

p

dx (1 + 1 x2

n exp[zn x])

1

:

(4.15)

= exp

modulated and unmodulated systems (zn

h

En kB T

i

and zn = jVn j =(kB T ): Eq. (4.15) can be applied to both 0). For …xed electron concentration ne = N=A the

above equation gives (B; T ) only implicitly and it can not be decoupled explicitly. Therefore we have solved it numerically to obtain the chemical potential (B; T ). Once the chemical potential and the density of states are known, the free energy F of the system can be calculated. From there on the thermodynamic properties are evaluated from the free energy by taking the appropriate derivatives. For non-interacting fermions, Helmholtz free energy is [80]:

1 Z 0

F = N

kB T

D(E) ln 1 + exp

E kB T

dE:

(4.16)

The density of states D(E) is the central quantity in the above expression. The expression for D(E) in graphene is di¤erent from that in conventional 2DEG due to the di¤erence in the energy spectrum in the two cases. This di¤erence will be re‡ ected in the thermodynamic properties of the two systems. The free energy for an electrically modulated graphene system is: F = N kB T A X 2 l2

@2F @T 2 1

n=0 1

Z1

p

dx ln 1 + 1 x2

n

1

exp ( zn x) :

(4.17) F )=T and the

From Eq. (4.17), we calculate the electronic contribution to the entropy S = (U electronic speci…c heat C = T

A;N

: The electron’ orbital motion in the presence of an s

@F @B A; N

external magnetic …eld produces the orbital magnetization M = magnetic susceptibility = @ 2 F=@B 2

A;N

as well as orbital

: These quantities are numerically evaluated and

87

results presented in the next section.

4.4

Results and Discussion

Numerical results for thermodynamic properties of a graphene monolayer subjected to electrical modulation and an external magnetic …eld are presented. The focus, here, is on the modulation induced changes in thermodynamic properties. For comparison with a 2DEG system we have chosen the following parameters: ne = N=A = 3:16 1015 m

2

and a = 382nm. The strength

of the electrical modulation is taken to be Ve = 1meV . These are the same parameters considered for a 2DEG in [58, 59]. Modulation induced e¤ects on thermodynamic quantities can be highlighted by calculating the di¤erence between the modulated and the unmodulated cases in each system. The Figures (4-2) through (4-6), depict the change in di¤erent thermodynamic properties due to electric modulation at temperatures 2K (solid curve) and 6K (broken curve) respectively. The Figure (4-2), shows the change in the chemical potential versus magnetic …eld at temperatures 2K(solid) and 6K(broken). For conventional 2DEG system, for B < 0:3T; oscillations depend very weakly on temperature, which is a clear signature of Weiss type oscillations whereas, for B > 0:3T , the oscillations depend strongly on temperature, in particular they die out at 6K, a clear signature of dHvA type oscillations. Furthermore, the zeros in the chemical potential are in close agreement with those predicted by the ‡ band condition Eq. (4.8). We at have discussed this point from the perspective of the bandwidth in the discussion of Figure (4-3) given below. In graphene, the value of B de…ning the boundary between the two oscillatory phenomena is quite low (It lies somewhere between 0:1 and 0:15T ). For smaller values of B; Weiss type oscillations are present and the amplitude of the oscillations remain essentially the same at di¤erent temperatures. For larger values of B; the familiar dHvA-type oscillations are present with the amplitude of oscillations reduced considerably at comparatively higher temperature. dHvA oscillations are found to start at a lower magnetic …eld in graphene compared to 2DEG. This di¤erence can be attributed to the large cyclotron gap in graphene compared to the cyclotron gap of conventional 2DEG systems since the condition }! g >> kB T is satis…ed at lower magnetic …elds in graphene compared to a 2DEG. Moreover, in a conventional 2DEG sys-

88

0.3

0.2

Ve = 1 meV a = 382 nm ne = 3.16 x 1015 m− 2

T=2K T=6K

0.1

∆η ( meV )

0.0

-0.1

-0.2 0.0 0.2 0.4

B(Tesla)

0.6

0.8

1.0

Figure 4-2: The change in the chemical potential ( ) due to 1D modulation versus B at two di¤erent temperatures 2K (solid curve) and 6K (broken curve).

89

tem oscillations completely die out at 6K whereas they persist in graphene at this temperature. In this section, we have derived analytic asymptotic expressions for the free energy in order to obtain temperature scales for both dHvA-type and Weiss type oscillations. Our …ndings show that for a graphene system comparatively higher temperatures are required for the damping of both the dHvA and Weiss type oscillations. This result holds for all the thermodynamic quantities considered here. The change in free energy ( F ) due to 1D modulation in graphene is shown in Figure (4-3).

To make y axis dimensionless, the free energy has been scaled using Fo = 1 N EF . It is observed 2 that at small values of magetic …eld, modulation induces weakly temperature dependent Weiss type oscillations. The zeros in F occurrs at their respective ‡ band conditions. To illustrate at

this point, we consider the bandwidth of the nth Landau band jVn j. For the modulation induced e¤ects considered here the magnetic …eld is small and in order to make an estimate of the minima of the bandwidth we take the large n limit of the Laguerre polynomial term such that p exp[ u=2]Ln (u) may be approximated by ( 2 nu) 1=4 cos(2 nu =4). If the bandwidth is plotted in both cases, for exact result and the asymptotic limit, minima of the bandwidth are found at B = 0:086; 0:106; 0:140; 0:184; :::. Modulation induced change in free energy vanishes at B = 0:084; 0:102; 0:130; 0:174; :::. which shows that vanishing of the change in free energy occurs at those magnetic …eld values where the minima of bandwidth occurs. Furthermore, Weiss oscillations are more pronounced in graphene system, signi…cantly the amplitude of Weiss oscillations for the graphene system remains unchanged at higher temperature, contrary to the 2DEG in which damping is observed. The familiar dHvA type oscillations are observed for higher values of B. As in the case of the chemical potential, the dHvA type oscillations start quite early. The …rst period for the dHvA type oscillations starts at B = 0:3T and extends up to 0:6T for the conventional 2DEG system, whereas for graphene the …rst

period of dHvA oscillations starts at B = 0:175T and terminates at 0:27T: In Figures (4-4) and (4-5) the changes in the magnetization M and the susceptibility

against the magnetic …eld are shown. The change in orbital magnetization and change in the susceptibility has been scaled using Mo = N

B B

and

o

=N

B =B

respectively where

2 = e }vF =(2EF ) = 5:021meV =T is e¤ective Bohr magneton in graphene, such that y axis

in both curves appears dimensionless. The orbital magnetization considered here is the Landau

90

0.24 0.20 0.16

Ve = 1 meV a = 382 nm ne = 3.16 x 1015 m−2

T=2K T=6K

−∆F/Fo(x10− 3)

0.12 0.08 0.04 0.00 -0.04 0.0 0.2 0.4

B(Tesla)

0.6

0.8

1.0

Figure 4-3: The F=Fo versus B at two di¤erent temperatures 2K (solid) and 6K (broken) using Fo = 1 N EF . 2

91

0.12

0.08

Ve = 1 meV a = 382 nm ne = 3.16 x 1015 m−2

T=2K T=6K

0.04

∆M/Mo

0.00

-0.04

-0.08 0.0 0.2 0.4 0.6 0.8 1.0

B(Tesla)

Figure 4-4: The M=Mo versus B at two di¤erent temperatures 2K (solid) and 6K (broken) using Mo = N B .

diamagnetic contribution. As the magnetic …eld is applied, the electron distribution breaks up into a series of Landau levels. This change in energy with …eld is equivalent to magnetization of a system. At low B, Weiss type oscillations are clearly visible while for higher values of B, dHvA oscillations are present. The Weiss oscillations are weakly dependent on temperature while dHvA type are strongly a¤ected. We observe that dHvA oscillations in M and are

less damped with temperature in the graphene system than in the 2DEG system. The change in the electronic speci…c heat capacity ( Cel ) due to 1D modulation in graphene is shown in Figure (4-6). To make y axis dimensionless, the speci…c heat has been scaled using Co = N kB . We …nd that the amplitude of the Weiss type oscillations in Cel =Co is

not large which suggests that the modulation induced e¤ects on the speci…c heat are small.

92

30

20

Ve = 1 meV a = 382 nm ne = 3.16 x 1015 m− 2

T=2K T=6K

∆χ/χo

10

0

-10 0.0 0.2 0.4 0.6 0.8 1.0

B(Tesla)

Figure 4-5: The = using o = N B =B.

o

versus B at two di¤erent temperatures 2K (solid) and 6K (broken)

93

0.04

0.03

Ve = 1 meV a = 382 nm ne = 3.16 x 1015 m− 2

T=2K T=6K

∆Cel/Co

0.02

0.01

0.00 0.0 0.2 0.4 0.6 0.8 1.0

B(Tesla)

Figure 4-6: The Cel =Co versus B at two di¤erent temperatures 2K (solid) and 6K (broken) using Co = N kB .

We also …nd that the speci…c heat at low B is enhanced when the modulation is introduced compared to the situation without modulation. This occurs due to the broadening of the Landau levels caused by the modulation resulting in the contribution of intra-Landau level thermal excitations to the electronic speci…c heat in addition to the contribution from interLandau level thermal excitations. We also observe damping with temperature of the dHvA type oscillations at higher B. In order to gain further physical insight into the results presented above, we analyze asymptotic expressions of the thermodynamic quantities in the next section.

4.4.1

Asymptotic Results

An analytic expression for the Helmholtz free (F ) energy is derived, which is true in the quasiclassical limit when many Landau bands are …lled. To obtain the asymptotic expression for

94

F , it is essential to …rst derive an asymptotic expression for the density of states. In the quasi-classical case, an approximated analytical formula for the density of states is given in the Appendix-D as D(E) = where e E A 1 + 2 cos l2 (}! g )2

2 E2 (}! g )2

1

eE

cos2

p

2

E Kl }! g

4

(4.18)

=

a Ve2 l

p !3 2 : }! g

Density of states consists of two parts, the terms outside the curly braces are for the limiting case of vanishing modulation potential. The additional modulation contribution to the density of states depends quadratically on the strength of the modulation through Substitution of Eq. (4.18) in Eq. (4.16) yields A kB T l2 (}! g )2

1 Z 0 1 Z E 1 + 2 cos 0 e

in Eq. (4.18).

F

N

2 E2 (}! g )2 p

ln 1 + exp

E kB T

dE

+

e

A kB T l2 (}! g )2

E 2 cos

2 E2 (}! g )2

cos2

2

E Kl }! g

4

ln 1 + exp

E kB T

dE : (4.19)

The …rst two terms on the right hand side correspond to the unmodulated free energy Fu while the third term arises due the modulation applied on the system such that Fmod we consider Fu A kB T l2 (}! g )2

1 Z 0

F . Firstly,

Fu = N

E 1 + 2 cos

2 E2 (}! g )2

ln 1 + exp

E kB T

dE:

(4.20)

95

Setting (E

)=kB T = 2 . At low temperatures, such that

coincides with the Fermi energy

EF and EF =kB T >> 1, the above equation can be expressed as A 2(kB T )2 EF l2 (}! g )2

1 Z 1 Z

Fu

N

ln[1 + exp( 2 )]d

EF =2kB T

A 4(kB T )2 EF l2 (}! g )2

cos

2 2 EF 8 kB T EF + 2 (}! g ) (}! g )2

ln[1 + exp( 2 )]d :

(4.21)

EF =2kB T

The second integral in the above equation is of the type:

1 Z o I( ; ;

o)

=

cos (

+ ) ln[1 + exp( 2 )]d

(4.22)

where

=

8 kB T EF (}! g )2

,

=

2 2 EF (}! g )2

and

o

= EF =2kB T >> 1. For large

o;

it is possible to

analytically solve the above integral. Twice integrating by parts, and then replacing the lower limit of integration by 2 1 as o o

>> 1, leads to 2

2

I( ; ;

o)

sin(

o)

+

cos(

o)

1

2

1 Z 1

cos ( + ) d : cosh2 ( )

(4.23)

Using the following identity [63]:

1 Z cos ( + ) d = 2 cosh2 ( ) 0

2 sinh(

2

)

:

(4.24)

we obtain

1 Z 1

cos( ) cos ( + ) d = 2 sinh( 2 ) cosh ( )

(4.25)

96

with the result A 2(kB T )2 EF l2 (}! g )2 )2

2 2 EF (}! g )2 1 Z

Fu

N

ln[1 + exp( 2 )]d

A EF sin l2 2

2 2 EF (}! g )2

2 2 EF (}! g )2

EF =2kB T

A (}! g cos l 2 8 2 EF

+

A kB T cos 2 sinh(T =T dHvA ) 2 l g

2k E B F

:

(4.26)

dHvA = (}! )2 =4 In the above expression, Tg g

de…nes the critical/characteristic tempera-

ture for dHvA-type oscillations appearing in a graphene, determining amplitude of these oscildHvA lations at small B. It is worth mentioning is that the critical temperature T2DEG = ~! c =2 2k B

for dHvA oscillations in the case of conventional 2DEG [58] is lower than the critical temperature for graphene. Further, the ratio has come out to be independent of the external magnetic …eld such that dHvA Tg dHvA T2DEG

vF me = p } 2 ne

4:

(4.27)

Comparatively higher value of critical temperature found in graphene can be attributed to the larger cyclotron gap }! g ; characteristic of Dirac fermions in graphene. Moreover, the e¤ective dHvA 2 dHvA ; which mass term in T2DEG is analogous to the EF =vF (similar to E = m c2 ) term in Tg

also con…rms the relativistic nature of Dirac fermions in graphene. Moreover, from Eq. (4.26) we …nd pure oscillatory behaviour of dHvA-type in Fu because of the presence of magnetic …eld dependent sine and cosine terms. If we compare the above result with the results for conventional 2DEG systems [58], we …nd that additional temperature independent and magnetic …eld dependent sine and cosine terms are present in the expression for graphene. Furthermore, at higher temperatures, the last term on the right hand side of Eq. (4.26) decays exponentially whereas the additional magnetic …eld dependent sine and cosine terms persist, contrary to the conventional 2DEG system. Along the same lines, the second integral on the right hand side of Eq. (4.19) may be solved for Fmod F: F =

2 A EF l2 2 !

Fmod = 1

e

sin 1 cos 2

2 2 EF (}! g )2 2 2 EF (}! g )2

+

2 2 EF = (}! g )2

dHvA T =Tg dHvA ) sinh(T =Tg

)

cos2

p EF 2 Kl }! g

4

:

(4.28)

97

The above expression accounts for dHvA-type oscillations in graphene under electric modulation. At small B, the cosine squared term give rise to Weiss-type oscillations in F which

results in the amplitude modulation of the dHvA-type oscillations, such that zeros of the oscilations occurs when the electric ‡ band condition gets satis…ed. The period of modulation at is found to be p ne B a e = p 2 2 }

(4.29)

which is the same as for the conventional 2DEG system. From Eq. (4.28) no temperature scale for the Weiss type oscillations is obtained. However, if we solve the second integral of Eq. (4.19) for Fmod by replacing all the energy terms by EF leaving the one which is included in the last cosine function (where small energy changes can in‡ uence the damping of Weiss type oscillations) we …nd an expression for type oscillations as well

2 EF A (kB T )2 cos l2 (}! g )2 2 2 EF (}! g )2 2 2 EF (}! g )2 W T =Tg eiss 1 W 2 sinh T =Tg eiss }! g pa 2 2 2 2lkB 2 2 EF (}! g )2 1 Z

F containing the critical temperature for the Weiss

Fmod =

F =

3 EF

e

ln[1 + exp( 2 )]d +

EF =2kB T e

A p cos l2 4 2Kl}! g E2 A p F cos l2 (2 2Kl)2

e

1 + 2 cos2

p EF 2 Kl }! g

4

(4.30)

p W where Tg eiss = }! g =(2 2 KlkB ) = of Weiss oscillations in graphene.

de…nes the critical temperature for the damping

We can now compare the critical temperature scales for damping of Weiss oscillations in

W W eiss graphene, Tg eiss ; and conventional 2DEG systems, T2DEG ; where electrons obey the standard

parabolic energy spectrum. The critical damping temperature in conventional 2DEG system

W eiss [36, 62] is T2DEG = p vF eBa ; 4 2 kB

p where vF = }kF =me is the Fermi velocity of electrons in 2DEG

with parabolic energy spectrum and me is the e¤ective mass of electron. On comparing the

W two temperature scales we …nd that the damping temperature Tg eiss for Weiss oscillations in W eiss graphene is higher than T2DEG of 2DEG system. The ratio of critical temperatures of Weiss

98

oscillations is found to be same as that of dHvA type oscillations,

W Tg eiss W eiss T2DEG

=

vF p vF

4;

(4.31)

which means comparatively higher temperature is also required for damping of Weiss type oscillations in graphene. The reason being the higher Fermi velocity of Dirac electrons in graphene compared to standard electrons in 2DEG systems. From the above discussion, it is evident that both dHvA and Weiss-type oscillations are dHvA ) factor in the last term of Eq. (4.26) is more enhanced in case of graphene. The sinh(T =Tg dHvA is replaced the damping factor with temperature of dHvA oscillations in graphene. If T =Tg dHvA =B for graphene and T = T dHvA is replaced by B dHvA =B for 2DEG system, where by Bg 2DEG 2DEG dHvA = 2 Bg 2 k k T =ev F B F dHvA and B2DEG = 2 2 k k T =ev p F B F

de…nes the critical magnetic …eld

for graphene and 2DEG respectively, then we are able to determine the amplitude of dHvA oscillations at a given magnetic …eld. In Figure (4-7) we have plotted (sinh(Bo =B))

1 dHvA for graphene and B = ; where Bo = Bg o

dHvA B2DEG for 2DEG systems respectively; versus the magnetic …eld B at a temperature of 2K.

The solid curve is for the graphene system whereas the broken curve is for a conventional 2DEG system. From the …gure it can be seen that the curve for the graphene system(solid) leaves the zero axis at a magnetic …eld of 0:12T while the curve for the conventional 2DEG system(broken) becomes non zero at a magnetic …eld of 0:35T , depicting minimum values of the magnetic …eld B for the occurrence of dHvA oscillations for the two systems. These minimum values of critical magnetic …elds are consistent with our numerical results. From the values of critical temperature and critical magnetic …eld Bo ; it is clear that conditions favorable for dHvA oscillations in a conventional 2DEG system persists to smaller magnetic …elds and higher temperatures in graphene.

4.5

Conclusions

We have presented in this chapter a study of the thermodynamic properties of a graphene monolayer subjected to a weak electric modulation in the presence of a magnetic …eld. The results obtained are compared with those of a conventional 2DEG system realized in semiconductor 99

1.6 1.4 1.2

Ve = 1 meV a = 382 nm ne = 3.16 x 1015 m−2 graphene 2DEG

T=2K

(Sinh[Bo/B])-1

1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

B(Tesla)

Figure 4-7: The (sinh(Bo =B)) 1 versus B at T = 2K for graphene system (solid curve) and for conventional 2DEG (broken curve).

100

heterostructures. As a result of the commensurability of modulation period and cyclotron orbit radius, Weiss type and dHvA type oscillations are re‡ ected in all the thermodynamic quantities under consideration in this work for the two systems. However, these e¤ects are more pronounced in the graphene system in the sense that the oscillations in the thermodynamic quantities are more robust against temperature as well as having a higher amplitude than in a conventional 2DEG system. This di¤erence arises due to the di¤erent nature of the quasiparticles (Dirac electrons and standard electrons) in the two systems with the result that the energy spectrum for the quasiparticles in the two systems as well as their Fermi velocities are di¤erent. For an electrically modulated graphene we are able to determine the critical temperature and critical magnetic …eld for damping of magnetic oscillations in the thermodynamic properties, on the basis of analytic asymptotic expressions.

101

Chapter 5

Thermal Magnetotransport in Graphene

5.1 Introduction

After studying thermodynamic properties of a graphene monolayer in the previous chapter, we investigate thermal magnetotransport in graphene in this chapter. We employ the modi…ed Kubo formalism required to study thermal transport in a magnetic …eld. The usual Kubo formula for thermal response functions is invalid in a magnetic …eld and needs to be modi…ed when calculating the transverse (Hall) thermal conductivity and the Nernst coe¢ cient [81, 82]. We use the phenomenological transport equations obtained from the modi…ed Kubo formalism [82, 83]. Further, in the scattering rate and the impurity broadening of the Landau levels the e¤ects of the carrier concentration that can be varied by the gate voltage are taken into account. In the …rst stage, we determine the components of magnetoelectrothermal (MET) power and MET conductivity of an unmodulated graphene in the presence of randomly distributed charged impurities. The results are as a function of the gate voltage for small and large magnetic …elds applied perpendicular to the graphene sheet. We determine both the Nernst and Seebeck coe¢ cients as well as longitudinal and transverse thermal conductivity. These results are then compared with experimental work. In addition, we have also carried out a detailed investigation of the MET transport properties of a graphene monolayer subjected to a weak one-dimensional periodic potential along with 102

a perpendicular magnetic …eld. Motivation for this has arisen from recent work, experimental and theoretical, that has shown that interaction with a substrate can lead to weak periodic modulation of the graphene spectrum. Furthermore, by applying patterned gate voltage or placing graphene on a pre-patterned substrate can also lead to modulated graphene [37, 38, 39]. Placing impurities or adatom deposition can do the same. In Chapter 2, we have computed the electric transport coe¢ cients of electrically modulated graphene [73]. It was shown that modulation turns the sharp Landau levels into bands and their widths start oscillating periodically with the magnetic …eld. This a¤ects the magnetoelectric transport coe¢ cients which exhibit commensurability (Weiss) oscillations. The origin of these Weiss oscillations is the commensurability of the two characteristic length scales of the system: The cyclotron diameter at the Fermi energy and the period of the modulation [62]. An interesting feature of electronic conduction in the modulated system is the opening of the di¤usive (band) transport channel in addition to hopping (collisional) transport. Both di¤usive and collisional contributions to MET transport are taken into account in this work. Recent measurements of thermoelectric power (TEP) on graphene samples in zero and nonzero magnetic …elds have shown a linear temperature dependence of TEP which suggest that the dominant contribution is that of di¤usive thermopower (Sd ). A comparison between the measured TEP and that predicted by the Mott formula shows general agreement, particularly at lower temperatures (T < 50K) [52]. However, at higher temperatures deviation from the Mott relation have been reported [48, 49]. In theoretical work, Yan et al., [57] have determined the TEP of Dirac fermions in graphene with in the self-consistent Born approximation. Also, Hwang et al., [51] in their calculation of TEP incorporate the energy dependence of various transport scattering rates and show that the dominant contribution is from the screened charged impurities in graphene’ environment. Further, Vaidya et al., [52] used Boltzmann transport s theory to calculate Sd in graphene after considering contributions of optical phonon and surface roughness scatterings. The quantum magnetic oscillations in electrical and thermal transport have been earlier investigated theoretically by Gusynin and Sharapov [72] and they obtained analytical results for longitudinal thermal conductivity and the Nernst coe¢ cient. However, they assumed a scattering rate that is constant in energy, independent of magnetic …eld and temperature.

103

Hence the self energy used is not self consistent. Moreover, they evaluated the longitudinal thermal conductivity as a function of the magnetic …eld at di¤erent temperatures, …xed chemical potential and constant impurity broadening. Further, they determined the Nernst coe¢ cient (signal) without recourse to the modi…ed Kubo formalism appropriate for thermal transport in a magnetic …eld. They neglected the dependence of impurity broadening on the chemical potential or on carrier concentration. Dora and Thalmeier extended the work presented in Ref. [72] and studied the electric and thermal response of two dimensional Dirac fermions in a quantizing magnetic …eld in the presence of localized disorder [84]. They evaluated the Seebeck coe¢ cient and the corresponding thermal conductivity as a function of the chemical potential and the magnetic …eld. They did not determine the Nernst coe¢ cient and the transverse thermal conductivity. What distinguishes our work on thermal magnetotransport of unmodulated graphene from others is that we employ the modi…ed Kubo formalism required to study thermal transport in a magnetic …eld. As has been discussed earlier, the usual Kubo formula for thermal response functions is invalid in a magnetic …eld and needs to be modi…ed when calculating the transverse (Hall) thermal conductivity and the Nernst coe¢ cient [81, 82]. We use the phenomenological transport equations obtained from the modi…ed Kubo formalism [82, 83]. Further, in the scattering rate and the impurity broadening of the Landau levels the e¤ects of the carrier concentration that can be varied by the gate voltage are taken into account. In the …rst stage, we determine the components of magnetoelectrothermal (MET) power and MET conductivity of an unmodulated graphene monolayer in the presence of randomly distributed charged impurities. The results are presented as a function of the gate voltage for small and large magnetic …elds applied perpendicular to the graphene sheet. We determine both the Nernst and Seebeck coe¢ cients as well as longitudinal and transverse thermal conductivity. These results are then compared with experimental work [47, 48, 49, 50]. In addition, we have also carried out a detailed investigation of the MET transport properties of a graphene monolayer which is modulated by one-dimensional (1D) weak periodic potential in the presence of a perpendicular magnetic …eld. In the following section, the general formulation of the magnetoelectrothermal transport problem is presented and the calculation of the thermopower and the thermal conductivity

104

of unmodulated graphene as well as graphene subjected to 1D weak periodic modulation is performed. The results for the transport coe¢ cients as a function of the gate voltage (Vg ) for unmodulated graphene are discussed in section 5.3, where we also make a comparison with experimental results. In section 5.4, the results for modulated graphene verses the gate voltage and the external magnetic …eld are presented. The chapter ends with conclusions.

5.2

Thermal Magnetotransport Coe¢ cients

As mentioned in the introduction, corrections to the usual Kubo formula for transport have to be made when studying thermal transport in a magnetic …eld. This was carried out by Luttinger, Smerka, Streda and Oji [83, 82]. We employ the modi…ed Kubo formalism to determine the thermal transport coe¢ cients. Let J1 = Je and J2 = JQ represent the electrical and thermal current densities of a graphene monolayer in equilibrium. The corresponding driving forces are respectively X1 =

1 e

(r

) and X2 = T r

1 T

. Here

=

+ eV with

the chemical

potential and V the scalar potential, e is the charge of an electron and T the temperature of the system. Linear response assumes that [83]: Ji = X j L(ij) Xj

(5.1)

The coe¢ cients L

(ij)

are measurable quantities which are de…ned in terms of correlation func(12)

tions. There are also Onsager relations which specify L

=L

(21)

. These relations are valid

for the choice of currents or forces for which @Sentropy =@t > 0: Here Sentropy is just the part of the entropy which is generated irreversibly. We require that entropy generation be expressed as: X @Sentropy = Ji Xi : @t i (5.2)

From Eq. (5.1) we get the following linear response equations:

(11) (21)

Je = L JQ =

1 (r e 1 (r e

) + ) +

L

(12)

e

(22)

Tr Tr

1 T 1 T

; ;

(5.3) (5.4)

L

e

L e2

105

and the rate of entropy generation is @Sentropy = @t 1 Je (r e ) + T JQ r 1 T : (5.5)

If the system has a temperature gradient rT and no particle current Je = 0, then Eq. (5.3) gives the relationship: (r and for no concentration gradients (r )= TL L

(12) (11)

[r (T )]

(5.6)

), we …nd voltage di¤erence as: 1 L rT: eT L(11)

(12)

rV =

(5.7)

If there is no particle ‡ ow, then JQ = Je and by substituting Eq. (5.6) into Eq. (5.4) we …nd the heat current as: JQ = e2 T 2 n o 3 (21) 2 L 7 rT: (11) 5 L

The thermoelectric coe¢ cient is de…ned as [83]: S= and the thermal conductivity

1 6 (22) 4L

(5.8)

rV rT

(5.9)

is usually written as: J= rT: (5.10) , ther-

Hence, the electrical and thermal transport coe¢ cients: the electrical conductivity mopower S and the thermal conductivity lowing [59, 82, 83, 85, 86], as = L(0) ; S = = 1 e2 T 1 [(L(0) ) eT [L(2)

1

can be obtained from the above expressions, fol-

(5.11) L(1) ] ; (5.12) (5.13)

eT (L(1) S) ]

106

with L( The L

( ) )

=

Z

dE

@f (E) (E @E

) ;

(E):

(5.14)

(

= i+j

2) are, in general, tensors where

= x; y. These phenomenological

( )

transport coe¢ cients satisfy the Onsager relation [83, 85] L Dirac distribution function with

(B) = L

( )

( B). The + 1)]

1

(E) is

the zero-temperature energy dependent conductivity and f (E) =

[exp( EB T k , =

is the Fermi

1

the chemical potential. The quantity xx = (L(0) ) yx is the with

resistivity tensor whose components are = xx yy xy yx .

=

yy =

,

yy

=

xx =

xy

=

yx =

In order to calculate the thermal transport coe¢ cients for a graphene system, we have considered a graphene monolayer in the xy plane with a magnetic …eld B applied along the z direction. In the Landau gauge, the unperturbed single particle Dirac-like Hamiltonian may be written as Ho = vF : ( i}r + eA) : where =f x; yg

(5.15)

represents 2D Pauli matrices, vF = 106 m=s is for fermi velocity of electrons

in graphene and A = (0; Bx; 0) is the chosen vector potential. The normalized eigenfunctions of the Hamiltonian given in Eq. (5.15) are eiky y =p 2Ly l i + xo )=l] ; n [(x + xo )=l] n 1 [(x

n;ky

(5.16)

where

length of two-dimensional graphene system in the y direction. The corresponding eigenvalues p p p are En = }! g n where ! g = vF 2eB=} = vF 2=l is the cyclotron frequency of the Dirac electrons in graphene. In order to investigate the e¤ects of modulation, we express the Hamiltonian in the presence of modulation as H = Ho + V (x). Here, V (x) is the one-dimensional periodic modulation potential along the x axis. It is given by V (x) = Ve cos Kx such that K =

2 a

are the harmonic oscillator wavefunctions centred at xo = l2 ky . Here q } n is the Landau level index., l represents the magnetic length given by eB and Ly is the n (x)

and

n 1 (x)

, a is the

modulation period and Ve is the constant modulation strength. To account for weak modulation, p Ve is taken to be very small as compared to the Fermi energy EF = vF }kF ; where kF = 2 ne

107

is the Fermi wave vector and ne is the density of electrons. Thus, energy eigenvalues for weak electric modulation (Ve order perterbation theory. In the presence of a periodic modulation, there are two contributions to magnetoconductivity: the collisional (hopping) contribution and the di¤usive (band) contribution. Since in the linear response regime, the conductivity tensor is a sum of a diagonal and a non diagonal part: (!) = d EF ) are En;ky = En + Vn;B cos Kx as derived in Chapter 2 using …rst

(!) +

nd (!),

;

= x; y and

d

(!) = nd (!)

dif f

(!) +

col (!),

accounting for both xx di¤usive and collisional contribution whereas and yy is the Hall contribution. Here,

=

col xx

=

col xx

+

dif f yy .

Similar to the conductivity tensors, the diagonal components of the

thermal transport coe¢ cients are evaluated by the following expressions: L( ) = L( ) col = L( ) col xx xx yy L( ) = L( )dif f + L( ) col : yy yy yy The …nite temperature conductivity components (5.17) (5.18)

have been determined in Chapter 2 for

scattering by random screened charged impurities having density NI with impurity broadenp 2 ing [73]. The screened potential (in Fourier space) is Uo = 2 e2 =" q 2 + ks ; which is valid for small wave vectors, q

( )

ks , ks being the inverse screening length and " the dielectric con-

stant. Therefore, from Eq. (5.14), we obtain the …nite-temperature phenomenological transport coe¢ cients L as L( )dif f yy X e2 u [Vn;B ]2 [E =2 h} n=0 1

] [

@f (E) ]E=En ; @E

(5.19)

L( ) col xx and e2 l2 X = h a

1 a=l2 Z

e2 NI U 2 X n h a n=0 1

a=l2 Z 0

dky [E

] fn;ky (1

fn;ky );

(5.20)

En+1;ky

L( ) yx

dky

1 En+1;ky En;ky =}! g

2

n=0 0

En;ky

Z

dE [E

] [

@f (E) ] @E

: (5.21)

En;ky

108

where

is the scattering time. Here, we have taken the scattering time to be independent of

Landau-level index n. The components of thermopower are given by the following equations: Sxx = 1 eT 1 eT yy L(1) + xx L(1) + yy

1 yx L(1) ; yx L(1) yx

(5.22)

Syy = and Sxy = Syx =

xx

1 yx (5.23)

1 eT 1 eT

yy

( L(1) ) + yx L(1) + yx

1 yx L(1) ; yy L(1) : xx

(5.24) (5.25)

xx

1 yx The components of the thermal conductivity are given by = 1 h (2) Lxx e2 T n eT L(1) Sxx xx L(1) Syx yx oi ; (5.26)

xx

yy

=

1 h (2) Lyy e2 T

n oi eT L(1) Sxy + L(1) Syy yx yy

(5.27)

and

n oi 1 h (2) (1) (1) Lyx eT Lxx Sxy Lyx Syy ; xy = 2 e T n oi 1 h (2) Lyx eT L(1) Sxx + L(1) Syx : yx = 2 yx yy e T

(5.28) (5.29)

5.3

Magnetothermoelectric Transport in Unmodulated Graphene

, we determine the phenomenological transport coe¢ cients

From the electrical conductivity

( )

Lyx employing Eqs. (5.19), (5.20) and (5.21). Employing these, the components of thermopower and thermal conductivity are numerically evaluated using Eqs. (5.22) through (5.29). The results for the magnetoelectrothermal transport properties of an unmodulated graphene monolayer as a function of the gate voltage are presented in this section. The number density ne is related to the gate voltage Vg through the relation ne = an electron, o o

Vg =te, where e is the charge of

and

= 3:9 are the permittivities for free space and the dielectric constant for 300nm) [5].

graphene on SiO2 substrate respectively. The thickness of the sample is t( 109

The components of thermopower (S ) and thermal conductivity (

), as the system moves

away from the charge neutral point on the electron side on changing the gate voltage, are shown in Figure (5-1) at a magnetic …eld of one Tesla. The lattice temperature of 10K and mobility of =

EF 2 evF

= 20m2 =V s [87] is chosen. The scattering time is related to the mobility as can be expressed in terms

in a graphene monolayer [88]. The impurity broadening (E) as (E) = 2 Im [

(E)] and also (E) = }= [89]. We use the p expression for Im [ (E)] derived in [73] to …nd = }(}! g )2 =(4 EF ). The electron number p p density is ne = 7:19Vg 1014 m 2 and Fermi energy is EF = ~vF 2 ne = 44:3 Vg meV .

2 The impurity density is related to through NI = l2 2 =Uo [34]. The scattering time of p p = 4:431 Vg 10 14 s, impurity broadening = 5:934 B=( Vg )meV and impurity density

of the self energy

NI =

2:46

1014 m

2

were employed in this part of work [34, 61, 67, 33, 87, 88, 89].

Moreover, the same study is carried out at a higher magnetic …eld of 8:8T for graphene with mobilities of = 1m2 =V s and = 20m2 =V s respectively and the results are shown in

Figure (5-2). Since Sxx and Syy are identical so only Sxx is depicted in these …gures. The longitudinal coe¢ cient of thermopower (Sxx ) is equivalent to the Seebeck coe¢ cient and our results provide a qualitative as well as quantitative understanding of the overall behavior of the observed Sxx (Vg ). The Sxx can have either sign and it is negative in our case since the charge carriers are electrons in this range of Vg . The transverse component of thermopower (Syx ) is also known as the Nernst signal and it arises due to the presence of the perpendicular magnetic …eld as the Lorentz force bends the trajectories of the thermally di¤using carriers. It can be p observed from Figure (5-1a) and Figure (5-2a) that Sxx follows 1= Vg (with Vg / ne ). Similar behavior of Sxx is observed in experiments [47, 48, 49]. Notice that we have presented results for di¤usive thermopower and we have ignored the phonon contribution to thermopower due to weak electron phonon coupling in graphene at low temperatures [48, 51]. Sxx and Syx ( Syx = Sxy ) both show Shubnikov-de Haas (SdH) type oscillations in the Landau quantizing

magnetic …eld. At the lower magnetic …eld of 1 Tesla (Figure (5-1)), the oscillations are more closely spaced since the separation between the Landau levels, which is proportional to the magnetic …eld strength, is smaller compared to the results for the higher magnetic …eld of 8.8 Tesla, (Figure (5-2)). We observe in Figures (5-1a) and (5-2a) that Sxx approach its minimum value at those values

110

6.0 4.0 2.0

(a) Syx

Sµν(µV/K)

0.0 -2.0 -4.0 -6.0 -8.0 -10.0 -12.0 10 4.0 20 30 40 50

Sxx T = 10K, B = 1T, µ = 20 m2 /Vs Vg(Volts) κyx/ 20 κxx

(b)

κµν(x10− 6(eV)2/hK)

3.0

2.0

1.0

0.0 10 20

Vg(Volts)

30

40

50

Figure 5-1: The components of (a) thermopower and (b) thermal conductivity of graphene versus gate voltage Vg at a …xed magnetic …eld B = 1T and temperature T = 10K:

111

20 10

Syx

Sµν( µV/K)

0 -10 -20 -30 -40 -50 1.6 1.4 20 30

Sxx (a) Vg(Volts) 40

50

(b) κyx/5 κxx

κµν(x10 − 6(eV)2/hK)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 20

30

Vg(Volts) 40

50

Figure 5-2: The components of (a) thermopower and (b) thermal conductivity of graphene with mobility 20m2 =V s (solid curve) and 1m2 =V s (broken curve) versus gate voltage Vg at a …xed magnetic …eld B = 8:8T and temperature T = 10K.

112

of Vg where there are boundaries of Landau Levels and no carriers are available to participate in transport. The peaks of Sxx are observed at the centre of Landau levels. However, Syx approaches zero at those values of Vg where there are boundaries of Landau levels as well as at the centers of Landau levels. With the increase in Vg and hence an increase in ne , higher Landau levels are occupied. The oscillations in Sxx and Sxy are damped as we increase Vg . The reason for this is that higher Vg corresponds to higher values of the Fermi energy and if the Fermi energy is much larger that the Landau level separation, Landau quantization e¤ects are lost. At B = 8:8T; (Figure (5-2a)), the oscillations in Sxx and Sxy show that the width of the peaks broaden compared to those for smaller magnetic …eld of B = 1T . At lower magnetic …eld, the separation between the Landau levels is smaller compared to higher …elds with the result that the peaks of Sxx are more closely spaced. Furthermore, the overall magnitude of Sxx and Sxy increases with increasing magnetic …eld strength (See Figure (5-1a) and Figure (5-2a)). In these …gures, we also present thermal conductivity as a function of the gate voltage. The longitudinal thermal conductivity xx shows oscillating behavior which damps out as Vg

increases, where Landau quantization e¤ects become less signi…cant. However, the transverse component of thermal conductivity yx rises monotonically with Vg as shown in Figure (5-1b)

and Figure (5-2b). At the higher magnetic …eld, quantum Hall steps have begun to appear. The behavior of longitudinal and transverse thermal conductivity follows that of the corresponding components of electrical conductivity. At higher magnetic …elds, the splitting of the peaks in the longitudinal thermal conductivity xx is seen in Figure (5-2b) which was also observed in

[84] where it is shown that the splitting occurs in such a way that they produce antiphase oscillations with respect to the electric one and lead to the violation of the Wiedemann-Franz law. For the unmodulated case, we …nd that Syx = Sxy , xx yy ( ) ( )

= yy xx

, Lyy = Lxx and using Eq. (5.22) through Eq. (5.29) xy =

and

=

yx .

Therefore, only

xx

and

yx

are shown

in the …gures. We …nd that the results for magnetothermal power obtained in our work at B = 8:8T with T = 10K are in good agreement, both qualitative and quantitative, with the experimental results obtained in [48, 49], see Figure (3) of [48]. These results do indicate that scattering from screened charged impurities is the dominant scattering mechanism required to explain the experimental results. We must add that our quantitative results for Syx depend

113

strongly on the mobility of the graphene system. The heights of the peaks of the Seebeck coe¢ cient increase with the increase in impurity in the sample. However, from Eq. (5.22) we can see that the major contribution to the thermopower comes from the second term as

(1) yx Lyx (1) yy Lxx

and at low temperatures the heights of the peak values of thermopower kB en ln2

approach the quantized value of

[48, 81, 90]. However, this universal result is only

obtained for charge carriers in the absence of impurity scattering. In fact, in the presence of impurity scattering we …nd a higher value of nSxx the universal value of kB en ln2

71 V K

1,

see Figure (5-2), compared to

60 V K

1

in the absence of impurity scattering.

5.4

Magnetothermoelectric Transport in Periodically Modulated Graphene

Now we consider the e¤ects of modulation. The one-dimensional modulation broadens the sharp Landau levels (LLs) into bands and gives rise to an additional di¤usive (or band) contribution to transport. This additional contribution is absent without modulation. We now focus on the modulation induced changes in the thermal magnetotransport coe¢ cients of graphene. Therefore, in the …rst part we present the thermopower and thermal conductivity of modulated graphene with mobility of 20m2 =V s versus the gate voltage as depicted in Figure (5-3) and Figure (5-5) respectively. These results are for a constant external magnetic …eld of B = 1T applied perpendicular to the graphene sheet, with electric modulation of strength Ve = 3meV applied in the x-direction at a temperature of T = 10K. In this case }! g = 36:3meV , such that Ve

1:3 = p meV and Vg

}! g to satisfy the requirements of weak modulation. The

period of modulation is a = 382nm. The results for Sxx and Syy are identical, so only Sxx is shown in these …gures. The amplitude of oscillations in Sxx ( Sxx ) is greater than that of Sxy ( Sxy ) which damps out with increasing gate voltage (Vg ). Both Sxx and Sxy show SdH-type oscillations and it veri…es that the system is Landau quantized. The modulations e¤ects are apparent in Sxx and Sxy which shows modulation of SdH-type oscillations and Figure (5-3). yx Sxx

Sxy ,

is greater than

xx

and

yy

as shown in Figure (5-5). These modulation

induced e¤ects on thermal transport coe¢ cients can be highlighted by calculating the di¤erence between the thermopower/thermal conductivity of modulated and unmodulated graphene.

114

12.0 8.0 4.0

(a)

Sxx(µV/K)

0.0 -4.0 -8.0 -12.0 -16.0 -20.0 10 4.0 20 Electric modulation Ve = 3meV, T = 10K a = 382nm, B = 1T, µ = 20m2/Vs

Vg(Volts)

30

40

50

(b)

2.0

Sxy(µV/K)

0.0

-2.0

-4.0 10 20

Vg(Volts)

30

40

50

Figure 5-3: (a) Longitudinal thermopower and (b) transverse thermo-power of modulated graphene versus gate voltage Vg at a …xed magnetic …eld B = 1T and temperature T = 10K.

115

12

(a)

8

∆Sxx(µV/K)

4 0 -4 -8 -12 10 2.0 20 30 40 50

Vg(Volts) (b)

1.0

∆Sxy(µV/K)

0.0 -1.0 -2.0 -3.0 -4.0 10 20 30 40 50

Electric modulation Ve = 3meV, T = 10K a = 382nm, B = 1T, µ= 20 m2/Vs

Vg(Volts)

Figure 5-4: The correction to (a) Longitudinal thermopower and (b) transverse thermopower of modulated graphene versus gate voltage Vg at a …xed magnetic …eld B = 1T and T = 10K.

116

κxx(x10− 6(eV)2/hK)

1.8 1.2 0.6 0.0

(a)

Electric modulation Ve = 3meV, T = 10K a = 382nm, B = 1T, µ = 20m2/Vs

κyy(x10− 6(eV)2/hK)

1.8

10

20

(b)

Vg(Volts)

30

40

50

1.2 0.6 0.0

κyx(x10− 6(eV)2/hK)

80 60 40 20 0

10

20

(c)

Vg(Volts)

30

40

50

10

20

Vg(Volts)

30

40

50

Figure 5-5: The comopents of thermal conductivity (a) xx , (b) yy and (c) yx of modulated graphene versus gate voltage Vg at a …xed magnetic …eld B = 1T and temperature T = 10K:

117

−6 2 ∆κyy(x10 − 6 (eV)2/hK) ∆κxx(x10 (eV) /hK)

0.20

(a)

0.10 0.00

-0.10 -0.20 0.60 10 20

(b)

0.40 0.20 0.00 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 10 10

Vg(Volts)

30

40

50

Electric modulation Ve = 3meV, T = 10K a = 382nm, B = 1T, µ = 20m2/Vs

∆κyx(x10− 6(eV)2/hK)

20

(c)

Vg(Volts)

30

40

50

20

Vg(Volts)

30

40

50

Figure 5-6: The correction to thermal conductivity due to 1D electric modulation; (a) xx and (b) yy (c) yx of modulated graphene versus gate voltage Vg at a …xed magnetic …eld B = 1T and temperature T = 10K.

118

The contribution of modulation to thermopower conductivity (Vg ) = (Ve )

S (Vg ) =

S (Ve )

S (0) and thermal

(0) are shown in Figure (5-4) and Figure (5-6) respec-

tively. These …gures clearly show the modulation of SdH oscillations in both the thermopower and the thermal conductivity. For an unmodulated case graphene xx xx

=

yy ,

however for modulated xx xx

6=

yy

and this expected behavior is seen in Figure (5-6) where yy 6=

yy . yx

The one-dimensional modulation gives a positive contribution to oscillate around zero. yy yy xx ,

while

and

because

xx

has only collisional contribution, whereas

has large contribution from band conduction along with the collisional part.

We also show the results when the magnetic …eld is varied and the electron density is …xed which corresponds to a gate voltage of Vg = 4:39V . The Fermi energy of p the system is EF = }vF 2 ne 92:3meV . We have taken the mobility of 20m2 =V s[87]. Hence p scattering time is taken to be = 1:86 10 12 s. Impurity broadening = 0:633 BmeV and impurity density NI = 1:23 1013 m

2

at ne = 3:16 1015 m

2

were employed in this part of the work. The strength of

the electrical modulation is taken to be Ve = 2meV with period a = 382nm and temperature T = 10K. The thermopower and the change in thermopower due to modulation in Figure (5-7) as a function of the magnetic …eld in the units of S (B) are shown

kB =e. When B is less than

0:2T Weiss oscillations are observed whereas SdH type oscillations dominate at higher magnetic …elds. It is also seen that these oscillations in Sxx are 90o out of phase with those in Sxy : The amplitude of the oscillations in Sxx Sxy and they are 90o out of phase. Again for B greater

than 0:2T the oscillations appear as envelopes of SdH oscillations. The di¤erent components of the thermal conductivity tensor and the correction to it due to one-dimensional modulation are shown in Figure (5-8). The magnetic …eld dependence of the thermal conductivity tensor is similar to that of the electrical conductivity tensor obtained in [73]. Electric modulation acting on the system results in a positive contribution to zero. In Figure (5-8) we see that yx yy yy yy

whereas xx ;

xx

and yx yx

oscillate around xx xx

such that xx ,

and

are 180o has only

out of phase from each other. We …nd that a collisional contribution, while yy is greater than

because

has large contribution from band conduction along with

the collisional part. The ‡ band condition for electrically modulated graphene is 2Rc = i 1 , at a 4 p with i = 1; 2; 3; :::, where Rc = l2 kF and 2 ne is the cyclotron orbit as given in Eq. (2.20)

119

0.16

(a)

0.12 0.08 0.04 0.00 -0.04

Electric modulation T = 10K, µ = 20 m2/Vs a = 382nm, ne = 3.16x1015m− 2

Sµν(-kB/e)

Sxx

Sxy

0.0 0.2 0.4

0.08

B(Tesla)

0.6

0.8

1.0

(b)

0.04

∆Sµν(-kB/e)

∆Sxx

0.00

∆Sxy

-0.04

-0.08 0.0 0.2 0.4

B(Tesla)

0.6

0.8

1.0

Figure 5-7: The components of (a) thermopower and (b) the correction to power due to 1D electric modulation versus magnetic …eld at gate voltage Vg = 4:39V and temperature T = 10K for Ve = 2meV (solid curve) and Ve = 3meV (broken curve) respectively.

120

800 700 600

(a)

Electric modulation T = 10K, µ = 20m2/Vs a = 382nm, ne = 3.16x1015m− 2

κµν (kB2K/h)

500 400 300 200 100 0 80 0.0 60 40 0.2 0.4

κyx/10

κxx

κyy

(b)

B(Tesla)

0.6

0.8

1.0

∆ κyy

∆κµν (kB2K/h)

20 0 -20 -40 -60 -80 0.0 0.2 0.4 0.6 0.8 1.0

∆ κxx ∆ κyx/4

B(Tesla)

Figure 5-8: The components of (a) thermal conductivity and (b) the correction to thermal conductivity due to 1D electric modulation versus magnetic …eld at gate voltage Vg = 4:39V and temperature T = 10K for Ve = 2meV (solid curve) and Ve = 3meV (broken curve) respectively

121

[73]. From …gures Figure (5-7b) and (5-8b) we observe that the zeros in

S (B) and

(B)

are in close agreement with the values of magnetic …eld predicted by the ‡ band condition. at With the increase in modulation strength Ve at a …xed gate voltage, the modulation of SdH oscillations increases. The amplitude of the oscillations in Sxx ( Sxx ) increases with increase in modulation whereas the amplitude of the oscillations in Sxy is weakly dependent on the yy modulation strength. Similarly the amplitudes of the oscillations in with increase in modulation. However, as compared to yy xx

and

yx

increase

is weakly dependent on the modulation strength

and

yx .

In order to clearly observe the e¤ects of modulation we have

to consider higher mobilities or use a higher amplitude of modulation Ve . In this regard, we have to bear in mind the condition of 20m2 V

1s 1

Ve

}! g for weak modulation. Although the mobility

used for the modulated system is still only achievable for suspended graphene

where the primary scattering mechanism is short range scattering due to sharp defects and not screened charged impurity scattering, the results obtained are qualitatively correct. As mentioned above, we can consider lower mobility for our work but that would require higher modulation amplitude in order to observe its e¤ects; qualitatively the results would be the same. Thermal magnetotransport of magnetically modulated graphene can also be studied. The electrical conductivity , calculated in Chapter 3, can be used to determine the phenom( )

enological transport coe¢ cients Lyx , of a graphene monolayer subjected to one-dimensional magnetic modulation, employing Eqs. (5.19), (5.20) and (5.21) respectively. Then the components of thermopower and thermal conductivity can be numerically evaluated using Eqs. (5.22) through (5.29). It is expected that the thermal power and corrections to the thermal power due to the electric modulation will be greater than those of magnetic modulation of a graphene monolayer, where the strength of both modulation is kept equal i.e. Ve = }! m . In both cases of modulations Sxx = Syy . However Sxx of electrically modulated graphene is expected to be 90o out of phase where as Sxy is in phase with that of magnetically modulated graphene. The thermal conductivity of magnetically modulated graphene is expected to be greater than of magnetically modulated graphene are xx that of electrically modulated graphene and

greater than those of the electrically modulated graphene with

and

yy

of magnetically

modulated graphene as a function of magnetic …eld are expected to be 180o where as that of

122

yx

are 90o out of phase. The magnetic …eld dependence of the thermal conductivity tensor

is expected to be similar to the electrical conductivity tensor obtained in Chapter 3.

5.5

Conclusions

In this chapter we have studied magnetothermoelectric transport in graphene in the linear response regime using the modi…ed Kubo formalism appropriate for thermal transport in a magnetic …eld. Results are presented for both unmodulated graphene as well as graphene that is weakly modulated by an electric modulation. We take into account scattering from screened charged impurities and our results indicate that these provide the most dominant scattering mechanism at low temperatures. The thermopower, the Seebeck coe¢ cient and the Nernst coe¢ cient are determined as a function of the gate voltage. Furthermore, we also determine the magnetothermal conductivity tensor, both the longitudinal and the transverse (Hall) components. For unmodulated graphene we were able to make a comparison of the thermopower with experimental results and …nd that they are in good agreement, both qualitative as well as quantitative, with experimental results. In the case of modulated graphene, we focus on the modulation induced e¤ects that appear as commensurability (Weiss-type) oscillations in the magnetothermoelectric coe¢ cients. The results are presented as both functions of the gate voltage and the magnetic …eld.

123

Chapter 6

Summary

This research was motivated by the realization that modulation of a graphene sheet has an important role in electron transport in graphene. Modulation of graphene can occur in various ways: periodic ripples in suspended graphene that is placed in a perpendicular electric …eld can result in a periodic potential, it can also be induced by interaction with a substrate, with adatom deposition and even with applied electric or magnetic gates to the substrate. The results obtained in this thesis apply to all these cases to a varying degree. The thesis was divided into four major parts. In the …rst part, a detailed investigation of the electrical transport properties of a graphene monolayer modulated by a one-dimensional weak periodic electric potential in the presence of a perpendicular magnetic …eld B was done. Then a detailed study of the transport properties of a graphene monolayer which was subject to simultaneous one dimensional periodic magnetic and electric modulations in the presence of a perpendicular magnetic …eld was performed. Thermodynamic study of an electrically modulated graphene monolayer subjected to a perpendicular magnetic …eld was conducted in the third part. Finally, a comprehensive formulation and study of magnetothermoelectric transport in a graphene monolayer, unmodulated as well as modulated, was carried out within the linear response regime using the modi…ed Kubo formalism for thermal transport in a magnetic …eld. First chapter included an overview of the electronic properties of a graphene monolayer. It also included motivation and objectives of this research work. Chapter 2 provided electrical magnetotransport in periodically modulated graphene. The e¤ects of a weak periodic electric modulation on the conductivities of a graphene monolayer subjected to a perpendicular mag124

netic …eld investigated were investigated. As a result of modulation a new length scale, period of modulation, enters the system leading to commensurability oscillations in the di¤usive, collisional and Hall contributions to conductivities/resistivities. These modulation induced e¤ects on graphene magnetotransport were discussed in detail in this chapter. In Chapter 3, we had determined the transport properties by calculating the magnetoconductivity tensor for a graphene monolayer that was subject to a weak periodic magnetic modulation in the presence of an external magnetic …eld. The results obtained were compared with those of an electrically modulated graphene monolayer. We found that the period of commensurability oscillations in the transport coe¢ cients is the same for both the electrically and the magnetically modulated graphene. Furthermore, for equal modulation strengths, the oscillations are greater in amplitude for magnetically modulated graphene and =2 out of phase with the electrically modulated graphene. We had also determined the collisional, di¤usive and Hall conductivity when electric and magnetic modulations were simultaneous applied to the graphene monolayer. For in-phase modulations, the position of the oscillations in the transport coe¢ cients is in‡ uenced by the comparative strength of these modulations. In addition, for out-of-phase modulated system, oscillations occur at the same position as in the case of magnetically modulated graphene and the Weiss oscillations can be completely suppressed by choosing particular values of the relative strengths of the two modulations. In Chapter 4, we studied the thermodynamic properties of a graphene monolayer subjected to one-dimensional weak electric modulation along with a magnetic …eld. Comparison of the results obtained were done with those of a conventional 2DEG system realized in semiconductor heterostructures. As a result of the commensurability of modulation period and the cyclotron orbit radius, Weiss type and dHvA type oscillations were re‡ ected in all the thermodynamic quantities under consideration in this work for the two systems. However, these e¤ects are more pronounced in graphene in the sense that the oscillations in the thermodynamic quantities are more robust against temperature as well as having a higher amplitude than in a conventional 2DEG system. This di¤erence arises due to the di¤erent nature of the quasiparticles (Dirac electrons and standard electrons) in the two systems with the result that the energy spectrum for the quasiparticles in the two systems as well as their Fermi velocities are di¤erent. On the

125

basis of analytic asymptotic expressions, we were able to determine the critical temperature and the critical magnetic …eld for damping of magnetic oscillations in the thermodynamic properties of graphene under weak periodic modulation. In Chapter 5, we investigated magnetothermoelectric transport in graphene in the linear response regime using the modi…ed Kubo formalism appropriate for thermal transport in a magnetic …eld. Results were presented for both unmodulated graphene as well as graphene that was weakly modulated by an electric modulation. We took into account scattering from screened charged impurities and our results indicated that these provide the most dominant scattering mechanism at low temperatures. The thermopower, the Seebeck coe¢ cient and the Nernst coe¢ cient were determined versus the gate voltage. Furthermore, we also determined the magnetothermal conductivity tensor, both the longitudinal and the transverse (Hall) components. For unmodulated graphene we were able to make a comparison of the thermopower with experimental results and …nd that they are in good agreement with experimental results. In the case of modulated graphene, our focus was on the modulation induced e¤ects that appear as commensurability (Weiss-type) oscillations in the magnetothermoelectric coe¢ cients. In terms of future work, we can study magnetotransport in a two-dimensional graphene superlattice. This is relevant to corrugated graphene and the expected Hofstadter-type miniband structure can lead to interesting consequences for transport. Further, the thermodynamic properties and thermal transport in a two-dimensional graphene superlattice subjected to perpendicular magnetic …eld can also be studied. As the work done in this thesis is for doped graphene with valence band states assumed to be completely …lled further improvements in this work can be carried out by considering undoped graphene and hence considering both the contributions of electrons and holes states. Another area in which we can extend the work presented here is the study of electrical and thermal magnetotransport of modulated graphene in the presence of microwave radiation. Moreover, future work can also be done on electrical as well as thermal magnetotransport of strained graphene under periodic modulations.

126

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133

Appendix-A Hall Conductivity of Unmodulated n-doped Graphene

Here we derive the expression for Hall conductivity for an unmodulated graphene at low temperatures and high magnetic …elds. When the modulaton is absent, Eq. (2.58) becomes e2 l2 X h a h n=0 1 2X 1 a=l2 Z n

= 0 and fn;ky

fn ,

yx

=

n=0 0

dky p

fn n+1

2

fn+1 p

n

2

(A.1)

= = = = = =

1 e2 X

p

fn n+1

fn+1 p

n

n=0 1 2X e n=0 1 2X e

e h h h

fn fn+1 p p 2n + 1 2 n + 1 n fn fn+1 p p 2n + 1 2 n + 1 n fn + 1)2 fn+1 ) fn+1 p p 2 n+1 n

2

p p 2n + 1 + 2 n + 1 n p p 2n + 1 + 2 n + 1 n p p 2n + 1 + 2 n + 1 n r 1 1+ n !

e h e h

n=0 (2n 1 2X

(fn (fn

2n + 1 + 2n

n=0 1 2X

fn+1 ) (4n + 2) fn+1 ) n+ 1 2

= 4

n=0 1 2X e

h

(fn

n=0

Further, at low temperatures and high magnetic …eld regime, we may take all fn (n = 0; 1:::N ) equal to unity while fn+1 0 for a certain N . Thus Eq. (2.58) leads to =4 N+ 1 2 e2 h (A.2)

yx

which is the well known result for Hall conductivity in graphene[71, 72].

134

Appendix-B Density of States of an Electrically Modulated Graphene Monolayer

In section 2.3.1 the analytic expression for the density of states (DOS) of a graphene monolayer in the presence of a magnetic …eld subjected to electric modulation was used to quantitatively analyze the results presented in the …gures of Chapter 2. Here, we will explicitly derive the density of states expression, Eq. (2.59) which hold in large n …eld and at …nite energies E >> }! g . We consider monolayer graphene in the presence of a uniform quantizing magnetic …eld B = B z with an additional one-dimensional weak periodic modulation potential. The energy ^ spectrum in the quasi classical approximation, i.e. when many Landau bands are …lled may be written as En;xo = where Vn;B =

Ve u=2 [L (u) n 2 e

limit for small magnetic

p

n}! g + Vn;B cos Kxo with Ln (u), Ln

1 (u)

(B.1) the Laguerre polynomials and Using the asymptotic p cos(2 nu 4 ) and taking

+ Ln

1 (u)]

u = K 2 l2 =2. For large n; Ln (u)

Ln

1 (u)

and Vn;B = Ve e u=2 L (u) n

u=2 L (u). n 1 p nu

expression for the Laguerre polynomials [63]; e

p 1 the continuum limit n ! 2 ( vlE} )2 , where vF = ! g l= 2 we get F Vn;B = Ve

1=2

!p

1 22 E 2 K l ( ) 2 }! g

1=4

cos

p

2Kl

E }! g

4

:

(B.2)

To obtain a more general result which will lead to the result that we require as a limiting case we consider the broadening of the Landau levels induced by impurities. The self energy may be expressed as (E) = which yields (E) =

2 o

X Z dxo a E n a 0

1 En;x0 (E)

(B.3)

Za

0

dxo X a 1E

1

2 o

(E)

Vn;B cos Kxo

p

n}! g

:

(B.4)

o

represents the broadening of the Landau levels induced by impurities. Also, the density of

135

states is related to the self energy through D(E) = Im (E)

2 l2 2 o 1 P

:

(B.5)

The residue theorem has been used to sum the series at all poles of f (n)g [91]. Here f (n) =

1 P bp c d n

f (n) =

2, o

1

fSum of residues of (cot n)f (n) c=E (E) Vn;B cos Kxo

with b =

and d = }! g . The function f (n) has a pole at c2 =d2 and the residue of ( (cot n)f (n)) at the 1 P 2 2 2bc pole is d2 cot( dc ). Hence f (n) = 2bc cot( dc ) and we obtain 2 2 d2

1

1

(E) =

Za

0

dxo 2 a Za

0

2 (E o

(E) Vn;B cos Kxo ) cot (}! g )2 E [E (}! g )2 2f

(E

(E) Vn;B cos Kxo )2 (}! g )2 (B.6)

2 2E o (}! g )2 Separating

dxo cot a

(E) + Vn;B cos(Kxo )g] :

(E) into real and imaginary parts (E) = (E) + i (E) ; 2 (B.7)

Eq. (B.6) takes the form (E) 2 2E o (E) + i = 2 (}! g )2 where u= E [E (}! g )2 2f (E) + Vn;B cos(Kxo )g] v= (E)E (}! g )2 (B.9) (B.10) Za

0

dxo sin 2u + i sinh 2v a cosh 2v cos 2u

(B.8)

136

Im

(E)

=

2 2E o (}! g )2 2 2E o (}! g )2

Za

0

sinh 2v dxo a cosh 2v cos 2u dxo a ( 1+2

1 X k=1

=

Za

0

cos(2ku) exp( 2kv) :

)

(B.11)

E If we de…ne dimensionless variables " = ( }!g )2 ,

=

E (}! g )2

and vB =

2VB E (}! g )2

the density of states

is obtained as 8 1 a < XZ dxo D(E; VB ) = Do (E) 1 + 2 cos[2 k(" : a k=1 0 2E . (}! g )2 l2

where Do (E) =

Taking Kxo = t in the above expression results in

9 = vB cos Ko x)] exp( 2 k ) ;

(B.12)

Solving the integral yields

8 2 1 < X 1 Z D(E; VB ) = Do (E) 1 + 2 cos[2 k(" : 2 k=1 0

9 = vB cos t)]dt exp( 2 k ) : ; )

(B.13)

D(E; VB ) = Do (E) 1 + 2

(

1 X k=1

cos(2 k")Jo (2 kvB ) exp( 2 k ) :

(B.14)

137

Appendix-C Density of States of Graphene under Electric and Magnetic Modulation

Here, we derive the expression for the density of states of a monolayer graphene, in a uniform quantizing magnetic …eld B = Bo z subjected to both periodic electric and magnetic modulation ^ potentials, used in Chapter 3. The energy spectrum in the quasi classical approximation, i.e. when many Landau bands are …lled of such a system may be written as En;xo = p

T n}! g + Vn;B (xo )

(C.1)

T T where Vn;B (xo ) = (Vn;B +Un;B ) cos Kxo , when the two modulations are in phase and Vn;B (xo ) =

Vn;B cos Kxo + Un;B sin Kxo when the two modulations are out of phase. Here Vn;B = }! m 2Kln e p Ve u=2 [L (u) n 2 e

+ Ln

1 (u)]

is due to the electric modulation and Un;B =

1 (u)

u=2 [L n 1 (u)

Ln (u)] is due to the magnetic modulation with Ln (u) , Ln

the Laguerre polynomials and u = K 2 l2 =2. For large n, the asymptotic expression for the p Laguerre polynomials is [63]; e u=2 Ln (u) ! p 1 cos(2 nu 4 ) and in the continuum limit p nu p n ! 1 ( vlE} )2 , where vF = ! g l= 2 we obtain 2 F Vn;B v u u V B = Ve t p 2 cos p 2Kl E }! g (C.2)

E Kl }!g

4

Un;B

To obtain a more general result which will lead to the result that we require as a limiting case we consider impurity induced broadening of the Landau levels. The self energy may be expressed as (E) = which yields (E) =

2 o

v p u p u 2 2 E UB = }! m t sin( 2Kl E }! g Kl }!g

4

):

(C.3)

X Z dxo a E n a 0

1 En;x0

(E)

(C.4)

Za

0

dxo X a 1E

1

(E) 138

2 o T (x ) Vn;B o

p

n}! g

:

(C.5)

o

is the broadening of the levels due to the presence of impurities. The density of states is

related to the self energy through D(E) = Im (E)

2 l2 2 o 1 P

:

(C.6)

The residue theorem has been used to sum the series at all poles of f (n)g [91]. Here f (n) =

1 P bp c d n

f (n) =

2, o

1

fSum of residues of (cot n)f (n) c=E (E)

T Vn;B (xo ) and

with b =

d = }! g . The function f (n) has a pole at c2 =d2 and the residue of ( (cot n)f (n)) at the pole 1 P 2 2 2bc is d2 cot( dc ). Hence f (n) = 2bc cot( dc ) and we obtain 2 2 d2

1

1

(E) =

Za

0

dxo 2 a Za

0

2 (E o

(E) (}! g )2

T Vn;B (xo ))

cot

(E

(E)

T Vn;B (xo ))2

(}! g )2

!

(C.7)

2 2E o (}! g )2 Separating

dxo cot a

E [E (}! g )2

2f

T (E) + Vn;B (xo )g] :

(E) into real and imaginary parts (E) = (E) + i (E) ; 2 (C.8)

Eq. (C.7) takes the form (E) 2 2E o (E) + i = 2 (}! g )2 where u= E E (}! g )2 v= 2

T (E) + Vn;B (xo )

Za

0

dxo sin 2u + i sinh 2v a cosh 2v cos 2u

(C.9)

;

(C.10) (C.11)

(E)E (}! g )2

139

and 2 2E o (}! g )2 2 2E o (}! g )2 Za

0

Im

(E)

=

dxo sinh 2v a cosh 2v cos 2u X dxo (1 + 2 cos(2ku) exp( 2kv): a k=1 1

=

Za

0

(C.12)

E If we de…ne dimensionless variables " = ( }!g )2 ,

=

E (}! g )2

and vB =

2VB E (}! g )2

and uB =

2UB E the (}! g )2

density of states is obtained as 8 1 a < XZ dxo = Do (E) 1 + 2 cos [2 k(" : a k=1 0 k=1 0

T D(E; VB )in

phase

T D(E; VB )outof

phase

where Do (E) =

2E . (}! g )2 l2

8 1 a < XZ dxo = Do (E) 1 + 2 cos[2 k(" : a 8 2 1 < X 1 Z = Do (E) 1 + 2 cos[2 k(" : 2 k=1 0

9 = (vB + uB ) cos Ko x)] exp( 2 k ) ; vB cos Ko x

If we let Kxo = t in the above expression, the result is

9 = uB sin Ko x)] exp( 2 k ) ; (C.14)

(C.13)

T D(E; VB )in

phase

and

9 = (vB + uB ) cos t]dt exp( 2 k ) ; ;

(C.15)

T D(E; VB )outof

phase

To get an expression of density of states of magnetically modulated graphene we take vB = 0 in above equations and then executing the integrals yields (

1 X k=1

8 2 1 < X 1 Z cos[2 k(" = Do (E) 1 + 2 : 2 k=1 0

vB cos t

9 = uB sin t)]dt exp( 2 k ) : ; (C.16)

D(E; UB ) = Do (E) 1 + 2

cos(2 k")Jo (2 kuB ) exp( 2 k ) :

)

(C.17)

140

Appendix-D Asymptotic Expression for the Density of States of Electrically Modulated Graphene

Here, we derive the asymptotic expression for the density of states; of a graphene monolayer in the presence of a uniform quantizing magnetic …eld B = B z and a weak periodic modulation ^ potential; appearing as Eq. (4.18) in Chapter 4. In the case of large collision broadening, >>

}! g ; we can expand with respect to the small quantity exp( v) and solve Eq. (B.8) by iteration. Up to …rst order, we obtain (E) 2 2E o 1 + 2 exp = 2 (}! g )2 1 where e eE

cos2

4 2E2 2 o cos (}! g )4 p E 2 Kl }! g 4 p !3 2 : }! g

2 E2 (}! g )2 ; (D.1)

=

a Ve2 l

(D.2)

Using Eq. (D.1) in Eq. (B.5), an expression for density of states may be obtained as

D(E) =

A 2E 4 E2 2 2 E2 o 1 + 2 exp cos l2 (~! g )2 (}! g )4 (}! g )2 p E 2 1 2 Kl : e E cos }! g 4

(D.3)

In the limit of vanishing impurity potential, considered in this work, we obtain

D(E) =

A E 1 + 2 cos 2 (}! )2 l g

2 E2 (}! g )2

1

eE

cos2

p

2

E Kl }! g

4

:

(D.4)

141

142

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