Author’s Accepted Manuscript

Electronic heat capacity and conductivity of gappedgraphene

Hamze Mousavi, Jabbar Khodadadi

PII: S1386-9477(13)00049-0DOI: http://dx.doi.org/10.1016/j.physe.2013.02.015Reference: PHYSE11182

To appear in: Physica E

Cite this article as: Hamze Mousavi and Jabbar Khodadadi, Electronic heat capacity andconductivity of gapped graphene, Physica E, http://dx.doi.org/10.1016/j.physe.2013.02.015

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Electronic heat capacity and conductivity of

gapped graphene

Hamze Mousavi1,2∗and Jabbar Khodadadi1

1Department of Physics, Razi University, Kermanshah, Iran

2Nano Science and Nano Technology Research Center, Razi University, Kermanshah, Iran

Abstract

It is investigated the effects of orderly substituted atoms on density of states, electronic

heat capacity and electrical conductivity of graphene plane within tight-binding Hamil-

tonian model and Green’s function method. The results reveal a band gap in the density

of states, leading to an acceptor or donor semiconductor. In presence of foreign atoms,

the heat capacity decreases (increases) before (after) the Schottky anomaly. Moreover,

the electrical conductivity of the gapped graphene reduces on all ranges of temperature

compared to the pristine case. Deductively, all changes in the electronic properties depend

on the difference between the on-site energies of the carbon and replaced atoms.

Keywords: Graphene; Tight-binding; Density of states; Heat capacity; Electrical conduc-

tivity.

∗Corresponding author. Tel./fax: +98 831 427 4556. E-mail: hamze.mousavi@gmail.com (H. Mousavi) .

1

Graphene [1] is an allotrope of carbon in two-dimensional (2D) honeycomb lattice, with one

atom thickness and unusual electronic properties, originated from its structure. It’s dispersion

relation between energy and momentum of the electrons presents a characteristic linear behavior

near the K point of the first Brillouin zone (FBZ) [2-4]. Moreover, as a transistor, the current

is modulated by a gate voltage but unable to switch off due to lack of a band gap in the energy

dispersion. Since the electronic properties of graphene are strongly concerned to itinerant

electrons, it will obviously get effect of any modification. Generally, substituting carbon atoms

by another ones will open a gap in the density of states (DOS) and results in an acceptor

(p) or donor (n) type semiconductor [5-17]. For example, by deposition of potassium in ultra

high vacuum, the density of charged impurities on clean graphene have been determined by

Chen et al. [5]. Besides, they explained the conductivity as a function of carrier density.

McCreary et al. [6] investigated the effects of transition metals on the electronic doping and

scattering in graphene using molecular beam epitaxy, suggesting that the room temperature

deposition of transition metals onto graphene produces clusters that dope n−type. Lherbier

et al. [12] considered the electronic transport in boron- and nitrogen-doped graphene using

ab initio calculations. They obtained that the conduction is affected by quantum interference

effects for any doping concentration.

Similar to electronic properties, thermal behavior of graphene is of high importance in theo-

retical research and applications [18-22]. For instance, thermal conductivity (TC) of isotopically

modified graphene has experimentally been measured by Chen et al. [18]. They studied the

TC of isotopically pure graphene via the optothermal Raman technique. The low-energy elec-

tronic structure and the temperature behavior of heat capacity (HC) of graphene strips has

been investigated by Yi et al. [20]. They found that its HC is similar to that of a two-level

system due to the finite width of the conduction and valence bands. The HC of fullerite doped

2

with deuteromethane has been reported in the temperature interval 1.2–120 K by Bagatskii et

al. [22]. They found that the HC has a minimum near T < 2.5K which indicates that the HC

passes through a maximum at T < 2.5K.

In our theoretical effort, DOS, electronic HC and electrical conductivity (EC) of graphene

sheet are considered through tight-binding (TB) Hamiltonian model and Green’s function tech-

nique. Using the band representation of the Green’s function, we calculate the EC of the system

by Kubo formula [23-25]. Then, we compare the mentioned quantities of the pristine graphene

with substituted case, in which one of the carbon atoms in each Bravais unit cell is replaced by

a foreign atom as a charge acceptor or donor.

The second quantization form of the Hamiltonian of TB model on a hexagonal honeycomb

lattice reads as [26],

H = −∑

α,β

Nc∑

i,j=1

tαβij cα†i cβj +

∑

α

Nc∑

i=1

εα0icα†i cαi , (1)

where α and β refer to A or B sub-sites inside the Bravais lattice unit cell (Fig. (1)). We note

that B sites include carbon atoms while A sites would be replaced by foreign ones. Indices i and

j denote the position of the Bravais lattice unit cells in the system, Nc indicates the number of

the Bravais lattice unit cells (the number of modes in the FBZ), tαβij describes the amplitude of

a π electron to hop from sub-site α within the Bravais lattice site i to the sub-site β of the site

j as a nearest-neighbor (NN), while cα†i (cβj ) shows the creation (annihilation) operator of an

electron there. εα0i displays the on-site energy of sub-sites α of Bravais lattice site i. We accept

such units that h = kB = me = e = 1 and take the chemical potential equal to zero which

corresponds to one electron per pz orbital contribution.

Since each Bravais lattice unit cell includes Na = 2 atoms, the Hamiltonian is represented

by a 2 × 2 matrix with following basis kets of Hilbert space: |Φαi 〉 = |ΦA

i 〉, |ΦBi 〉. So, the

3

Green’s function is written as a 2× 2 matrix,

G(i, j; τ) =

GAA(i, j; τ) GAB(i, j; τ)

GBA(i, j; τ) GBB(i, j; τ)

, (2)

with Gαβ(i, j; τ) = −〈T cαi (τ)cβ†j (0)〉 in which τ = ıt implies imaginary time and T points the

time ordering operator. Here, 〈· · ·〉 plays as ensemble averaging on the ground state of the

system.

Using the Hamiltonian in Eq. (1), Heisenberg equation, ∂cαi (τ)/∂τ = [H(τ), cαi (τ)], the

Green’s function formalism and calculating ∂Gαβ(i, j; τ)/∂τ , the equation of motion for elec-

trons gets the shape [27],

∑

`

[

−δi`

(

I∂

∂τ+ ε0i

)

+ ti`

]

G(`, j; τ) = δ(τ)δijI, (3)

where δi` (δij) presents the Kronecker symbol, I serves as a 2 × 2 unit matrix and a Dirac

δ-function is introduced by δ(τ). It’s imaginary time Fourier transformation yields,

∑

`

[(ıωnI− ε0i) δi` + ti`]G(`, j; ıωn) = δijI, (4)

in which ωn = π(2n + 1)T describe the fermionic Matsubara frequencies, n notifies an integer

number and T remarks the temperature of the system. Analytical continuation, ıωn → E =

E + ı0+, of Eq. (4) leads to the following equation,

∑

`

(E − εA0i)δi` tAB〈i`〉

tBA〈i`〉 (E − εB0i)δi`

GAA(`, j;E) GAB(`, j;E)

GBA(`, j;E) GBB(`, j;E)

=

1 0

0 1

δij , (5)

where the index 〈· · ·〉 indicates NN sites. The k-space Fourier transformation of Eq. (5) is

given by,

G(i, j;E) =1

Ω

FBZ∑

k

eık·rij

E − εA0 εk

ε∗k

E − εB0

−1

, (6)

4

so that Ω = NcΩc implies the total area of the system, Ωc displays the area of the unit cell and

k = kxex + kyey exhibits a 2D wave vector in the FBZ with following components,

kx ∈[

− 2√3(π

a),+

2√3(π

a)

]

, ky ∈[

−4

3(π

a),+

4

3(π

a)]

, (7)

a = |a1| = |a2| =√3a0, being a0 and a1, a2 as interatomic distance and primitive vectors

(Fig. (1)) respectively,

a1 =a

2(√3ex − ey), a2 = aey, (8)

wherein ex, ey specify unit vectors in xy graphene plane. Also, rij perform as three vectors

that connect an A (B) site to it’s NN B (A) ones (Fig. (1)),

r01 = a0ex, r02 =a02(−ex +

√3ey), r03 = −a0

2(ex +

√3ey). (9)

In Eq. (6), εk refers to the Fourier transformation of tAB〈ij〉, achievable as,

εk = t0

[

exp

(

ıkxa√3

)

+ 2 exp

(

−ıkxa

2√3

)

cos

(

kya

2

)]

, (10)

in which t0 ≡ tAB〈ij〉 = tBA

〈ij〉.

The electronic HC, is defined by following expression [28],

Ce(T ) =∫ +∞

−∞dE [∂Tf(E , T )] ζ(E), (11)

where f(E , T ) = [1 + exp(E/T )]−1 shows the Fermi-Dirac distribution function and ζ(E) =

ED(E). The DOS of the system is obtained by,

D(E) = − 1

πNaNcΩc

FBZ∑

k

∑

α

=Gαα(k;E). (12)

In this equation, Gαα(k;E) are derived from Eq. (6) as follows,

GAA(k;E) =E − εB0

(E − εA0 ) (E − εB0 )− |εk|2, (13)

5

GBB(k;E) =E − εA0

(E − εA0 ) (E − εB0 )− |εk|2, (14)

so that

|εk| = t0

1 + 4[

cos(√3kx

a

2) + cos(ky

a

2)]

cos(kya

2) 1

2

. (15)

To calculate the EC of the graphene plane, we use Kubo formula [23-25]. For this purpose,

we exhibit the Green’s function in the band representation by which the Hamiltonian gets

diagonal form. Therefore, the k-space Green’s function of the system is:

G(k;E) =

E − E (1)0 (k) 0

0 E − E (2)0 (k)

−1

, (16)

where E (b)0 (k) are eigenvalues of the Hamiltonian and b hints the band index, both calculated

by,

E (1)0 (k) =

εA0 + εB02

+

√

√

√

√

(

εA0 − εB02

)2

+ |εk|2, (17)

E (2)0 (k) =

εA0 + εB02

−

√

√

√

√

(

εA0 − εB02

)2

+ |εk|2. (18)

Our starting point for EC tensor is the so-called Kubo formula [23-25],

σµν(T ) =∫ +∞

−∞dE [−∂Ef(E , T )] ξµν(E), (19)

that µ, ν indicates Cartesian components. In band representation, the energy-dependent EC

is defined as,

ξµν(E) =1

πNbNcΩc

FBZ∑

k

Nb∑

b=1

v(b)µ (k)v(b)ν (k)[

=G(b)(k;E)]2, (20)

Nb = Na displays the number of the bands and v(b)µ (k) = ∂kµE (b)0 (k) presents a b band Cartesian

component of the velocity operator written by,

vµ(k) =

v(1)µ (k) 0

0 v(2)µ (k)

. (21)

6

Using Eq. (16), G(b)(k;E) turns out to be,

G(b)(k;E) =1

E − E (b)0 (k)

. (22)

Eqs. (15), (17) and (18) yield the x-component velocity of the electrons as,

v(2)x (k) = −v(1)x (k) =√3at20

sin(√3kx

a2) cos(ky

a2)

√

(

εA0−εB

0

2

)2

+ |εk|2. (23)

Consequently, by Eqs. (15), (17), (18) and (20)-(23), the x-component of the energy-

dependent EC is specified,

ξxx(E) =6a2t40πNcΩc

FBZ∑

k

sin2(√3kx

a2) cos2(ky

a2)

(εA0 − εB0 )2 + 4|εk|2

2∑

b=1

[

=(

1

E − E (b)0 (k)

)]2

. (24)

In summary, using Green’s function approach and Kubo formula through the TB Hamilto-

nian model, the EC of graphene is calculated. We also studied DOS and electronic HC of the

system based on the same manner. We are to compare the DOS (Eqs. (12)-(15)), electronic

HC (Eqs. (11)-(15)) and the EC (Eqs. (15), (17)-(19) and (22)-(24)) of pristine and substi-

tuted graphene. The hopping to the NN sites and interatomic distance are t0 ' 2.8 eV and

a0 = 1.42A [29-31]. In the numerical calculations, we insert Ωc = 1, a0 = 1 and t0 = 0.28. The

sub-lattices A and B are supposed to be occupied by foreign and carbon atoms respectively

and the on-site energy of carbon is set as reference, namely εB0 = 0. The on-site energy of

impurities gets εA0 = δ which δ is chosen to be −0.25t0, −0.50t0 and −0.75t0 when replaced

atoms act as p-type and 0.25t0, 0.50t0 and 0.75t0 for n-type cases. It is noted that different

values of impurity strength refer to the various impurity types. Figs. (2) and (3) show the

results. In Figs. (2a)-(3a) the DOS, in Figs. (2b)-(3b) HC, and in Figs. (2c)-(3c) the EC are

plotted in both pure and substituted cases. In Fig. (2), foreign atoms act as p-type, while

perform as n-type in Fig. (3).

7

As it is shown in Figs. (2a) and (3a), the presence of substituted atoms opens a gap in the

DOS of the graphene sheet because of difference between the on-site energies of carbon and

impurity atoms, i.e. εA0 − εB0 which directly affects the value of the opened gap. Furthermore,

the role of created potential by dopants could be positive (negative) which implies a barrier

(well) for electrons. So the positive (negative) potential will inherently move the DOS towards

higher (lower) absolute values of energy. This shift increases by raise in the foreign atoms’

on-site energy. We see that van-Hove singularities in the DOS get sharper with increase in the

on-site energy of the dopant. This could be interpreted as follows. The qualitative shape of

the DOS depends on impurities. It is also understood that the total number of the available

states corresponds to the area under the curve of the DOS and is proportional to the electron

density of the system. Therefore, in spite of the relative changes in DOS, the area under the

curve remains constant. Furthermore, impurities open a gap close to the zero energy whereby

we expect the van-Hove singularities to get much sharper rather than the pristine case. The

more increase in the on-site energy of the impurities, the more sharpness in the van-Hove

singularities.

Figs. (2b) and (3b) illustrate an anomalous cross over in the HC of the system, because

it usually exhibits an increasing or constantly behavior with temperature. This irregular low

temperature peak is called as Schottky anomaly [32] owing to the thermal population of discrete

energy levels as the temperature gets raise. More importantly, in presence of the dopants, the

position of this anomaly moves towards higher temperatures and its height shifts down as

the on-site energy of the impurity gets raise. In other words, by existence of foreign atoms,

the HC represents different behaviors in two temperature regions. Before (after) the Schottky

anomaly the HC decreases (increases) in comparison with the pristine graphene. Increasing the

impurities’ on-site energy, this phenomenon will be of more notability.

8

In Figs. (2c) and (3c), the EC of the pristine and doped graphene could be compared.

This figures show that the EC of the system decreases by doping. As we mentioned above the

potential created by impurities behaves as a barrier or well for electrons. In other words, the

role of impurity strength, δ, is a potential barrier (well) against the motion of the electrons.

The effect of scattering raises by increasing the impurity strength that causes increase in the

scattering amplitude of the pure system’s electrons originated from impurity substance. Since

the EC of the semiconducting state is less than semimetal one, then the EC of the doped

graphene is less than pristine case. The more the dopant’s on-site energy increases, the more

decrease in the EC occurs because of a growth in both scattering effects and the energy gap

which the latter one reveals itself as the quantity εA0 − εB0 in the denominator of the Eq. (24).

This phenomenon is of more recognition if the on-site energy of the impurities goes up.

It is mentioned that random doping by impurities leads graphene to a p- or n-type case [12-

17,33]. However, we have achieved that orderly doping leads to a similar p- or n-type graphene,

but with an appeared band gap [5-11] and reasonably a decrease in the EC of the system.

In conclusion, dopants open a band gap in the DOS of the graphene, leading to a p-type or

n-type semiconductor. The results show that the electronic HC of the graphene changes in two

temperature regions due to displacement of the Schottky anomaly towards higher temperatures

and lower amounts. Also, the EC of the system reduces on all ranges of temperature because

of the scattering effect as well as appearing an energy gap. All changes in the electronic

properties of graphene plane depend on the on-site energy of the dopants. So by introducing

different dopants, it’s electronic properties could be manipulated.

9

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12

Figure 1: Geometry of graphene plane. The dashed lines illustrate the Bravais lattice unit

cell. Each cell includes Na = 2 atoms shown by A and B. The Primitive vectors are denoted

by a1 and a2, while, rij are three vectors that connect an A (B) site to it’s NN B (A) ones and

a0 denotes interatomic distance.

Figure 2: DOS, HC and EC of the graphene sheet shown in pristine (solid line) and doped

(dashed lines) cases in panel (a), (b) and (c). The on-site energies of the p-type dopants, δ, are

chosen to be −0.25t0, −0.50t0 and −0.75t0. The dopants open a gap in the DOS (panel (a)),

part the behavior of the HC in two temperature regions (panel (b)) and decrease the EC of the

system on all ranges of temperature (panel (c)). The more increase in the on-site energy of the

dopants, the more changes in the electronic properties of the system.

Figure 3: As Fig. (2) but the dopants act as n-type with opposite signs of the on-site

energies.

13

a

a

1

2 r01y

x

a 0

A B

B

BA

B

B

B

A

A

B

B

A

A

A

B

A

A

A

B

B

B

A

A

A

B

B

B

B

A

A B

BA

A

A

A

r02

r03

B

Figure 1:

14

/t0ε

D(

)ε

δδδ

(a)Pristine

=−0.25t=−0.50t=−0.75t

0

0

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

−3 −2 −1 0 1 2 3

T/t

(T)

0

eC

(b)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5 3

σ(T

)xx

T/t 0

(c)

0

2

4

6

8

10

12

0 0.5 1 1.5 2 2.5 3

Figure 2:

15

/t

D(

)

=0.25t=0.50t=0.75t

(a)

ε

ε

δδδ

0

0

0

0

Pristine

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

−3 −2 −1 0 1 2 3

T/t 0

(T)

Ce

(b)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5 3

σ(T

)

T/t0

xx

(c)

0

2

4

6

8

10

12

0 0.5 1 1.5 2 2.5 3

Figure 3:

16

The effects of dopants on the electronic properties of graphene are investigated.

The tight-binding model and Green's function technique are implemented.

Dopants open a band gap in the density of states.