- Email: [email protected]
Dual electronic thermal conductivity on graphene: Gate-potential and ripples
Physica B 577 (2020) 411828
Contents lists available at ScienceDirect
Physica B: Physics of Condensed Matter journal homepage: www.elsevier.com/locate/physb
Direct/Dual electronic thermal conductivity on graphene: Gate-potential and ripples L. Palma-Chilla a ,∗, J.C. Flores b a b
Departamento de Física y Astronomía, Universidad de La Serena, Av. Juan Cisternas 1200, La Serena, Chile Departamento de Física, FACI, Universidad de Tarapacá, Casilla 7-D, Arica, Chile
ARTICLE Keywords: Graphene Massless fermions Negative temperature Fractal dimensions Dual systems Ripples
INFO
ABSTRACT A functional relationship allows to connect the thermodynamic properties between the Direct and Dual bands of a graphene sheet. For low temperature, analytical expressions for internal energy, heat capacity and entropy are obtained when a positive gate potential is considered. Through a numerical study, it is shown that thermal conductivity decreases in both branches when the gate potential is applied, whereas ripples increase electronic thermal conductivity.
1. Introduction
henceforth the Direct, is given by
By graphene (monolayer) we understand a network of carbon atoms with geometric hexagonal at two dimensions (𝐷 = 2), which allows to make materials with graphite in other dimensions. For example, fullerenes (𝐷 = 0), nanotubes (𝐷 = 1) or several layers of graphene forming three dimensional (𝐷 = 3) structures [1–3]. Experimentally it has been shown that the carriers in graphene behave like Dirac fermions [4,5] and that these crystals are not only continuous but also of high quality [5–8]. Interestingly, graphene when approaching the dimension 𝐷 = 3, for example through the formation of ripples, becomes intrinsically stable on a lateral scale of approximately 10 nm [8]. This deformation of graphene 𝐷 = 2 generates important thermal properties that need to be studied [9]. Generically, graphene is considered a semiconductor without gap between the valence (holes) and conduction (electrons) bands, also as a semimetal of superposition zero [10–14]. Previous studies on heat conduction showed an exotic behavior, a very high thermal conductivity due to phonons compared to other materials [15,16]. In this work we consider an external gate-potential that has consequences in the thermodynamic behavior of this semiconductor. The conduction and valence bands, respectively Direct and Dual, are connected through an additive functional relationship [17]: ⃖⃗ + 𝜔(𝑘) ⃖⃗ = 0. 𝜔′ (𝑘)
(1)
In this work, we use the prime (′ ) to denote the Dual band. At first order in the wave-vector expansion [10–12,18,19], for graphene or carbon monolayer with gate-potential, the “massless” electrons spectrum,
| | 𝐸 = 𝑞𝑉𝑔 + ℏ𝑣𝐹 |𝑘⃖⃗| , | |
(2)
| | where 𝑉𝑔 corresponds to the positive gate-potential and ℏ𝑣𝐹 |𝑘⃖⃗| to the | | 6 usual graphene spectrum, 𝑣𝐹 ∼ 10 (m/s) being the Fermi velocity and ℏ the Planck constant. It is also assumed 𝐸 = ℏ𝜔. From Eq. (1) , the spectrum of the Dual band will be | | 𝐸 ′ = −(𝑞𝑉𝑔 + ℏ𝑣𝐹 |𝑘⃖⃗|), | |
(3)
Where, naturally, 𝐸 ′ = ℏ𝜔′ . Eq. (1) allows to connect the thermodynamic properties of both bands. As long as the number of states is the same for the Direct and Dual bands (sphere, Eqs. (2)–(3)). Then, using Boltzmann′ s definition for entropy, these branches are connected by 𝑆 ′ (𝑢) = 𝑆(−𝑢),
(4)
where 𝑢 = 𝑈 ∕𝑈0 is the dimensionless energy of the thermodynamic system, 𝑈0 = ℏ𝑣𝐹 ∕𝑎 ∼ 7. 429 6 × 10−19 (𝐽 ) denotes an intrinsic energy parameter related to a fundamental energy, and 𝑎 ∼ 1.42𝐴̊ corresponds to the approximate distance between neighboring carbon atoms [11,19]. From the Direct band, Eq. (4) allows to deduce the behavior of the Dual band: entropy, heat capacity, and others [20,21]. Through differential equation 𝜕𝑆∕𝜕𝑈 = 1∕𝑇 [22] it can be noted that the temperature is negative in the Dual band and positive in the Direct band.
∗ Corresponding author. E-mail address: [email protected] (L. Palma-Chilla).
https://doi.org/10.1016/j.physb.2019.411828 Received 29 August 2019; Received in revised form 11 October 2019; Accepted 25 October 2019 Available online 29 October 2019 0921-4526/© 2019 Elsevier B.V. All rights reserved.
Physica B: Physics of Condensed Matter 577 (2020) 411828
L. Palma-Chilla and J.C. Flores
𝑆(𝑇 ) 2 = 𝑘𝐵 𝜋
Considering Fermi–Dirac statistics [22–25], the internal energy is expressed as ∑ 𝐸𝑖 𝑈 = 𝑞𝑉𝑔 + (5) 𝛽(𝐸𝑖 −𝜇) + 1 𝑖 𝑒
𝑈 ′ = −𝑞𝑉𝑔 + 2 ×
−∞
(
𝐸+𝑞𝑉𝑔 𝐷−1 ) 𝐸 𝑈0
2𝐷 𝛤 (𝐷∕2 + 1)𝑈0 ∫−𝑞𝑉𝑔 exp
(𝐸−𝜇) 𝑘𝐵 𝑇
𝑑𝐸,
(10)
𝑒
𝑈 ′ = −𝑈 (−𝑇 )
(11)
𝐶 ′ = 𝐶(−𝑇 )
(12) (13)
𝑆 = 𝑆(−𝑇 ).
Fig. 1 shows the internal energy, heat capacity and entropy as function of temperature, for the Direct (𝑇 > 0) and Dual (𝑇 < 0) bands. Note that the low temperature range corresponds to the interval −1 < 𝑘𝐵 𝑇 ∕𝑈0 < 1. As expected, the gate-potential generates a gap in the energy spectrum between the Direct and Dual bands (see Fig. 1(a)). For the Direct and in the low temperature range, internal energy is larger than in the case without gate-potential. Conversely, for temperatures 𝑇 ≫ 𝑈0 ∕𝑘𝐵 and for the fixed value of 𝑉𝑔 , the internal energy is smaller than in the case without gate-potential. Fig. 1(b) and (c) show that heat capacity and entropy are lower when the gate potential is considered in both branches. This difference arises when temperature increases. Further, when the gate-potential is ignored, the expression for the internal energy, heat capacity and entropy obtained in reference [17] is recovered.
(6)
𝐵
𝐷𝜋 −𝐷∕2
𝑞𝑉𝑔 −𝑘 𝑇 𝐵
′
𝐸−𝑞𝑉
𝑔 𝐷−1 ) 𝐸 ∞ ( 𝑈0 𝐷𝜋 −𝐷∕2 𝑑𝐸, 𝐷 (𝐸−𝜇) 2 𝛤 (𝐷∕2 + 1)𝑈0 ∫𝑞𝑉𝑔 exp +1 𝑘 𝑇
𝑘𝐵 𝑇 𝑈0
)2
Taking into account that the temperature is negative for the Dual and the change of sign in the charges (electron to hole), from Eqs. (8)–(10)– (9) for this band, internal energy, heat capacity and entropy can be obtained directly as:
where notably the term referred to as gate-potential 𝑞𝑉𝑔 , 𝜇 is the chemical potential and 𝛽 = 1∕𝑘𝐵 𝑇 , 𝑘𝐵 the Boltzmann′ s constant. 𝐸±𝑞𝑉 Eqs. (2) and (3) define a hypersphere of radio 𝑅∕𝐿 = ( 𝑈 𝑔 ) 𝜋1 in 0 the wave-vector space, where 𝐿 is the size of system. From a generic point of view, using microscopic properties, thermodynamic aspects can be obtained from standard method [22–25]. Then, from Eq. (5) and in the continuous limit, the internal energy for the Direct and Dual bands (electrons and holes in graphene) when a positive gate-potential is considered, is obtained respectively for example through 𝑈 = 𝑞𝑉𝑔 + 2 ×
(
(7)
+1
where 𝐷 is the spatial dimension with spin-degeneration two, 𝛤 the Gamma function. Note that 𝑇 is negative in the Dual band (Eq. (7)). In this work, the thermodynamics for graphene are studied in the following cases: (a) Graphene without gate-potential (𝑞𝑉𝑔 = 0) (b) Graphene with positive gate-potential, which generates a gap between the Direct and Dual bands (𝑞𝑉𝑔 ≠ 0). Experimentally it is measured that a graphene mono-layer properties [26] gap does not exist. Nevertheless, for graphene on a SiC substrate [27], a gap is generated for mono-layer graphene under specific conditions. This gap is possibly related to interaction-breaking local symmetries [28]. In our model, the generation of the gap between both bands (Direct/Dual) requires to break the electron–hole Coulomb energy which is larger than 𝑈0 . In this article we consider only the lattice's effect on the electron dynamics (eventually, holes) and associated thermodynamics. Interaction effects like electron–electron, electron–phonon or electron–hole are not considered. Nevertheless, our estimations of the thermal conductivity for ‘‘free electrons’’ are reasonable when compared with quantum full calculations [29] in Section 3. In Section 2, for the Direct band of graphene the internal energy, heat capacity and entropy are obtained in the limit of low temperature. The effect to consider a gate potential in the dispersion relationship is studied numerically. Mainly in Section 3 electronic thermal conductivity is obtained and studied. Moreover, a graphene sheet with aleatory broken bonds, ribbons, ripples or others [17,30,31] is modeled through the fractional spatial dimension. Conclusions appear in the last section.
3. Electronic thermal conductivity for Dirac fermions at low temperature (𝑫 = 𝟐 and 𝑫 = 𝟐.𝟏𝟖) The heat flux in a hypercube of dimension 𝐷 and edge 𝑎 is defined 𝐷−1 = 𝜅 𝑎 𝑎 △ 𝑇 , where 𝜅 is the thermal conductivity [35,36]. as △𝑄 △𝑡 Note that this expression defines a differential equation as long as 𝐷−1 𝑣 = 𝑎∕ △ 𝑡 (i.e. △𝑄 = 𝜅 𝑎 𝑣 △𝑇 ). Besides, as △𝑄 = 𝑁𝐶 △ 𝑇 with △𝑡 △𝑡 𝑁 as the particles number, the thermal conductivity can be expressed 𝐶 , where, in our case, 𝑣 corresponds to the Fermi velocity as 𝜅 = ( 𝐷𝑣 ) 𝑎𝐷−1 𝑣𝐹 . From Eq. (6) at any dimension, and using the above expression for the thermal conductivity, for the Direct band the electronic thermal conductivity (per particle) is then expressed formally as
𝜅𝑔𝑟𝑎𝑝ℎ = 2 ×
𝜋 −𝐷∕2 𝑣𝐹
∞
𝑎𝐷−1 2𝐷 𝛤 (𝐷∕2 + 1)𝑈0 𝑘𝐵 𝑇 2 ∫𝑞𝑉𝑔
(
𝐸−𝑞𝑉𝑔 𝐷−1 2 ) 𝐸 𝑈0
exp( 𝑘 𝐸𝑇 )𝑑𝐸 𝐵 . ( )2 exp( 𝑘 𝐸𝑇 ) + 1 𝐵
(14) Similarly, from Eq. (7) the electronic thermal conductivity for the Dual band can be obtained. Then, considering Eq. (14) in the low temperature limit (𝑘𝐵 𝑇 ≪ 𝑈0 ) and for 𝐷 = 2, the electronic thermal conductivity becomes: ( ) 𝑘 𝑣 𝑘𝐵 𝑇 2 − 𝑘𝑞𝑉𝑔𝑇 (15) 𝜅𝑔𝑟𝑎𝑝ℎ (𝑇 ) = 3 𝐵 𝐹 𝑒 𝐵 𝑎𝜋 𝑈0
2. Internal energy, heat capacity and entropy for Dirac fermions: to low temperature (𝑫 = 𝟐) The positive parameter 𝐷 will be 2 for fermions in a graphene monolayer without ripples [18,32–34], among other systems. Usually, for low temperature in fermions, the chemical potential is considered equal to the Fermi energy, particularly considered zero for graphene (𝜇 = 𝜀𝐹 = 0) From Eq. (6), in the limit of low temperature 𝑘𝐵 𝑇 ≪ 𝑈0 , the internal energy is obtained. Then, using 𝐶𝑣 = (𝜕𝑈 ∕𝜕𝑇 )𝑣 and 𝑆 = 𝑇 ∫0 𝑑𝑇 𝐶𝑣 ∕𝑇 , heat capacity and entropy are obtained. At this limit, these thermodynamic quantities are respectively expressed as ( )3 𝑞𝑉𝑔 𝑞𝑉𝑔 − 𝑈 (𝑇 ) 2 𝑘𝐵 𝑇 = + 𝑒 𝑘𝐵 𝑇 (8) 𝑈0 𝑈0 𝜋 𝑈0 ( )2 𝑞𝑉𝑔 − 𝐶(𝑇 ) 6 𝑘𝐵 𝑇 = 𝑒 𝑘𝐵 𝑇 (9) 𝑘𝐵 𝜋 𝑈0
Mainly, due to the exponential factor, when a gate potential on graphene is considered, the electronic thermal conductivity is inferior compared to the case without (Fig. 2). Further, from Eq. (15), when the gate potential is considered, the electronic thermal conductivity tends to zero exponentially (as exp(−𝑞𝑉𝑔 ∕𝑘𝐵 𝑇 )). Just to compare, from Eq. (14) without the gate potential (i.e. 𝑉𝑔 = 0) and for low temperature, the electronic thermal conductivity (per particle) for graphene (𝐷 = 2) is expressed as ( ) 𝑘 𝑣 𝑘𝐵 𝑇 2 (16) 𝜅𝑔𝑟𝑎𝑝ℎ (𝑇 ) = 3 𝐵 𝐹 𝑎𝜋 𝑈0 And in this case the electronic thermal conductivity tends to zero as 𝑇 2 . Note that the above expression has a similar structure as photons 2
Physica B: Physics of Condensed Matter 577 (2020) 411828
L. Palma-Chilla and J.C. Flores
Fig. 1. Thermodynamic properties as function of temperature for the Direct and Dual bands (𝑇 > 0 and 𝑇 < 0, respectively), for massless fermions in graphene (without and with gate-potential). (a) Internal energy, (b) Heat capacity and (c) Entropy. For the numerical calculation 𝐷 = 2, 𝐿 = 1, 𝑞𝑉𝑔 ∕𝑈0 = 0 (continuous line) and 𝑞𝑉𝑔 ∕𝑈0 = 2 (segmented line) were considered. The gap corresponds to 2𝑉𝑔 .
at dimension two, because at the Dirac point for graphene the mass of carriers is formally zero. In this sense we conjecture for graphene that for any dimension, including fractional ones related to ripples, 𝜅𝑔𝑟𝑎𝑝ℎ ∼ 𝑇 𝐷. Studies in graphene, theoretical and experimental, show that the thermal conductivity due to phonons is of order 2000 (W/m K) at room temperature, and greater than the thermal conductivity of electrons [15,16]. Using Eq. (16) the numerical calculation for graphene without the gate potential shows that the thermal conductivity (per particle) is of order 𝜅𝑔𝑟𝑎𝑝ℎ ∼ 2. 8856 × 10−12 (W/K) for 𝑇 = 300 (K). To realize some estimates and compare with experiments, consider that our evaluation is strictly in dimension two where the units are (W/K). Experimental and analytical calculations are at order of 𝜅𝑒𝑥𝑝 ∼ 300 (W/m K) [29]. Thus, to compare, from (14), we must divide by 𝜋 and the size of our macroscopic sample (approx. 1 cm), and multiply by 2∕3 𝐾𝑇 ∕𝑈0 (at 300 K) and the Avogadro number at dimension two 𝑁𝐴 . Our estimation gives approximately: 𝜅𝐷=3
⎛ ∼ 𝜅𝐷=2 ⋅ ⎜ ⎜ ⎝
Fig. 2. Electronic thermal conductivity as function of temperature for Direct and Dual bands (𝑇 > 0 and 𝑇 < 0, respectively) for massless fermions in graphene (without and with gate-potential). For the numerical calculus was considered 𝐷 = 2 with 𝑞𝑉𝑔 = 0 (continuous line) and 𝑞𝑉𝑔 = 2 (segmented line). (Note that 𝛼 = 𝑘𝐵 𝑣𝐹 ∕𝑎.)
2∕3 𝑁𝐴 ⎞ ⎟ ∼ 63, 662 (W/m K), 𝜋𝐿 ⎟ ⎠
𝐾𝑇 𝑈0
in reasonable agreement with experiments. The growth of ripples in graphene becomes related to temperature rises supporting the fractional spatial dimension [8,37,38], so the thermodynamic behavior of graphene should be affected. Eq. (14), Fig. 3 shows that electronic thermal conductivity increases when ripples are considered or, equivalently, the spatial dimension changes from 𝐷 = 2 to 𝐷 = 2.18.
4. Conclusions For graphene the upper and lower bands (Direct/Dual) are interrelated through a functional relationship (Eq.(1)) including a gate potential. Hence, the thermodynamic properties of both bands are strongly correlated through entropy (Eq. (4)). 3
Physica B: Physics of Condensed Matter 577 (2020) 411828
L. Palma-Chilla and J.C. Flores
Fig. 3. Electronic thermal conductivity for the Direct and Dual bands as function of temperature for two dimension values: 𝐷 = 2 (continuous line) and ripples with 𝐷 = 2.18 (segmented line). (a) 𝑞𝑉𝑔 = 0 and (b) 𝑞𝑉𝑔 = 2. (𝛼 = 𝑘𝐵 𝑣𝐹 ∕𝑎).
Analytical expressions at the limit of low temperature for internal energy, heat capacity and entropy for the Direct (conduction) band in graphene were obtained. In the low temperature regime, the gate potential increases the internal energy and the other way around in the high energy regime. Further, heat capacity and entropy decrease when a (positive) gate potential is considered. When the gate potential does not act on graphene, these thermodynamic quantities match well those obtained in reference [17]. More important, a gate potential in the dispersion relationship of graphene reduces its electronic thermal conductivity, which also increases with temperature (Fig. 3). Additionally, when ripples are modeled through changes in the fractional dimension, from 𝐷 = 2 to 𝐷 = 2.18, electronic thermal conductivity increases.
[10] [11] [12] [13] [14] [15] [16]
[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment
[28] [29] [30] [31]
L. Palma-Chilla acknowledges the financial support of DIDULS/ULS, through the project PR192132.
[32]
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Contents lists available at ScienceDirect
Physica B: Physics of Condensed Matter journal homepage: www.elsevier.com/locate/physb
Direct/Dual electronic thermal conductivity on graphene: Gate-potential and ripples L. Palma-Chilla a ,∗, J.C. Flores b a b
Departamento de Física y Astronomía, Universidad de La Serena, Av. Juan Cisternas 1200, La Serena, Chile Departamento de Física, FACI, Universidad de Tarapacá, Casilla 7-D, Arica, Chile
ARTICLE Keywords: Graphene Massless fermions Negative temperature Fractal dimensions Dual systems Ripples
INFO
ABSTRACT A functional relationship allows to connect the thermodynamic properties between the Direct and Dual bands of a graphene sheet. For low temperature, analytical expressions for internal energy, heat capacity and entropy are obtained when a positive gate potential is considered. Through a numerical study, it is shown that thermal conductivity decreases in both branches when the gate potential is applied, whereas ripples increase electronic thermal conductivity.
1. Introduction
henceforth the Direct, is given by
By graphene (monolayer) we understand a network of carbon atoms with geometric hexagonal at two dimensions (𝐷 = 2), which allows to make materials with graphite in other dimensions. For example, fullerenes (𝐷 = 0), nanotubes (𝐷 = 1) or several layers of graphene forming three dimensional (𝐷 = 3) structures [1–3]. Experimentally it has been shown that the carriers in graphene behave like Dirac fermions [4,5] and that these crystals are not only continuous but also of high quality [5–8]. Interestingly, graphene when approaching the dimension 𝐷 = 3, for example through the formation of ripples, becomes intrinsically stable on a lateral scale of approximately 10 nm [8]. This deformation of graphene 𝐷 = 2 generates important thermal properties that need to be studied [9]. Generically, graphene is considered a semiconductor without gap between the valence (holes) and conduction (electrons) bands, also as a semimetal of superposition zero [10–14]. Previous studies on heat conduction showed an exotic behavior, a very high thermal conductivity due to phonons compared to other materials [15,16]. In this work we consider an external gate-potential that has consequences in the thermodynamic behavior of this semiconductor. The conduction and valence bands, respectively Direct and Dual, are connected through an additive functional relationship [17]: ⃖⃗ + 𝜔(𝑘) ⃖⃗ = 0. 𝜔′ (𝑘)
(1)
In this work, we use the prime (′ ) to denote the Dual band. At first order in the wave-vector expansion [10–12,18,19], for graphene or carbon monolayer with gate-potential, the “massless” electrons spectrum,
| | 𝐸 = 𝑞𝑉𝑔 + ℏ𝑣𝐹 |𝑘⃖⃗| , | |
(2)
| | where 𝑉𝑔 corresponds to the positive gate-potential and ℏ𝑣𝐹 |𝑘⃖⃗| to the | | 6 usual graphene spectrum, 𝑣𝐹 ∼ 10 (m/s) being the Fermi velocity and ℏ the Planck constant. It is also assumed 𝐸 = ℏ𝜔. From Eq. (1) , the spectrum of the Dual band will be | | 𝐸 ′ = −(𝑞𝑉𝑔 + ℏ𝑣𝐹 |𝑘⃖⃗|), | |
(3)
Where, naturally, 𝐸 ′ = ℏ𝜔′ . Eq. (1) allows to connect the thermodynamic properties of both bands. As long as the number of states is the same for the Direct and Dual bands (sphere, Eqs. (2)–(3)). Then, using Boltzmann′ s definition for entropy, these branches are connected by 𝑆 ′ (𝑢) = 𝑆(−𝑢),
(4)
where 𝑢 = 𝑈 ∕𝑈0 is the dimensionless energy of the thermodynamic system, 𝑈0 = ℏ𝑣𝐹 ∕𝑎 ∼ 7. 429 6 × 10−19 (𝐽 ) denotes an intrinsic energy parameter related to a fundamental energy, and 𝑎 ∼ 1.42𝐴̊ corresponds to the approximate distance between neighboring carbon atoms [11,19]. From the Direct band, Eq. (4) allows to deduce the behavior of the Dual band: entropy, heat capacity, and others [20,21]. Through differential equation 𝜕𝑆∕𝜕𝑈 = 1∕𝑇 [22] it can be noted that the temperature is negative in the Dual band and positive in the Direct band.
∗ Corresponding author. E-mail address: [email protected] (L. Palma-Chilla).
https://doi.org/10.1016/j.physb.2019.411828 Received 29 August 2019; Received in revised form 11 October 2019; Accepted 25 October 2019 Available online 29 October 2019 0921-4526/© 2019 Elsevier B.V. All rights reserved.
Physica B: Physics of Condensed Matter 577 (2020) 411828
L. Palma-Chilla and J.C. Flores
𝑆(𝑇 ) 2 = 𝑘𝐵 𝜋
Considering Fermi–Dirac statistics [22–25], the internal energy is expressed as ∑ 𝐸𝑖 𝑈 = 𝑞𝑉𝑔 + (5) 𝛽(𝐸𝑖 −𝜇) + 1 𝑖 𝑒
𝑈 ′ = −𝑞𝑉𝑔 + 2 ×
−∞
(
𝐸+𝑞𝑉𝑔 𝐷−1 ) 𝐸 𝑈0
2𝐷 𝛤 (𝐷∕2 + 1)𝑈0 ∫−𝑞𝑉𝑔 exp
(𝐸−𝜇) 𝑘𝐵 𝑇
𝑑𝐸,
(10)
𝑒
𝑈 ′ = −𝑈 (−𝑇 )
(11)
𝐶 ′ = 𝐶(−𝑇 )
(12) (13)
𝑆 = 𝑆(−𝑇 ).
Fig. 1 shows the internal energy, heat capacity and entropy as function of temperature, for the Direct (𝑇 > 0) and Dual (𝑇 < 0) bands. Note that the low temperature range corresponds to the interval −1 < 𝑘𝐵 𝑇 ∕𝑈0 < 1. As expected, the gate-potential generates a gap in the energy spectrum between the Direct and Dual bands (see Fig. 1(a)). For the Direct and in the low temperature range, internal energy is larger than in the case without gate-potential. Conversely, for temperatures 𝑇 ≫ 𝑈0 ∕𝑘𝐵 and for the fixed value of 𝑉𝑔 , the internal energy is smaller than in the case without gate-potential. Fig. 1(b) and (c) show that heat capacity and entropy are lower when the gate potential is considered in both branches. This difference arises when temperature increases. Further, when the gate-potential is ignored, the expression for the internal energy, heat capacity and entropy obtained in reference [17] is recovered.
(6)
𝐵
𝐷𝜋 −𝐷∕2
𝑞𝑉𝑔 −𝑘 𝑇 𝐵
′
𝐸−𝑞𝑉
𝑔 𝐷−1 ) 𝐸 ∞ ( 𝑈0 𝐷𝜋 −𝐷∕2 𝑑𝐸, 𝐷 (𝐸−𝜇) 2 𝛤 (𝐷∕2 + 1)𝑈0 ∫𝑞𝑉𝑔 exp +1 𝑘 𝑇
𝑘𝐵 𝑇 𝑈0
)2
Taking into account that the temperature is negative for the Dual and the change of sign in the charges (electron to hole), from Eqs. (8)–(10)– (9) for this band, internal energy, heat capacity and entropy can be obtained directly as:
where notably the term referred to as gate-potential 𝑞𝑉𝑔 , 𝜇 is the chemical potential and 𝛽 = 1∕𝑘𝐵 𝑇 , 𝑘𝐵 the Boltzmann′ s constant. 𝐸±𝑞𝑉 Eqs. (2) and (3) define a hypersphere of radio 𝑅∕𝐿 = ( 𝑈 𝑔 ) 𝜋1 in 0 the wave-vector space, where 𝐿 is the size of system. From a generic point of view, using microscopic properties, thermodynamic aspects can be obtained from standard method [22–25]. Then, from Eq. (5) and in the continuous limit, the internal energy for the Direct and Dual bands (electrons and holes in graphene) when a positive gate-potential is considered, is obtained respectively for example through 𝑈 = 𝑞𝑉𝑔 + 2 ×
(
(7)
+1
where 𝐷 is the spatial dimension with spin-degeneration two, 𝛤 the Gamma function. Note that 𝑇 is negative in the Dual band (Eq. (7)). In this work, the thermodynamics for graphene are studied in the following cases: (a) Graphene without gate-potential (𝑞𝑉𝑔 = 0) (b) Graphene with positive gate-potential, which generates a gap between the Direct and Dual bands (𝑞𝑉𝑔 ≠ 0). Experimentally it is measured that a graphene mono-layer properties [26] gap does not exist. Nevertheless, for graphene on a SiC substrate [27], a gap is generated for mono-layer graphene under specific conditions. This gap is possibly related to interaction-breaking local symmetries [28]. In our model, the generation of the gap between both bands (Direct/Dual) requires to break the electron–hole Coulomb energy which is larger than 𝑈0 . In this article we consider only the lattice's effect on the electron dynamics (eventually, holes) and associated thermodynamics. Interaction effects like electron–electron, electron–phonon or electron–hole are not considered. Nevertheless, our estimations of the thermal conductivity for ‘‘free electrons’’ are reasonable when compared with quantum full calculations [29] in Section 3. In Section 2, for the Direct band of graphene the internal energy, heat capacity and entropy are obtained in the limit of low temperature. The effect to consider a gate potential in the dispersion relationship is studied numerically. Mainly in Section 3 electronic thermal conductivity is obtained and studied. Moreover, a graphene sheet with aleatory broken bonds, ribbons, ripples or others [17,30,31] is modeled through the fractional spatial dimension. Conclusions appear in the last section.
3. Electronic thermal conductivity for Dirac fermions at low temperature (𝑫 = 𝟐 and 𝑫 = 𝟐.𝟏𝟖) The heat flux in a hypercube of dimension 𝐷 and edge 𝑎 is defined 𝐷−1 = 𝜅 𝑎 𝑎 △ 𝑇 , where 𝜅 is the thermal conductivity [35,36]. as △𝑄 △𝑡 Note that this expression defines a differential equation as long as 𝐷−1 𝑣 = 𝑎∕ △ 𝑡 (i.e. △𝑄 = 𝜅 𝑎 𝑣 △𝑇 ). Besides, as △𝑄 = 𝑁𝐶 △ 𝑇 with △𝑡 △𝑡 𝑁 as the particles number, the thermal conductivity can be expressed 𝐶 , where, in our case, 𝑣 corresponds to the Fermi velocity as 𝜅 = ( 𝐷𝑣 ) 𝑎𝐷−1 𝑣𝐹 . From Eq. (6) at any dimension, and using the above expression for the thermal conductivity, for the Direct band the electronic thermal conductivity (per particle) is then expressed formally as
𝜅𝑔𝑟𝑎𝑝ℎ = 2 ×
𝜋 −𝐷∕2 𝑣𝐹
∞
𝑎𝐷−1 2𝐷 𝛤 (𝐷∕2 + 1)𝑈0 𝑘𝐵 𝑇 2 ∫𝑞𝑉𝑔
(
𝐸−𝑞𝑉𝑔 𝐷−1 2 ) 𝐸 𝑈0
exp( 𝑘 𝐸𝑇 )𝑑𝐸 𝐵 . ( )2 exp( 𝑘 𝐸𝑇 ) + 1 𝐵
(14) Similarly, from Eq. (7) the electronic thermal conductivity for the Dual band can be obtained. Then, considering Eq. (14) in the low temperature limit (𝑘𝐵 𝑇 ≪ 𝑈0 ) and for 𝐷 = 2, the electronic thermal conductivity becomes: ( ) 𝑘 𝑣 𝑘𝐵 𝑇 2 − 𝑘𝑞𝑉𝑔𝑇 (15) 𝜅𝑔𝑟𝑎𝑝ℎ (𝑇 ) = 3 𝐵 𝐹 𝑒 𝐵 𝑎𝜋 𝑈0
2. Internal energy, heat capacity and entropy for Dirac fermions: to low temperature (𝑫 = 𝟐) The positive parameter 𝐷 will be 2 for fermions in a graphene monolayer without ripples [18,32–34], among other systems. Usually, for low temperature in fermions, the chemical potential is considered equal to the Fermi energy, particularly considered zero for graphene (𝜇 = 𝜀𝐹 = 0) From Eq. (6), in the limit of low temperature 𝑘𝐵 𝑇 ≪ 𝑈0 , the internal energy is obtained. Then, using 𝐶𝑣 = (𝜕𝑈 ∕𝜕𝑇 )𝑣 and 𝑆 = 𝑇 ∫0 𝑑𝑇 𝐶𝑣 ∕𝑇 , heat capacity and entropy are obtained. At this limit, these thermodynamic quantities are respectively expressed as ( )3 𝑞𝑉𝑔 𝑞𝑉𝑔 − 𝑈 (𝑇 ) 2 𝑘𝐵 𝑇 = + 𝑒 𝑘𝐵 𝑇 (8) 𝑈0 𝑈0 𝜋 𝑈0 ( )2 𝑞𝑉𝑔 − 𝐶(𝑇 ) 6 𝑘𝐵 𝑇 = 𝑒 𝑘𝐵 𝑇 (9) 𝑘𝐵 𝜋 𝑈0
Mainly, due to the exponential factor, when a gate potential on graphene is considered, the electronic thermal conductivity is inferior compared to the case without (Fig. 2). Further, from Eq. (15), when the gate potential is considered, the electronic thermal conductivity tends to zero exponentially (as exp(−𝑞𝑉𝑔 ∕𝑘𝐵 𝑇 )). Just to compare, from Eq. (14) without the gate potential (i.e. 𝑉𝑔 = 0) and for low temperature, the electronic thermal conductivity (per particle) for graphene (𝐷 = 2) is expressed as ( ) 𝑘 𝑣 𝑘𝐵 𝑇 2 (16) 𝜅𝑔𝑟𝑎𝑝ℎ (𝑇 ) = 3 𝐵 𝐹 𝑎𝜋 𝑈0 And in this case the electronic thermal conductivity tends to zero as 𝑇 2 . Note that the above expression has a similar structure as photons 2
Physica B: Physics of Condensed Matter 577 (2020) 411828
L. Palma-Chilla and J.C. Flores
Fig. 1. Thermodynamic properties as function of temperature for the Direct and Dual bands (𝑇 > 0 and 𝑇 < 0, respectively), for massless fermions in graphene (without and with gate-potential). (a) Internal energy, (b) Heat capacity and (c) Entropy. For the numerical calculation 𝐷 = 2, 𝐿 = 1, 𝑞𝑉𝑔 ∕𝑈0 = 0 (continuous line) and 𝑞𝑉𝑔 ∕𝑈0 = 2 (segmented line) were considered. The gap corresponds to 2𝑉𝑔 .
at dimension two, because at the Dirac point for graphene the mass of carriers is formally zero. In this sense we conjecture for graphene that for any dimension, including fractional ones related to ripples, 𝜅𝑔𝑟𝑎𝑝ℎ ∼ 𝑇 𝐷. Studies in graphene, theoretical and experimental, show that the thermal conductivity due to phonons is of order 2000 (W/m K) at room temperature, and greater than the thermal conductivity of electrons [15,16]. Using Eq. (16) the numerical calculation for graphene without the gate potential shows that the thermal conductivity (per particle) is of order 𝜅𝑔𝑟𝑎𝑝ℎ ∼ 2. 8856 × 10−12 (W/K) for 𝑇 = 300 (K). To realize some estimates and compare with experiments, consider that our evaluation is strictly in dimension two where the units are (W/K). Experimental and analytical calculations are at order of 𝜅𝑒𝑥𝑝 ∼ 300 (W/m K) [29]. Thus, to compare, from (14), we must divide by 𝜋 and the size of our macroscopic sample (approx. 1 cm), and multiply by 2∕3 𝐾𝑇 ∕𝑈0 (at 300 K) and the Avogadro number at dimension two 𝑁𝐴 . Our estimation gives approximately: 𝜅𝐷=3
⎛ ∼ 𝜅𝐷=2 ⋅ ⎜ ⎜ ⎝
Fig. 2. Electronic thermal conductivity as function of temperature for Direct and Dual bands (𝑇 > 0 and 𝑇 < 0, respectively) for massless fermions in graphene (without and with gate-potential). For the numerical calculus was considered 𝐷 = 2 with 𝑞𝑉𝑔 = 0 (continuous line) and 𝑞𝑉𝑔 = 2 (segmented line). (Note that 𝛼 = 𝑘𝐵 𝑣𝐹 ∕𝑎.)
2∕3 𝑁𝐴 ⎞ ⎟ ∼ 63, 662 (W/m K), 𝜋𝐿 ⎟ ⎠
𝐾𝑇 𝑈0
in reasonable agreement with experiments. The growth of ripples in graphene becomes related to temperature rises supporting the fractional spatial dimension [8,37,38], so the thermodynamic behavior of graphene should be affected. Eq. (14), Fig. 3 shows that electronic thermal conductivity increases when ripples are considered or, equivalently, the spatial dimension changes from 𝐷 = 2 to 𝐷 = 2.18.
4. Conclusions For graphene the upper and lower bands (Direct/Dual) are interrelated through a functional relationship (Eq.(1)) including a gate potential. Hence, the thermodynamic properties of both bands are strongly correlated through entropy (Eq. (4)). 3
Physica B: Physics of Condensed Matter 577 (2020) 411828
L. Palma-Chilla and J.C. Flores
Fig. 3. Electronic thermal conductivity for the Direct and Dual bands as function of temperature for two dimension values: 𝐷 = 2 (continuous line) and ripples with 𝐷 = 2.18 (segmented line). (a) 𝑞𝑉𝑔 = 0 and (b) 𝑞𝑉𝑔 = 2. (𝛼 = 𝑘𝐵 𝑣𝐹 ∕𝑎).
Analytical expressions at the limit of low temperature for internal energy, heat capacity and entropy for the Direct (conduction) band in graphene were obtained. In the low temperature regime, the gate potential increases the internal energy and the other way around in the high energy regime. Further, heat capacity and entropy decrease when a (positive) gate potential is considered. When the gate potential does not act on graphene, these thermodynamic quantities match well those obtained in reference [17]. More important, a gate potential in the dispersion relationship of graphene reduces its electronic thermal conductivity, which also increases with temperature (Fig. 3). Additionally, when ripples are modeled through changes in the fractional dimension, from 𝐷 = 2 to 𝐷 = 2.18, electronic thermal conductivity increases.
[10] [11] [12] [13] [14] [15] [16]
[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment
[28] [29] [30] [31]
L. Palma-Chilla acknowledges the financial support of DIDULS/ULS, through the project PR192132.
[32]
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