Tuning thermoelectric transport in phosphorene through a perpendicular magnetic field

Chemical Physics 519 (2019) 1–5

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Tuning thermoelectric transport in phosphorene through a perpendicular magnetic field

T

P.T.T. Lea,b, M. Yarmohammadic,

⁎

a

Laboratory of Magnetism and Magnetic Materials, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam c Lehrstuhl für Theoretische Physik I, Technische Universität Dortmund, Otto-Hahn Straße 4, 44221 Dortmund, Germany b

ARTICLE INFO

ABSTRACT

Keywords: A. Phosphorene B. Green’s function B. Tight-binding model D. Thermoelectric properties C. Perpendicular magnetic field

We address perpendicular magnetic field effects on the electrical conductivity, electronic thermal conductivity, thermopower, and figure of merit in phosphorene. By means of continuum Hamiltonian model and the Green’s function approach, we report a significant difference in the magneto-thermoelectric properties. Results showcase an irregular (regular) behavior in AC (ZZ) direction for electrical conductivity in the presence of a magnetic field. Furthermore, we found an oscillating (increasing) treatment for the Drude weight with increasing magnetic field in AC (ZZ) direction at low temperatures. Also, the electronic thermal conductivity findings introduce the ZZ direction as the right direction in thermoelectric real applications of phosphorene because electronic thermal conductivity in ZZ direction decreases with the magnetic field. In contrast to the electrical conductivity and electronic thermal conductivity, thermopower in AC direction satisfies the requirements of efficient thermoelectric properties at strong magnetic fields. These dependency aspects of the electronic transport coefficients provide insights to modulate the real applications in nanoelectronics.

1. Introduction After the successful isolation of graphene, the potential applications of other two-dimensional (2D) materials as novel functional devices have attracted great attention in the last decade [1–8]. Actually, 2D materials are important due to their extraordinary electronic, optical, and mechanical properties compared with their bulk counterparts. Each of these 2D materials has been synthesized and studied one after another due to shortcomings of the previous one. Although the transition metal dichalgonides (TMDs) were leaders of 2D materials until 2014, due to their high effective masses, the practical applications of them are the rarity. Recently, monolayer black phosphorus (BP), so-called phosphorene was revisited as a new 2D material with high mobility of carriers and moderate ON/OFF ratio [9–14]. Phosphorene with lower effective masses of carriers than TMDs is more applicable. The inherent anisotropic property of BP structure due to its puckered nature of in-plane lattice [see Fig. 1] moves BP into the practical applications in novel electronic and optoelectronic devices [11,15–17]. The structure of phosphorene consists of layers of six-membered rings in the chair conformation, in which each layer phosphorous atoms are threefold connected. Pristine phosphorene is inherently non-magnetic material

⁎

and a good candidate for spintronic applications in the presence of external electronic and magnetic perturbations [18–23]. Thermoelectric (TE) materials, which can directly convert heat into electricity and vice versa, have attracted much interest from the scientific community due to the current critical energy and environmental issues around the world. To decrease the lost produced energy, mostly in the form of waste heat as an alarming issue in the past decay, one promising approach is using thermoelectric (TE) materials [24,25]. The efficiency of a TE material is quantified by the dimensionless figure of merit ZT , which will be described later. This quantity depends strongly on the electronic and thermal transport coefficients such as electrical conductivity (EC), electronic and phononic thermal conductivity, and thermopower (TP). The competition between these quantities determines the best performance of the system. Although there are some works on the electrical and thermal properties of BP and phosphorene [26–29], researches on the TE properties of phosphorene as monolayer BP are very limited [30–33]. In this work, a systematic theoretical study of direction-dependent thermoelectric properties of phosphorene in the presence of Zeeman magnetic field within the Green’s function approach and the tightbinding model has been surveyed. Our results report the properties along both armchair (AC) and zigzag (ZZ) directions in the magnetic

Corresponding author. E-mail addresses: [email protected] (P.T.T. Le), [email protected] (M. Yarmohammadi).

https://doi.org/10.1016/j.chemphys.2018.11.016 Received 4 October 2018; Received in revised form 11 November 2018; Accepted 24 November 2018 Available online 24 November 2018 0301-0104/ © 2018 Published by Elsevier B.V.

Chemical Physics 519 (2019) 1–5

P.T.T. Le, M. Yarmohammadi

are the energies from the bottom of the conduction band and the top of the valence band at point, respectively, making a direct energy gap Eg = 1.52 eV [34]. k = (k x , k y ) is the momenta belonging to the first Brillouin zone (FBZ). On the basis of the DFT calculations [37], other coefficients in Eq. (3) are: c = 0.008187, 2 v = 0.038068, c = 0.030726, v = 0.004849 in units of eVnm , and = 0.48 in unit of eVnm. By considering and time-reversal invariant (TRI) in the system, one might delete terms including and and the new Hamiltonian by considering the magnetic field can be rewritten as

Fig. 1. (a) Top view geometry of monolayer BP. Each unit cell (the pink rectangle) consists of four atoms, as illustrated by red and black circles. The first Brillouin zone of monolayer BP is illustrated in (b) with high-symmetry points , X , S , and Y. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Hc Hcv HZ 0 Hcv Hv 0 HZ H (k) = , HZ 0 Hc Hcv 0 HZ Hcv Hv

field-induced phosphorene. The rest of this manuscript is organized as follows: In Section 2, the theoretical framework is introduced and also the transport coefficients are explained. Section 3 presents the numerical results and the final section (Section 4) contains the summary.

(4)

where

Hc =

c

2 c kx

+

+

2 c ky ,

(5a)

2. Theory and method

Hv =

2.1. Model description and non-interacting Green’s function

Hcv = k x ,

(5c)

Let us now consider the case of monolayer BP with an orthorhombic crystal structure, emphasizing the in-plane translational symmetry. The top view of the real space of monolayer BP with phosphorus (P) atoms in the upper and lower sublayers are represented by red and black circles in Fig. 1. To proceed, we take into account the C2h group invariance-included [34] tight-binding model proposed in Ref. [35] as

HZ = gµB B /2.

(5d)

H (k) =

tAD (k) [tAB (k) + tAC (k)]

tAB (k) + tAC (k) , tAD (k)

(1)

iky a1cos( 1/2)

+ (2t3cos[k x a1sin( 1/2)] e iky [a1cos(

tAC (k) = t2 e iky a2cos( ) + t5 e

1/2) + 2a2cos( )]),

(2a)

(2c)

G0 (k , E ) =

where a1 = 2.22 Å is the distance between nearest-neighbor P atoms in each sublayer with the hopping parameter t1 and a2 = 2.24 Å is the distance between one P atom in upper (lower) sublayer with nearestneighbor atom in lower (upper) sublayer with the hopping parameter t2 . The bond angles are also given by 1 = 96.5°, 2 = 101.9° and cos( ) = cos( 2)/cos( 1) . The hopping parameters taken from Ref. [35] are listed in Table 1. Using the expanded tight-binding model proposed by Rodin et al. [36] around the point and retaining the terms up to second-order in momentum, the continuum approximation of monolayer BP model Hamiltonian would be achieved as [36] 2 2 c kx + c k y 2 2 k x + kx + k y

c

+

k x + k x2 + k y2 v

2 v kx

2 v ky

,

1.220 +3.665 0.205 0.105 0.055

, (6)

2.2. Transport coefficients In this part, we intend to introduce the electronic transport coefficients. In so doing, we employ the linear response theory with the aid of Onsager transport coefficients L [38]. We apply an electric field E and a temperature gradient T along the Cartesian direction to the components of the respective system. From these potentials, the fluxes J1 and J2 can be given through [39]:

Table 1 The hopping parameters taken from Ref. [35].

t1 t2 t3 t4 t5

G031 G032 G033 G034

Using the solution of G0 (k , E ) the thermoelectric properties can be expressed.

which is in the basis of envelope wave functions associated with the probability amplitude at the respective sublattice sites. In the equation above, c = 2t1 + t2 + 2t3 + 4t4 + t5 and v = (2t1 + t2 + 2t3 4t4 + t5)

Value [eV]

G021 G022 G023 G024 G041 G042 G043 G044

(3)

Parameter

(5b)

G011 G012 G013 G014

(2b)

iky [2a1cos( 1/2) + a2cos( )],

tAD (k) = 4t4cos[k x a1sin( 1/2)] × cos(k y [a1cos( 1/2) + a2cos( )]).

H (k) =

v

2 v ky ,

where gµB B is the applied magnetic field potential referring to the magnetic field strength B , the degeneracy level g, and the Bohr magneton µB . By diagonalizing the Hamiltonian above, one obtains the energy dispersion relation for electrons and holes. Since the unit cell of monolayer BP includes four atoms, the Green’s function can be written as a 4 × 4 matrix, but using TRI and G0 (k , E ) = [E + i H (k)] 1 for = 5 meV ( stands for the broadening factor of singularities in the Green’s function elements, which affects the sharpness of peaks only. There is no way to determine its value and it can be determined phenomenologically only), we deduce the following non-interacting Green’s function matrix in the momentum space

with structure factors given by

tAB (k) = 2t1cos[k x a1sin( 1/2)] e

2 v kx

J1 =

L11 E

L12 T

T ,

J2 =

L12 E

L22 T

T .

(7a)

(7b)

for m , n {1, 2} . The Onsager transport coefficients Lmn are related to the transport parameters via: (8a)

= L11 , (T ) =

2

1 L22 T2

(L12 ) 2

L11

,

(8b)

Chemical Physics 519 (2019) 1–5

P.T.T. Le, M. Yarmohammadi

S

L12

(T ) =

TL11

.

(8c)

where , , and S refer to the electrical conductivity, thermal conductivity, and the thermopower, respectively. Finally, using these parameters, the efficiency of TE materials as the key parameter at a temperature T is typically quantified by the dimensionless figure of merit

ZT =

S2

T.

(9)

It is easy to conclude that while reducing the thermal conductivity, the S and should be increased in order to achieve a high efficiency of TE materials at a temperature T (8):

L11 (T ) = L12 (T ) = L22 (T ) =

+

+

+

dE

(E )[

dEE

(E )[

dEE 2

(E )[

Ef

(E , T )],

Ef

(E , T )],

Ef

(E , T )].

(10a) (10b) (10c)

in which

(E ) =

1 2 Na

v (k) v (k)[IG0 (k, E )]2 . ,

k FBZ

(11)

where v (k) is the group velocity of electrons and can easily be obtained by differentiating from dispersion energy relation. On the other hand, f (E , T ) = 1/[eE / k B T + 1] (kB is the Boltzmann constant) represents the Fermi–Dirac distribution function. We confined our investigations on xx and yy components of coefficients for AC and ZZ direction, respectively. In addition, we set = g = µB = kB = e = me = 1 in this work for simplicity because in the present work, the perpendicular magnetic-field dependent treatment of TE quantities is mattered qualitatively, not quantitatively.

Fig. 2. The electrical conductivity along the (a) AC and (b) ZZ direction in the presence of applied Zeeman magnetic field.

magnetic field becomes strong, the EC gets non-zero value and the Drude weight like peak appears [see the inset panel in (a)]. Moreover, when the gµ B B becomes stronger, the EC tends to the initial state, i.e. zero value at fairly low temperatures. After the point kB T 1.5 eV, EC decreases with temperature at all strengths and shows an irregular behavior with the magnetic field. In contrast to these exotic findings in the AC direction, there is a regular behavior in the ZZ direction for the magnetic field-dependent EC. As shown in panel (b), the Drude weight increases slightly with the magnetic field. Also, the EC does not change with the magnetic field after the critical temperature kB T 0.5 eV. It is necessary to mention that in all magnetic field strengths, all curves of EC fall on each other at enough high temperatures because of the lack of quantum effects. As a useful report, one concludes that the ZZ direction is a promising direction for TE applications of phosphorene because to have a high ZT , one needs a large EC. In fact, the physical reasons for this discrepancy between different directions at the microscopy level stem from different anisotropic effective masses in phosphorene in the presence and absence of the magnetic field [40,41]. The wavefunctions corresponding to electrons and holes in different directions deal with different couplings with the magnetic field. This, in turn, leads to anisotropic behaviors for EC mentioned above. In order to characterize the TE properties, we need a low electronic thermal conductivity (ETC). Fig. 3 shows the ETC of phosphorene in both AC (a) and ZZ (b) directions in the presence of Zeeman magnetic field. Interestingly, the ETC of both directions increases gradually at very low temperatures with magnetic field [see the inset panels], whereas there is an irregular behavior for AC direction at kB T 1.0 eV. At high temperatures, all curves with the same reason mentioned before fall on each other. As for the ZZ direction, one can see a regular decreasing behavior for ETC with the magnetic field at kB T 0.75 eV. Moreover, the ETC does not change with the magnetic field after the critical temperature kB T 1.5 eV. Again, the ZZ direction is the best choice for TE real applications due to the regular decreased ETC with the magnetic field. The dependence of the magnetic field in the temperature

3. Results and discussions We begin our discussion with approximations applied in this work. It should be noted that throughout the paper the phonon contributions of the thermal conductivity have not been considered because we assume that the system is under high electron concentrations conditions and decreased phonon mean free path intrinsically. This stems from the 1000 × 1000 unit cell simulated in our numerical calculations. In fact, the overlap of electronic waves in 1000 × 1000 unit cells lead to the weakening of lattice thermal fluctuations. This, in turn, results in the reduction of the mean free path of phonons. Of course, the phonon contribution is not completely zero, but it’s neglectable compared to the electronic one. Furthermore, it is approximated that the structure of the system in the absence and presence of Zeeman magnetic field is in the stable state. Fig. 2 shows the calculated EC of monolayer BP as a function of temperature in the presence of different Zeeman magnetic field strengths in (a) AC and (b) ZZ direction. From the physical point of view, EC is the measure of how easily electricity flows through a material. As electrons move through a material, it comes into contact with atoms in the material. External perturbations behave as collisions, leading to slow the electrons down. The magnetic field acts as a perturbation in phosphorene, affecting the spatial distribution of electronic waves of carriers. The results show that in all strengths, there is a broadening peak around kB T 1.5 eV with opposite treatments before and after that. Before this critical point, the EC in AC direction starts at zero and increases with temperature at weak magnetic field potentials [0 eV and 0.2 eV]. This can be understood well from the movement of carriers. At low temperatures, the quantum effects are dominant and EC increases with T, while at high temperatures, thermal effects are dominant and EC decreases with T. Interestingly, as soon as the 3

Chemical Physics 519 (2019) 1–5

P.T.T. Le, M. Yarmohammadi

Fig. 5. The figure of merit of magnetic field-induced phosphorene along the AC (a) and ZZ (b) direction.

Fig. 3. The electronic contribution of thermal conductivity along the (a) AC and (b) ZZ direction in the presence of applied Zeeman magnetic field.

slightly with gµ B B and at high temperatures, all curves fall on each other as expected. As for the efficient TE property, the TP should be increased with perturbations. It means that in our calculations, the AC direction is the effective direction for TP part of the TE applications of phosphorene because it increases with magnetic field [inset panel of (a)], while in the ZZ direction, the TP decreases with gµ B B . According to Eq. (8), ETC has two terms containing the absolute temperature T. A denominator T 2 and an exponential function of T in Onsager coefficients. Whereas in TP relation, only the denominator T is the temperature-dependent term because the exponential factors in Onsager coefficients cancel each other. For this reason, different behaviors for Tdependent ETC and TP and eventually for their critical temperatures are expected. As for the physical nature of these discrepancies, one can say that the competition between thermal energy kB T and the energy E of carriers in Onsager elements lead to different trends for the critical temperatures. In the last paragraph, we discuss the dimensionless temperaturedependent ZT , as shown in Fig. 5. According to the Eq. (9) and Figs. 2–4, one expects the presented behaviors for ZT in both directions. In short, one observes that the ZT takes place in different points depending on the Zeeman magnetic field strength in AC direction with oscillatory trends, while there is a slightly decreasing behavior in ZZ direction with the magnetic field. Also, at high temperatures, ZT ZZ (T ) approaches zero faster than ZT AC (T ) . The results show that the ZZ direction is a better direction than AC direction because of the irregular and regular treatments of electronic transport coefficients in AC and ZZ direction, respectively.

Fig. 4. (a) S AC (T ) and (b) S ZZ (T ) at various magnetic fields in monolayer BP.

4. Conclusions

dependence of TP is quite subtle and not yet fully reported. For this reason, in Fig. 4, we compare our numerical results of TP along the AC and ZZ directions of phosphorene. Formally, the TP is defined as the ratio of the thermoelectric voltage added to the matter to the resulting V / T ]. It has different behaviors with diftemperature change [ ferent signs for different materials. Generally, at very low temperatures, it has a critical temperature and at high temperatures vanishes [42]. The critical temperature in AC (ZZ) direction decreases (increases) very

In summary, we have reported theoretically and numerically the significant different treatments for the electronic transport coefficients of magnetic field-induced phosphorene by employing the tight-binding model and the Green’s function technique. We found the exotic and irregular findings in the AC direction, whereas there is a regular behavior in the ZZ direction for the magnetic field-dependent EC. Also, the appeared Drude weight in the EC curves increases slightly with the 4

Chemical Physics 519 (2019) 1–5

P.T.T. Le, M. Yarmohammadi

magnetic field. In addition, we presented that the ETC of phosphorene in ZZ direction is more efficient than the AC one. Furthermore, we report that the Seebeck coefficient in the AC is the right electronic property needed for TE applications. Our results will help the engineers dealing with real applications that require the controllable anisotropic TE properties.

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