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Interlayer exciton-polaron effect in transition metal dichalcogenides van der Waals heterostructures
Journal of Physics and Chemistry of Solids 134 (2019) 1–4
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Interlayer exciton-polaron effect in transition metal dichalcogenides van der Waals heterostructures
T
Xi-Ying Dong, Run-Ze Li, Jia-Pei Deng, Zi-Wu Wang∗ Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology, School of Science, Department of Applied Physics, Tianjin University, Tianjin, 300354, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Interlayer exciton Exciton-phonon coupling van der Waals heterostructures
Interlayer exciton-polaron is formed arising from the exciton-surface optical (SO) phonons coupling in transition metal dichalcogenides (TMDS) van der Waals heterostructures, in which SO phonons are induced by the inserted hexagonal boron nitride (h-BN) and the coupling effect is evaluated by means of two unitary transformations. We find that the binding energy of interlayer exciton decreases obviously due to this exciton-polaron effect. The binding energy can be tuned by the numbers of inserted h-BN layers and the interlayer distance between two TMDS layers. Moreover, the critical temperature of quantum Bose gases composing of these interlayer excitons increases in a large scale due to the variations of binding energy. These theoretical results provide possible channels to adjust the properties of interlayer excitons in two-dimensional van der Waals heterostructures.
1. Introduction Vertical stacking of different two-dimensional transition metal dichalcogenides materials, permits the fabrication of a wide variety of so called van der Waals (vdW) heterostructures due to the relaxed constraint for the lattice matching [1,2]. In particular, these vdW heterostructures can host the formation of the interlayer exciton which is composed of an electron and a hole confined in spatially separated double layers. This new type of exciton with remarkably high binding energy is more stabler than in III-V or II-VI semiconductor heterostructures, which provides a wide platform for developing room-temperature exciton devices [2–4] and for studying a wealth of novel physical phenomena [5–10], such as the quantum Bose gas and superfluidity in high temperature and light-induced exciton spin Hall effect. Large binding energy is the unique property of the interlayer exciton in vdW heterostructure, determining these potential applications above mentioned. A number of recent experiments [11–26] have proved that binding energies of the interlayer excitons vary from tens of meV to several hundreds of meV and can be easily tuned by many external ways, such as the external electric field [20], the interlayer distance, the thickness of inserted insulator materials and twist angles between two TMDS layers [5,6,11,18,24,26]. However, the divergence exists between the experimental measurements and the theoretical calculations for the values of binding energy in the same heterostructure. To explain
∗
the divergence, the screening effect arising from the dielectric environment around the vdW heterostructure has been extensively proposed [27–31]. Nevertheless, regarding the mechanism of screening effect and the determining factors of binding energy are still ambiguous. In the present paper, we investigate the interlayer exciton in TMDS vdW heterostructure schemed in Fig. 1, in which the insulator h-BN layers are inserted between two monolayer TMDS. On the one hand, these inserted h-BN layers increase the spatial separation between electrons and holes, resulting in the dramatic decreasing of recombination of the electron-hole pairs and thus enhancing the lifetime of interlayer exciton distinctly. On the other hand, surface optical (SO) phonon modes are induced by the inserted layer and couples strongly with interlayer excitons. We employ the method of unitary transforms to evaluate the exciton-SO phonon coupling and find that the binding energies vary in tens of meV range due to this exciton-polaron effect. This indicates that the exciton-SO phonons coupling gives the significant contribution to the mechanism of screening effect. The binding energy can be modulated by the interlayer distance between two TMDS layers, internal distance between the monolayer TMDS and the thickness of the inserted h-BN (numbers of the h-BN layer). These modulating channels provide insight for both the determining factors of binding energy and the modulation of interlayer exciton in experiments. Furthermore, we present the dependence of the critical temperature of the quantum Bose gas, composing of the interlayer excitons,
Corresponding author. E-mail address: [email protected] (Z.-W. Wang).
https://doi.org/10.1016/j.jpcs.2019.05.022 Received 18 April 2019; Received in revised form 14 May 2019; Accepted 15 May 2019 Available online 19 May 2019 0022-3697/ © 2019 Elsevier Ltd. All rights reserved.
Journal of Physics and Chemistry of Solids 134 (2019) 1–4
X.-Y. Dong, et al.
are given in details in the supplementary materials. Through the intricate calculations, Eq. (2) becomes
H˜ =
ℏ2Q2 2M
− ∑k, ν +
−
ℏ2 2 ∇ 2μ r
+ V ′ (r ) + ∑k, ν
(
ℏ2k2 2M
)
+ ℏωSOν ak†, ν ak, ν
|M (h) |2 + | M (e ) |2 −M(*h)M (e ) ξk (r )
(ℏ2k2 / 2M ) + ℏωSO ν 2 2 * 2 2 * ℏ2k2 β1 | M (h) | | +β2 | M (e ) | +β1 β2 M(h)M (e ) ξk (r ) , ∑k, ν 2μ 2 2 2 [(ℏ k / 2M ) + ℏωSO, ν]
(4)
with
ξk (r ) = e−i (β1+ β2) k ⋅ r + ei (β1+ β2) k ⋅ r . Here, we emphasize that many terms including the multiphonon processes with the same phonon modes and interactions between two different phonon modes are neglected in Eq. (4), because these terms give the little contribution to the binding energy. The exciton binding energy with influence of exciton-SO phonons coupling in the ground state is calculated by Refs. ∼ Eb = 〈Φ|H˜ |Φ〉[37–39], in which |Φ〉 = |ϕ1s 〉|0〉ph is the ground state of the system. |ϕ1s 〉 = 1/( 2π a)exp(−r 2/4a2) is the exciton ground state in the relative motion with the parameter a = [ℏ/(2 2μγ )]1/2 (see supplementary material); |0〉ph is the zero phonon state. The corrected binding energy can be expressed as
Fig. 1. Schematic diagram of the interlayer exciton in the van der Waals heterostructure MoS2 /h-BN/MoSe2 , in which D is the interlayer distance between two TMDS layers and z 0 (z1) is the internal distance between the monolayer TMDS and the inserted h-BN layer.
on the binding energy in the frame of Berezinskii-Kosterlitz-Thouless (BKT) model. 2. Theoretical model Interlayer exciton is formed in the vdW heterostructure that is composed of two parallel TMDS monolayers inserted by the h-BN layers schemed Fig. 1. The total Hamiltonian for an interlayer exciton including the exciton-SO phonons coupling is
H=
ℏ2 − 2M ∇2R
−
ℏ 2 ∇ 2μ r
+ V (r ) +
+
∑k, ν {eik ⋅ R [M(h) e−iβ1 k ⋅ r
+
e−ik ⋅ R [M(h) eiβ1 k ⋅ r
∼ Eb = Eb(0) −
+
∑k, ν ℏωSOν ak†, ν ak, ν
− M(e) eiβ2 k ⋅ r ] ak, ν
∑k,ν
∑k,ν
|M(h) |2 + |M(e) |2 − M (*h)M(e) R (k ) (ℏ2k 2/2M ) + ℏωSOν
2 2 * 2 2 * ℏ2k 2 β1 |M(h) | | + β2 |M(e) | + β1 β2 M (h)M(e) R (k ) , 2 2 2 2μ [(ℏ k /2M ) + ℏωSO, ν ]
with the parameters
− M(e) e−iβ2 k ⋅ r ] ak†, ν}
R (k )* = R (k ) = exp(−a2k 2/2),
(1)
in which Eb(0) is the ground binding energy of exciton for the relative motion without the exciton-SO phonon coupling and has been solved in previous studies [37–39], more detailed processes are presented in supplementary material. In order to keep the interlayer exciton in the lowest state, the center-of-mass motion is neglected (Q = 0 ) in the numerical calculations. The corrected binding energies can be obtained after carrying out the summation of wave vector into the integral in Eq. (5), in which a cut-off value kc = 5 × 109 is fixed for the wave vector of SO phonon modes [44]. In the processes of numerical simulation, some parameters for the inserting h-BN layers are adopted as following [32,39]: κ 0 = 5.1(ε0) , κ∞ = 4.1(ε0) , ℏωSO1 = 101 meV, ℏωSO2 = 195 meV, the thickness of monolayer h-BN is 0.33 nm and the dielectric constant εd = 4.6(ε0) . The parameters for electron and hole effective masses are listed in Table S1 in the supplementary material. In fact, two kinds of interlayer excitons are formed due to the strong spin-orbit splitting in the TMDS materials [37–39]. In the present paper, we only study one kind of exciton and assume the coupling effects are similar for them. In addition, the stacking angles (or twist angles) between different layers cause dramatic changes in the properties of interlayer exciton and have been proved in recent experiments [40–43]. These effects are not taken into account in this paper.
In general, the motion of exciton can be divided into two parts: the center-of-mass motion is denoted by the first term with the total mass M = me + mh and center-of-mass coordinate R, me (mh ) is the electron (hole) effective mass; the relative motion is described by the second and third terms with the reduced mass μ = me−1 + mh−1 and relative coordinate r. The fourth term describes the energies of SO phonons, which are induced by the inserted h-BN layers, including two branches with the energy ℏωSOν = 1,2 . ak, ν (ak†, ν ) is annihilation (creation) operator for the νth branch phonon mode with the wave vector k. The last term is the coupling between exciton and SO phonons with the coupling element [32–34]M(e),(h) = ∑k, ν (e 2ηℏωSOν )/(2ε 0 k ) e−kz in which the parameter η = (κ 0 − κ∞)/[(κ∞ + 1)(κ 0 + 1)] is related to the high (κ∞) and low (κ 0 ) frequency dielectric constants of the h-BN layer. z is the internal distance between the h-BN and the monolayer TMDS, which is replaced by the z 0 and z1 in Fig. 1. and ε0 are the area of the monolayer TMDS and the permittivity of vacuum, respectively. The parameter β1 = me / M (β2 = mh / M ) describes the ratio of the electron (hole) effective mass to the total mass. The Lee-Low-Pines unitary transformations [35,36] are adopted to merge the fourth and last terms into a simple form. The processes of transformations for the total Hamiltonian via
H˜ = W2−1 W1−1 HW1 W2,
(2)
3. Results and discussion
with
⎡ ⎛ W1 = exp ⎢i ⎜Q − ⎢ ⎣ ⎝
According to Eq. (5), we present the binding energies of interlayer excitons for different TMDS vdW heterostructures inserted by h-BN without and with exciton-SO phonon coupling in Table 1 and Table 2, respectively. From the comparison, one find that the binding energies can be corrected by several tens of meV due to these coupling. The values of binding energies in Table 2 are very closed to these results in Ref. 39. In fact, the larger binding energies above 100 meV for the interlayer exciton have been predicted in previous theories [29,31,45,46] and observed in recent experiments [10–21]. In the present paper, the
⎞ ⎤
∑ kak†,ν ak,ν⎟⋅R⎥, k, ν
⎠ ⎥ ⎦
(3a)
⎛ ⎞ W2 = exp ⎜∑ (Fk, ν ak†, ν − Fk*, νak, ν ) ⎟, k , ν ⎝ ⎠
(3b)
Fk*, ν
in which the variational functions Fk, ν and play a dominate role to solve the exciton-SO phonons coupling. The transformation processes 2
Journal of Physics and Chemistry of Solids 134 (2019) 1–4
X.-Y. Dong, et al.
interlayer distance. One can see that the binding energies decrease markedly with increasing the numbers of the inserted h-BN layers. This because of the internal distances (z 0 and z1) become small with increasing the numbers of the inserted h-BN layers, enhancing the coupling between exciton and SO phonons. This differs from the results that the numbers of the inserted h-BN layers give the very tiny correction to binding energies Berman et al. predicted [39]. These results need to be proved by the following experiments. To show the influence of the internal distance on the binding energy clearly, the dependences of binding energy on the internal distances in MoS2 (e)/h-BN/WS2 (h) are illustrated in Fig. 2 (c). One can conclude that the internal distance plays a dominate role to determine the correction of binding energy, which arises from the strength of the exciton-SO phonon coupling follows the exponential decay with the internal distances [44]. Due to the large binding energy of interlayer exciton in TMDS vdW heterostructures, the room temperature Bose-Einstein condensation and superfluid have been proposed based on the dilute interlayer exciton gas in recent years [5,8,37,39,48,49]. In the BTK model, the relation of the critical temperature of their quantum Bose gases with the binding energy satisfies [5,48] TBKT = C (μ/ M )(1/ Eb) , in which the constant C = 2.6ne 2/(4kB ε02) , n is the exciton density; kB is the Boltzmann constant and Eb is the exciton binding energy. According to Eq. (5), the variation of the critical temperature due to the correction of binding ∼ energy follows ΔTBKT = C (μ/ M )(1/ Eb − 1/ Eb0) . In Fig. 2, we have dis∼ cussed the variation of binding energy ( Eb ) in the present of exciton-SO phonon coupling, which can be obviously tuned by the interlayer distance, the internal distance and the number of the inserted h-BN layers. Therefore, the critical temperature can be modulated by these external ways. In the MoS2 (e)/h-BN/WS2 (h), the variation of the critical temperature (ΔTBKT ) as functions of two internal distances are presented in Fig. 3 (a); the dependences of the variation of the critical temperature on the interlayer distance and the number of the inserted h-BN layers are shown in Fig. 3 (b). From them, one can see that the critical temperature can be tuned in a very large scale by these external ways, which provides the potential methods to control the critical temperature of quantum Bose gas composing of these interlayer excitons in experiments. Here, we must admit that we only adopted the unitary transformations and a very simple variational calculation to roughly estimate the correction of binding energies arising from exciton-SO phonons coupling. In fact, several advanced methods basing on the first-principle calculations have been adopted in the studies of interlayer exciton [22,28–31]. However, our results can provide the comprehensive comparisons for the exciton binding energies between them.
Table 1 Interlayer exciton binding energies (meV) in different TMDS vdW heterostructures without the exciton-phonons coupling for the interlayer distance D = 4.0 nm and the thickness of the inserted h-BN is 2.0 nm (about six layers of h-BN). The subscripts (e) and (h) denote the positions for electron and hole in the corresponding monolayer materials, respectively.
MoS2 (h) MoSe2 (h) WS2 (h) WSe2 (h)
MoS2 (e)
MoSe2 (e)
WS2 (e)
WSe2 (e)
73.98 74.72 75.17 68.39
75.61 76.69 77.16 70.03
67.99 68.79 69.30 63.08
69.04 69.79 70.27 64.10
Table 2 Interlayer exciton binding energies (meV) in different TMDS vdW heterostructures with the exciton-phonon coupling at D = 4.0 nm and z 0 = z1=1.0 nm.
MoS2 (h) MoSe2 (h) WS2 (h) WSe2 (h)
MoS2 (e)
MoSe2 (e)
WS2 (e)
WSe2 (e)
45.50 46.29 46.78 38.35
47.52 48.90 49.47 40.02
37.23 37.90 37.89 32.09
38.72 39.35 39.79 33.38
binding energies of interlayer exciton are smaller than 100 meV for different TMDS vdW heterostructures, which can be attributed to the following reasons: (1) the larger interlayer distance D ≥ 3.0 nm between two TMDS layers are assumed, leading to the Coulomb interaction between electrons and holes are reduced directly in spatial; (2) the inserted h-BN layers enhance the screening effect and reduce the interaction of electron-hole pairs further. In addition, the exciton-intrinsic longitudinal optical phonons coupling also give the obvious correction to binding energies if the strong coupling regime exists in our recent study [47]. The SO phonon modes are induced by the inserted h-BN layers, which results in the strength of exciton-SO phonon coupling depending on the modulation of distances between different layers. In Fig. 2 (a), the dependences of the binding energies on the interlayer distance without (red lines) and with exciton-SO phonon coupling (blue lines) are shown for three types of vdW heterostructures. It can be seen that binding energies decrease with increasing the interlayer distance. The amplitudes are enhanced obviously if exciton-SO phonon coupling is taken into account shown in blue lines. Fig. 2 (b) shows the influence of the numbers of the inserted h-BN layers (or the thickness of h-BN layers) on binding energies with exciton-SO phonon coupling at fixed
Fig. 2. (a) The binding energies as a function of the interlayer distance without (red lines) and with (blue lines) exciton-SO phonon coupling for three types of heterostructures at z 0 = z1 = 1.0 nm. (b) The binding energies as a function of the numbers of h-BN layers with exciton-SO phonon coupling at D = 4.0 nm. (c) The binding energies as functions of two internal distances in MoS2 (e)/h-BN/WS2 (h) at D = 4.0 nm. 3
Journal of Physics and Chemistry of Solids 134 (2019) 1–4
X.-Y. Dong, et al.
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Fig. 3. (a) The variation of critical temperature as functions of two internal distances in MoS2 (e)/h-BN/WS2 (h) at D = 4.0 nm; (b) the variation of critical temperature as functions of the interlayer distance and the number of the inserted h-BN layers at z 0 = z1= 1.0 nm. The density of exciton gas n = 1010cm2 is assumed.
4. Conclusion In summary, the binding energies of interlayer excitons in TMDS vdW heterostructures varied in several tens of meV range due to the exciton-SO phonon coupling, depending on the interlayer distance, the numbers of inserted h-BN layers and the internal distance. The markedly variation of interlayer exciton binding energies result in the critical temperature of its quantum Bose gas can be tuned in a very large scale. Funding This work was supported by National Natural Science Foundation of China (No 11674241). Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.jpcs.2019.05.022. References [1] A.K. Geim, I.V. Grigorieva, Nature 499 (2013) 419. [2] K.S. Novoselov, A. Mishchenko, A. Carvalho, A.H. Castro Neto, Science 353 (2016) 6298. [3] L.V. Butov, Superlattice. Microst. 108 (2017) 22. [4] T. Mueller, E. Malic, npj 2D Mater. Appl. 2 (2018) 29. [5] M.M. Fogler, L.V. Butov, K.S. Novoselov, Nat. Commun. 5 (2014) 4555. [6] E.V. Calman, C.J. Dorow, M.M. Fogler, L.V. Butov, S. Hu, A. Mishchenko, A.K. Geim, Appl. Phys. Lett. 108 (2016) 101901.
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Interlayer exciton-polaron effect in transition metal dichalcogenides van der Waals heterostructures
T
Xi-Ying Dong, Run-Ze Li, Jia-Pei Deng, Zi-Wu Wang∗ Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology, School of Science, Department of Applied Physics, Tianjin University, Tianjin, 300354, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Interlayer exciton Exciton-phonon coupling van der Waals heterostructures
Interlayer exciton-polaron is formed arising from the exciton-surface optical (SO) phonons coupling in transition metal dichalcogenides (TMDS) van der Waals heterostructures, in which SO phonons are induced by the inserted hexagonal boron nitride (h-BN) and the coupling effect is evaluated by means of two unitary transformations. We find that the binding energy of interlayer exciton decreases obviously due to this exciton-polaron effect. The binding energy can be tuned by the numbers of inserted h-BN layers and the interlayer distance between two TMDS layers. Moreover, the critical temperature of quantum Bose gases composing of these interlayer excitons increases in a large scale due to the variations of binding energy. These theoretical results provide possible channels to adjust the properties of interlayer excitons in two-dimensional van der Waals heterostructures.
1. Introduction Vertical stacking of different two-dimensional transition metal dichalcogenides materials, permits the fabrication of a wide variety of so called van der Waals (vdW) heterostructures due to the relaxed constraint for the lattice matching [1,2]. In particular, these vdW heterostructures can host the formation of the interlayer exciton which is composed of an electron and a hole confined in spatially separated double layers. This new type of exciton with remarkably high binding energy is more stabler than in III-V or II-VI semiconductor heterostructures, which provides a wide platform for developing room-temperature exciton devices [2–4] and for studying a wealth of novel physical phenomena [5–10], such as the quantum Bose gas and superfluidity in high temperature and light-induced exciton spin Hall effect. Large binding energy is the unique property of the interlayer exciton in vdW heterostructure, determining these potential applications above mentioned. A number of recent experiments [11–26] have proved that binding energies of the interlayer excitons vary from tens of meV to several hundreds of meV and can be easily tuned by many external ways, such as the external electric field [20], the interlayer distance, the thickness of inserted insulator materials and twist angles between two TMDS layers [5,6,11,18,24,26]. However, the divergence exists between the experimental measurements and the theoretical calculations for the values of binding energy in the same heterostructure. To explain
∗
the divergence, the screening effect arising from the dielectric environment around the vdW heterostructure has been extensively proposed [27–31]. Nevertheless, regarding the mechanism of screening effect and the determining factors of binding energy are still ambiguous. In the present paper, we investigate the interlayer exciton in TMDS vdW heterostructure schemed in Fig. 1, in which the insulator h-BN layers are inserted between two monolayer TMDS. On the one hand, these inserted h-BN layers increase the spatial separation between electrons and holes, resulting in the dramatic decreasing of recombination of the electron-hole pairs and thus enhancing the lifetime of interlayer exciton distinctly. On the other hand, surface optical (SO) phonon modes are induced by the inserted layer and couples strongly with interlayer excitons. We employ the method of unitary transforms to evaluate the exciton-SO phonon coupling and find that the binding energies vary in tens of meV range due to this exciton-polaron effect. This indicates that the exciton-SO phonons coupling gives the significant contribution to the mechanism of screening effect. The binding energy can be modulated by the interlayer distance between two TMDS layers, internal distance between the monolayer TMDS and the thickness of the inserted h-BN (numbers of the h-BN layer). These modulating channels provide insight for both the determining factors of binding energy and the modulation of interlayer exciton in experiments. Furthermore, we present the dependence of the critical temperature of the quantum Bose gas, composing of the interlayer excitons,
Corresponding author. E-mail address: [email protected] (Z.-W. Wang).
https://doi.org/10.1016/j.jpcs.2019.05.022 Received 18 April 2019; Received in revised form 14 May 2019; Accepted 15 May 2019 Available online 19 May 2019 0022-3697/ © 2019 Elsevier Ltd. All rights reserved.
Journal of Physics and Chemistry of Solids 134 (2019) 1–4
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are given in details in the supplementary materials. Through the intricate calculations, Eq. (2) becomes
H˜ =
ℏ2Q2 2M
− ∑k, ν +
−
ℏ2 2 ∇ 2μ r
+ V ′ (r ) + ∑k, ν
(
ℏ2k2 2M
)
+ ℏωSOν ak†, ν ak, ν
|M (h) |2 + | M (e ) |2 −M(*h)M (e ) ξk (r )
(ℏ2k2 / 2M ) + ℏωSO ν 2 2 * 2 2 * ℏ2k2 β1 | M (h) | | +β2 | M (e ) | +β1 β2 M(h)M (e ) ξk (r ) , ∑k, ν 2μ 2 2 2 [(ℏ k / 2M ) + ℏωSO, ν]
(4)
with
ξk (r ) = e−i (β1+ β2) k ⋅ r + ei (β1+ β2) k ⋅ r . Here, we emphasize that many terms including the multiphonon processes with the same phonon modes and interactions between two different phonon modes are neglected in Eq. (4), because these terms give the little contribution to the binding energy. The exciton binding energy with influence of exciton-SO phonons coupling in the ground state is calculated by Refs. ∼ Eb = 〈Φ|H˜ |Φ〉[37–39], in which |Φ〉 = |ϕ1s 〉|0〉ph is the ground state of the system. |ϕ1s 〉 = 1/( 2π a)exp(−r 2/4a2) is the exciton ground state in the relative motion with the parameter a = [ℏ/(2 2μγ )]1/2 (see supplementary material); |0〉ph is the zero phonon state. The corrected binding energy can be expressed as
Fig. 1. Schematic diagram of the interlayer exciton in the van der Waals heterostructure MoS2 /h-BN/MoSe2 , in which D is the interlayer distance between two TMDS layers and z 0 (z1) is the internal distance between the monolayer TMDS and the inserted h-BN layer.
on the binding energy in the frame of Berezinskii-Kosterlitz-Thouless (BKT) model. 2. Theoretical model Interlayer exciton is formed in the vdW heterostructure that is composed of two parallel TMDS monolayers inserted by the h-BN layers schemed Fig. 1. The total Hamiltonian for an interlayer exciton including the exciton-SO phonons coupling is
H=
ℏ2 − 2M ∇2R
−
ℏ 2 ∇ 2μ r
+ V (r ) +
+
∑k, ν {eik ⋅ R [M(h) e−iβ1 k ⋅ r
+
e−ik ⋅ R [M(h) eiβ1 k ⋅ r
∼ Eb = Eb(0) −
+
∑k, ν ℏωSOν ak†, ν ak, ν
− M(e) eiβ2 k ⋅ r ] ak, ν
∑k,ν
∑k,ν
|M(h) |2 + |M(e) |2 − M (*h)M(e) R (k ) (ℏ2k 2/2M ) + ℏωSOν
2 2 * 2 2 * ℏ2k 2 β1 |M(h) | | + β2 |M(e) | + β1 β2 M (h)M(e) R (k ) , 2 2 2 2μ [(ℏ k /2M ) + ℏωSO, ν ]
with the parameters
− M(e) e−iβ2 k ⋅ r ] ak†, ν}
R (k )* = R (k ) = exp(−a2k 2/2),
(1)
in which Eb(0) is the ground binding energy of exciton for the relative motion without the exciton-SO phonon coupling and has been solved in previous studies [37–39], more detailed processes are presented in supplementary material. In order to keep the interlayer exciton in the lowest state, the center-of-mass motion is neglected (Q = 0 ) in the numerical calculations. The corrected binding energies can be obtained after carrying out the summation of wave vector into the integral in Eq. (5), in which a cut-off value kc = 5 × 109 is fixed for the wave vector of SO phonon modes [44]. In the processes of numerical simulation, some parameters for the inserting h-BN layers are adopted as following [32,39]: κ 0 = 5.1(ε0) , κ∞ = 4.1(ε0) , ℏωSO1 = 101 meV, ℏωSO2 = 195 meV, the thickness of monolayer h-BN is 0.33 nm and the dielectric constant εd = 4.6(ε0) . The parameters for electron and hole effective masses are listed in Table S1 in the supplementary material. In fact, two kinds of interlayer excitons are formed due to the strong spin-orbit splitting in the TMDS materials [37–39]. In the present paper, we only study one kind of exciton and assume the coupling effects are similar for them. In addition, the stacking angles (or twist angles) between different layers cause dramatic changes in the properties of interlayer exciton and have been proved in recent experiments [40–43]. These effects are not taken into account in this paper.
In general, the motion of exciton can be divided into two parts: the center-of-mass motion is denoted by the first term with the total mass M = me + mh and center-of-mass coordinate R, me (mh ) is the electron (hole) effective mass; the relative motion is described by the second and third terms with the reduced mass μ = me−1 + mh−1 and relative coordinate r. The fourth term describes the energies of SO phonons, which are induced by the inserted h-BN layers, including two branches with the energy ℏωSOν = 1,2 . ak, ν (ak†, ν ) is annihilation (creation) operator for the νth branch phonon mode with the wave vector k. The last term is the coupling between exciton and SO phonons with the coupling element [32–34]M(e),(h) = ∑k, ν (e 2ηℏωSOν )/(2ε 0 k ) e−kz in which the parameter η = (κ 0 − κ∞)/[(κ∞ + 1)(κ 0 + 1)] is related to the high (κ∞) and low (κ 0 ) frequency dielectric constants of the h-BN layer. z is the internal distance between the h-BN and the monolayer TMDS, which is replaced by the z 0 and z1 in Fig. 1. and ε0 are the area of the monolayer TMDS and the permittivity of vacuum, respectively. The parameter β1 = me / M (β2 = mh / M ) describes the ratio of the electron (hole) effective mass to the total mass. The Lee-Low-Pines unitary transformations [35,36] are adopted to merge the fourth and last terms into a simple form. The processes of transformations for the total Hamiltonian via
H˜ = W2−1 W1−1 HW1 W2,
(2)
3. Results and discussion
with
⎡ ⎛ W1 = exp ⎢i ⎜Q − ⎢ ⎣ ⎝
According to Eq. (5), we present the binding energies of interlayer excitons for different TMDS vdW heterostructures inserted by h-BN without and with exciton-SO phonon coupling in Table 1 and Table 2, respectively. From the comparison, one find that the binding energies can be corrected by several tens of meV due to these coupling. The values of binding energies in Table 2 are very closed to these results in Ref. 39. In fact, the larger binding energies above 100 meV for the interlayer exciton have been predicted in previous theories [29,31,45,46] and observed in recent experiments [10–21]. In the present paper, the
⎞ ⎤
∑ kak†,ν ak,ν⎟⋅R⎥, k, ν
⎠ ⎥ ⎦
(3a)
⎛ ⎞ W2 = exp ⎜∑ (Fk, ν ak†, ν − Fk*, νak, ν ) ⎟, k , ν ⎝ ⎠
(3b)
Fk*, ν
in which the variational functions Fk, ν and play a dominate role to solve the exciton-SO phonons coupling. The transformation processes 2
Journal of Physics and Chemistry of Solids 134 (2019) 1–4
X.-Y. Dong, et al.
interlayer distance. One can see that the binding energies decrease markedly with increasing the numbers of the inserted h-BN layers. This because of the internal distances (z 0 and z1) become small with increasing the numbers of the inserted h-BN layers, enhancing the coupling between exciton and SO phonons. This differs from the results that the numbers of the inserted h-BN layers give the very tiny correction to binding energies Berman et al. predicted [39]. These results need to be proved by the following experiments. To show the influence of the internal distance on the binding energy clearly, the dependences of binding energy on the internal distances in MoS2 (e)/h-BN/WS2 (h) are illustrated in Fig. 2 (c). One can conclude that the internal distance plays a dominate role to determine the correction of binding energy, which arises from the strength of the exciton-SO phonon coupling follows the exponential decay with the internal distances [44]. Due to the large binding energy of interlayer exciton in TMDS vdW heterostructures, the room temperature Bose-Einstein condensation and superfluid have been proposed based on the dilute interlayer exciton gas in recent years [5,8,37,39,48,49]. In the BTK model, the relation of the critical temperature of their quantum Bose gases with the binding energy satisfies [5,48] TBKT = C (μ/ M )(1/ Eb) , in which the constant C = 2.6ne 2/(4kB ε02) , n is the exciton density; kB is the Boltzmann constant and Eb is the exciton binding energy. According to Eq. (5), the variation of the critical temperature due to the correction of binding ∼ energy follows ΔTBKT = C (μ/ M )(1/ Eb − 1/ Eb0) . In Fig. 2, we have dis∼ cussed the variation of binding energy ( Eb ) in the present of exciton-SO phonon coupling, which can be obviously tuned by the interlayer distance, the internal distance and the number of the inserted h-BN layers. Therefore, the critical temperature can be modulated by these external ways. In the MoS2 (e)/h-BN/WS2 (h), the variation of the critical temperature (ΔTBKT ) as functions of two internal distances are presented in Fig. 3 (a); the dependences of the variation of the critical temperature on the interlayer distance and the number of the inserted h-BN layers are shown in Fig. 3 (b). From them, one can see that the critical temperature can be tuned in a very large scale by these external ways, which provides the potential methods to control the critical temperature of quantum Bose gas composing of these interlayer excitons in experiments. Here, we must admit that we only adopted the unitary transformations and a very simple variational calculation to roughly estimate the correction of binding energies arising from exciton-SO phonons coupling. In fact, several advanced methods basing on the first-principle calculations have been adopted in the studies of interlayer exciton [22,28–31]. However, our results can provide the comprehensive comparisons for the exciton binding energies between them.
Table 1 Interlayer exciton binding energies (meV) in different TMDS vdW heterostructures without the exciton-phonons coupling for the interlayer distance D = 4.0 nm and the thickness of the inserted h-BN is 2.0 nm (about six layers of h-BN). The subscripts (e) and (h) denote the positions for electron and hole in the corresponding monolayer materials, respectively.
MoS2 (h) MoSe2 (h) WS2 (h) WSe2 (h)
MoS2 (e)
MoSe2 (e)
WS2 (e)
WSe2 (e)
73.98 74.72 75.17 68.39
75.61 76.69 77.16 70.03
67.99 68.79 69.30 63.08
69.04 69.79 70.27 64.10
Table 2 Interlayer exciton binding energies (meV) in different TMDS vdW heterostructures with the exciton-phonon coupling at D = 4.0 nm and z 0 = z1=1.0 nm.
MoS2 (h) MoSe2 (h) WS2 (h) WSe2 (h)
MoS2 (e)
MoSe2 (e)
WS2 (e)
WSe2 (e)
45.50 46.29 46.78 38.35
47.52 48.90 49.47 40.02
37.23 37.90 37.89 32.09
38.72 39.35 39.79 33.38
binding energies of interlayer exciton are smaller than 100 meV for different TMDS vdW heterostructures, which can be attributed to the following reasons: (1) the larger interlayer distance D ≥ 3.0 nm between two TMDS layers are assumed, leading to the Coulomb interaction between electrons and holes are reduced directly in spatial; (2) the inserted h-BN layers enhance the screening effect and reduce the interaction of electron-hole pairs further. In addition, the exciton-intrinsic longitudinal optical phonons coupling also give the obvious correction to binding energies if the strong coupling regime exists in our recent study [47]. The SO phonon modes are induced by the inserted h-BN layers, which results in the strength of exciton-SO phonon coupling depending on the modulation of distances between different layers. In Fig. 2 (a), the dependences of the binding energies on the interlayer distance without (red lines) and with exciton-SO phonon coupling (blue lines) are shown for three types of vdW heterostructures. It can be seen that binding energies decrease with increasing the interlayer distance. The amplitudes are enhanced obviously if exciton-SO phonon coupling is taken into account shown in blue lines. Fig. 2 (b) shows the influence of the numbers of the inserted h-BN layers (or the thickness of h-BN layers) on binding energies with exciton-SO phonon coupling at fixed
Fig. 2. (a) The binding energies as a function of the interlayer distance without (red lines) and with (blue lines) exciton-SO phonon coupling for three types of heterostructures at z 0 = z1 = 1.0 nm. (b) The binding energies as a function of the numbers of h-BN layers with exciton-SO phonon coupling at D = 4.0 nm. (c) The binding energies as functions of two internal distances in MoS2 (e)/h-BN/WS2 (h) at D = 4.0 nm. 3
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Fig. 3. (a) The variation of critical temperature as functions of two internal distances in MoS2 (e)/h-BN/WS2 (h) at D = 4.0 nm; (b) the variation of critical temperature as functions of the interlayer distance and the number of the inserted h-BN layers at z 0 = z1= 1.0 nm. The density of exciton gas n = 1010cm2 is assumed.
4. Conclusion In summary, the binding energies of interlayer excitons in TMDS vdW heterostructures varied in several tens of meV range due to the exciton-SO phonon coupling, depending on the interlayer distance, the numbers of inserted h-BN layers and the internal distance. The markedly variation of interlayer exciton binding energies result in the critical temperature of its quantum Bose gas can be tuned in a very large scale. Funding This work was supported by National Natural Science Foundation of China (No 11674241). Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.jpcs.2019.05.022. References [1] A.K. Geim, I.V. Grigorieva, Nature 499 (2013) 419. [2] K.S. Novoselov, A. Mishchenko, A. Carvalho, A.H. Castro Neto, Science 353 (2016) 6298. [3] L.V. Butov, Superlattice. Microst. 108 (2017) 22. [4] T. Mueller, E. Malic, npj 2D Mater. Appl. 2 (2018) 29. [5] M.M. Fogler, L.V. Butov, K.S. Novoselov, Nat. Commun. 5 (2014) 4555. [6] E.V. Calman, C.J. Dorow, M.M. Fogler, L.V. Butov, S. Hu, A. Mishchenko, A.K. Geim, Appl. Phys. Lett. 108 (2016) 101901.
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