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Nondegenerate chiral phonons in graphene/hexagonal boron nitride heterostructure from first-principles calculations Mengnan Gao, Wei Zhang, and Lifa Zhang Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b01487 • Publication Date (Web): 25 Jun 2018 Downloaded from http://pubs.acs.org on June 25, 2018
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Nano Letters
Nondegenerate chiral phonons in graphene/hexagonal boron nitride heterostructure from first-principles calculations Mengnan Gao,† Wei Zhang,∗,†,‡ and Lifa Zhang∗,† †Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 210023, China ‡Physicochemical Group of College of Criminal Science and Technology, Nanjing Forest Police College, Nanjing, 210023, China E-mail:
[email protected];
[email protected] Phone: +86 025 85878792; +86 025 85891245
Abstract Triggered by recent successful observation of previously predicted phonon chirality in the monolayer tungsten diselenide (Science 359, 579, (2018)), we systematically study the chiral phonons in the classical heterostructure of graphene/hexagonal boron nitride (G/h-BN) by the first-principles calculations. It is found that the broken inversion symmetry and the interlayer interaction of G/h-BN not only open the phononic gaps but also lift the degeneracy of left-handed and right-handed chiral phonons at the firstBrillouin-zone corners (valleys). At valleys, the hybridization makes chiral phonons modes solely contributed from one individual layer. Moreover, we demonstrate that the vertical stress is effective to tune the degenerated phononic gap while keeping the valley-phonon chirality of G/h-BN heterostructure, which is favorable for the Raman or ultrafast infrared spectroscopy measurement. We also analyze the pseudo-angular momentum of valley 1
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phonon modes, which provide important references for the excitation and measurement of the chiral phonons in the process of electronic intervalley scattering. Collectively, our results on the chiral phonons in the G/h-BN heterostructure system could stimulate more experimental and theoretical studies and promote the future applications on the phononchirality based phononics.
Keywords: chiral phonons, graphene/hexagonal boron nitride heterostructure, firstprinciples calculations, pseudo-angular momentum
Chirality is the asymmetric property of an object or a system that is different from its mirror image, which is well known for some molecules and for elementary. The electronic chirality in graphene can induce unique transport properties such as unconventional Landau quantization and Klein tunneling. 1,2 Optical chirality displays in the valley-electron interband scattering which results the emergence of valleytronics; 3,4 however, the intervalley scattering of electrons will involve valley phonons, 5 then a natural question arise: whether can the phonons involved attain chirality? The phonons in the magnetic materials can be distorted by the spin-phonon interaction and thus leads to the phonon Hall effect 6–8 and phonon angular momentum. 9 The nonzero phonon angular momentum means that the phonon can be circularly or elliptically polarized, that is, the phonon can attain chirality in magnetic materials. In the nonmagnetic hexagonal AB lattice with broken inversion symmetry, the non-degenerate chiral phonon was theoretically predicted recently, which is different from the circular polarization in non-chiral media by the superposition of linear modes. 10,11 Very recently, the predicted circularly rotating chiral phonons were experimentally verified in monolayer WSe2 , 12 which is a significant progress since it reveals that phonon, a kind of Bosonic collective excitation, can attain the chirality. Considering that we can vastly excite the chiral phonons by the optical pump-probe technique , 12 the manipulation
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of phonon chirality becomes possible, which is very important for the future applications. The chiral phonons are also important for the control of topological states, 13 the electronic phase transition, 14,15 the intervalley scattering, 4,5,10,16–19 as well as the solid state quantum information applications. 20,21 The finding of chiral phonons also promotes the progress of other fields such as the valley transport of sonic crystals, 19 the polarization in twodimensional (2D) materials 22 and so on. The monolayer AB lattices play a key role in the study of valleytronics and chiral phonics. 3,4,12,23 Although graphene can obtain the AB structure by isotope doping or staggering potentials, it is difficult to fabricate. Encouragingly, another research field called the van der Waals (vdW) heterostructure, has emerged as a fast growing field in the past years. 24 This atomic scale Lego is based on the integration of 2D materials. Among various 2D materials, the hexagonal boron nitride (h-BN) was demonstrated to serve as an excellent substrate for graphene. 25 h-BN has an atomically flat surface without any hanging bonds and charge traps, and the lattice mismatch between h-BN and graphene is small. 26 The high mobility of h-BN has been regarded as the highest among the insulating substrates. 27 The electron mobility and drain current switching ratios of h-BN based graphene (G/h-BN) transistor is three times larger than that without the h-BN substrate. 28 Therefore the G/hBN heterostructure is generally considered as a model vdW heterostructure in this field. 24 Especially, the electronic band gap of G/h-BN was observed by a recent experiment. 29 It is well known that the phonons participates in the process of electrons scatting between valleys. Therefore, it is highly desirable to investigate the phonons chirality in the G/h-BN heterostructure, and to explore the possible manipulation of chiral phonons.
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4π/3
2π/3
b 0 a c
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(c)
&1
&2 2π/3
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&2 b
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1
a
4π/3
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Figure 1: (a)-(b) Top view and side view of a ball and stick model of graphene/h-BN, with B in green, N in gray and C in brown. The black dotted line is the boundary of the primitive cell. (c)-(d) phase correlation of the intercell part for C(B) atom and N atom at valley K. The C atom above the B atom is named as C1 , the other is C2 . In this letter, by the first-principles calculations, we study the chiral phonons in the classical G/h-BN heterostructure. We found that the broken inversion symmetry and the interlayer interaction of G/h-BN not only open the phononic gaps but also lift the degeneracy of left-handed and right-handed chiral phonons at the valleys. Moreover, it is found that the vertical stress can be an effective way to tune the degenerated phononic gap while keeping the valley-phonon chirality, which is favorable for the Raman or ultrafast infrared spectroscopy measurement. The pseudo-angular momentum of valley phonon modes are systematically explored to provide important references for the excitation and 4
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measurement of the chiral phonons through electronic intervalley scattering.
Structure and Method Firstly we constructed three kinds of G/h-BN heterostructures. The first stacking pattern is hexagonal (AA) stacking arrangement with all C atoms right above B or N atoms. The second type has Bernal stacking (AB) feature, with one carbon atom sits on top of a boron atom and one sits in the middle of a BN hexagon. The third type is 0
AB configuration, with one carbon atom sits on top of one nitrogen atom. Total energy calculations show that the AB type is most favorable, which is consistent with previous work. 30 Thus only the AB structure is considered hereafter, as shown in Fig.1(a)-(b). The phonon dispersions are calculated by the Quantum Espresso Code. 31 The local density approximation (LDA) is used for the exchange and correlation energy functional. The normcons pseudopotentials are used and the kinetic energy cutoff for wavefunctions is 65 Ry. Within the framework of density functional perturbation theory (DFPT), the dynamical matrices are calculated using a 7×7×1 q-point mesh. A vacuum space of 18 Å is used in the direction normal to the layers. For the G/h-BN heterostructure, the lattice parameter and interlayer distance are 2.49 Å and 3.35 Å, respectively. To illustrate the phonon chirality, we characterize it by the polarization of phonons, which can be calculated from the phonon eigenvector e by first principle calculation. The phonon eigenvector e can be represented (here we consider that each unit cell has n atoms) as e = ( x1 y1 z1 x2 y2 z2 · · · · · · xn yn zn ) T . We define a new basis where one of sublattices has right-handed or left-handed circular polarization as | R1 i ≡ √1 (1 2 1 √ (0 2
− i 0 · · · · · · 0)T , | Z1 i ≡ (0 0 1 · · · · · · 0)T , | Rn i ≡
√1 (1 2
√1 (0 2
i 0 · · · · · · 0) T , | L1 i ≡
· · · · · · 1 i 0) T , | L n i ≡
· · · · · · 1 − i 0)T , | Zn i ≡ (0 · · · · · · 0 0 1)T , on which the phonon eigenvector e can be
written as n
e=
∑ (eRα | Rα i + eLα | Lα i + zα |Zα i),
(1)
α =1
where eRα = h Rα |ei, e Lα = h Lα |ei. So the phonon circular polarization operator along the z
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direction can be written as Sˆz ≡
n
∑ Sˆαz ≡
α =1
n
∑ (| Rα ih Rα | − | Lα ih Lα |),
(2)
α =1
and the phonon polarization is Szph = e† Sˆz e¯h =
n
∑ Sαz =
α =1
n
∑ (|eRα |2 − |eLα |2 )h¯ ,
(3)
α =1
where Sαz is the polarization of α-th sublattice in the unit cell.
Monolayers of graphene and h-BN We start our analysis from the monolayers of graphene and h-BN. The phonon dispersions are shown in Fig.2, where the insets denote the circular motions of each atom. In graphene (Fig.2(a)), due to the presence of spatial inversion symmetry, there are two degenerate points at the K point. The first degeneracy point appears at the cross of ZA(K) and ZO(K) modes, the second one occurs at the branches of LA(K) and LO(K). From the insets of Fig.2(a), we can see that two carbon atoms of the two modes (TA(K) and LO(K)) create opposite circular vibration with same magnitudes. Thus for this two modes the total phonon circular polarization Szph is zero at valleys. By superposition of the degenerate modes (ZA(K) and ZO(K), or LA(K) and LO(K)), one can obtain left-handed or right-handed chiral phonons with the same energy.
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Figure 2: Phonon dispersions of (a) graphene and (b) h-BN along the Γ-K directions. The circular polarizations at the K point are marked accordingly. Brown, green and gray balls represent the C, B and N atoms, respectively, the dots on the balls denote the linear motion along z direction.
Fig.2(b) shows the phonon dispersion of monolayer h-BN. The spatial inversion symmetry is broken since B and N atoms have different masses, which makes the degenerate points split at the K point. Fig.2 (b) also shows ZA(K) of and ZO(K) are linearly polarized and only one of the two atoms vibrates linearly along the z axis with the other atom being still. For TA(K) and LO(K), the two atoms do the opposite circular polarization, and the amplitudes of N atoms are smaller than that of the B atoms. For phonon modes of LA(K) and TO(K), only one N or B atom in the unit cell is circularly polarized, which leads to the nonzero Szph . Here the phonon chirality is the same with those from model calculation in honeycomb AB lattice. 23 Since the atoms of B and N correspond to the A and B atoms in honeycomb AB lattice, the three combinations for circular polarization are: SzB < 0, SzN > 0, SzB = 0, SzN < 0, SzB > 0, SzN = 0, which can be explained from the three fold symmetry of system. 23 At the Γ point of both graphene and h-BN, where the time reversal symmetry is pre7
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served, the doubly degenerate acoustic modes (LA, TA) and optical modes (LO, TO) modes become degenerate, and we can always get the circular polarized phonons by superimposing the degenerate modes, which can resolve optical helicity in the electronic interband scattering at valleys. 11 ZA and ZO modes are linearly polarized along the z direction, which have no phase change under threefold rotation.
Chiral phonons in G/h-BN heterostructure Although chiral phonons can be attained in monolayer h-BN, the insulating properties of which limit its application. Next we turn to the investigation of chiral phonons in excellent candidate of vdW heterostructure of G/h-BN. The calculated phonon spectra along the Γ-K direction is shown in Fig.3(a). Comparing with the single layers, we can clearly see from (Fig.3(b)-(c)) that the broken spatial inversion symmetry and the interlayer interaction lift the degeneracy of ZO1 (K) and ZO3 (K), LO1 (K) and LO3 (K), which are mainly contributed by graphene, thus two phononic gaps are opened. Fig.3(d) shows that the phonon polarization of heterostructure has sharp peak when close to the K point (we only draw the bands which reach the peak at K point), whereas the phonon polarization approaches zero at the vicinity of Γ. Interestingly, there are four modes: TO2 (K), LA(K), LO1 (K), LO3 (K) which have circular polarizations with maximum values of h¯ .
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ZO (K)
3
570
ZO (K)
1
1143.0
LO (K)
3
LO (K)
1
1142.5
560
1142.0
550
(c)
(b)
Figure 3: (a) Phonon dispersion of G/h-BN heterostructure along the Γ-K direction. (b)(c) Magnified dispersion diagrams for ZO1 (K), ZO3 (K) and LO1 (K) LO3 (K). (d) Phonon polarization of G/h-BN along the Γ-K direction. To get more insight about the phonon chirality in the G/h-BN heterostructure, we show the eigenmodes at the Γ and K points in Fig.4. From the vibration modes at Γ, we can denote phonon branches by ZA, TA, LA, ZO1 , TO1 , LO1 , ..., ZOn−1 , TOn−1 ,LOn−1 , where n is the number of sublattices in one unit cell. Since the C and B atom overlap in the top view, we separate the upper and lower layers to facilitate the marking of the vibration of each
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atom. As shown in Fig.4(a), the first column displays phonons of three acoustic modes (TA, LA, ZA), and the others are optic branches. The second column shows optical modes of ZO1 , TO1 , LO1 , where the atoms in upper layer (graphene) and the lower ones (h-BN) vibrates in opposite directions. In the third column for optical modes of ZO2 , TO2 , LO2 , only two atoms in the h-BN layer vibrates opposite to each other while the two ones in upper layer keep still. The fourth column plots optical modes in which only the graphene layer vibrate. In each row of eigenmodes, the phonon energy is increasing from left to right. Therefore, due to the interlayer interaction, the phonon modes has been hybridized, such as the ZO1 , TO1 , LO1 are superposed from acoustic modes of individual layers. According the above analysis for phonon modes at Γ, we can denote all the phonon branches. For phonon modes at valleys, the phonons are not as linearly polarized as those at Γ, thus they cannot be strictly called as ZA, TA, LA, ZO, TO, LO, ..., however, for convenience we denote the modes by using the analysis at Γ but add ’(K)’ behind. The hybridization makes phonons modes solely contributed from one individual layer as shown in Fig.4(b). The first four bands are linearly polarized along the z axis where every modes only involves only one atom vibration along the positive or negative direction of the z axis, so their magnitude of Szph are zero. Except the lowest linearly polarized modes along the z direction, the other eight modes is circular polarized.
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ZA
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TA (K)
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TO (K)
TO (K)
LA (K)
LO (K)
LO (K)
LO (K)
1
1
1
2
2
2
3
3
3
(b)
Figure 4: Phonon vibration modes of B (Green ball), N (Gray ball), C1 and C2 (Brown ball) atoms of G/h-BN heterostructure at Γ (a) and K points (b), respectively. The arrow denotes the rotation direction. The points and forks on the balls represent vibrations along the positive and negative directions of z axis, respectively. 11 ACS Paragon Plus Environment
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In the eight modes with sublattices vibrating circularly, by hybridization of valley modes from two individual layers, the four circular modes from h-BN remains with atoms in the graphene layer keeping still, see TA(K), TO2 (K), LA(K) and LO2 (K) in Fig.4(b), which are contributed from the modes of TA(K), TO(K), LA(K) and LO(K) of h-BN in Fig.2(b), respectively. In monolayer graphene, phonon branches LA and LO degenerate at K, thus chiral phonons can be attained by superposition of the degenerated modes, but they have the same energy. Due to the Van der Waals interlayer interaction and the broken inversion symmetry, the degenerated chiral phonons contributed from the graphene layer are separated but the chirality remains, see LO1 (K) and LO3 (K), where only C1 atoms does left-handed circular vibration (LO1 (K)) and only C2 atom has a right-handed circular vibration (LO3 (K)) while all the other atoms keep still. For the TO1 (K) and TO3 (K), its phonon polarization are contributed from the modes of TA(K) and TO(K) in monolayer graphene. Here the total polarization is near to zero but not zero, which is because the two carbon atoms do the opposite circular vibration with different magnitudes due to the distinct force acting on them from h-BN layer. The three combinations for sublattice polarization are SCz 1 > 0, SCz 2 < 0, SCz 1 < 0, SCz 2 = 0, SCz 1 = 0, SCz 2 > 0, which is opposite to those in h-BN layer since the location of atoms C1 and C2 correspond to atoms B and A in a honeycomb AB lattice, respectively. For the AA and AB’ stacking types (Supporting Information), the phonon energies of corresponding modes are very close to the case of AB type. The degenerated chiral phonons contributed from the graphene layer are also separated and the chirality remains. However, since from AB to AB’ the relative locations of the two layers exchange, the chiralities for all the atoms change to their opposite signs. For AA lattice, the chiralities of B and N in the layer of BN are the same with those in AB lattice; and the chiralities of the two atoms C1 and C2 are the same with those of B and N atoms, respectively, since the two layers coincide in the top view.
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1143
1142
1141
1140
(b)
Figure 5: (a) Phonon dispersion relation of graphene/h-BN heterostructure with interlayer distances d = 2.7, 3.0 and 3.35 Å. (b) Magnified dispersion diagram for LO1 (K) and LO3 (K), which is denoted by the small rectangle.
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Vertical stress effect According to the first principle calculation, the most stable interlayer distance (d) is d=3.35 Å. We apply vertical stress on the heterostructure by decreasing the interlayer distance. In Fig.5 we calculate the phonon dispersions of G/h-BN heterostructure with d=2.7, 3 and 3.35 Å. We find that at Γ with decreasing d, the frequencies of lowest optical modes will increase (marked by the lowest large rectangles in Fig.5 (a)); it is also found that at valley K the phononic gap between ZO1 (K)(ZO3 (K)) increases (marked by circles). Most remarkably, as the stress increasing with shortening interlayer distance, the gap between LO1 (K)and LO3 (K) enlarges while the phonons chirality remains with one of carbon atoms doing circular polarized vibration. Therefore with applying vertical stress perpendicular to the plane of the layered heterostructure of G/h-BN, the phononic gaps can be effectively tuned while keeping the robust phonon chirality, which is favorable for the Raman or ultrafast infrared spectroscopy measurement since the larger gap will make the phonons with different chirality more distinguishable.
Phonon pseudoangular momentum The valley chiral phonons take part in the scatting of valley electrons, which decide the selection rules. The threefold rotational symmetry at the high-symmetry K valley and Γ point endows the phonon eigenmodes with a quantized pseudoangular momentum (PAM), 23 the conservation of which is proposed 23 and verified by the experiment. 12 And we find there are definite combination of sublattice circular polarization, which can also be explained from the PAM of the phonons. Therefore it is important to study the PAM of every phonon mode. Under the threefold discrete rotation, we can get
