Thickness and Stacking Dependent Polarizability and Dielectric

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Thickness and Stacking Dependent Polarizability and Dielectric Constant of hBN-Graphene Composite Stacks Piyush Kumar, Yogesh Singh Chauhan, Amit Agarwal, and Somnath Bhowmick J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b05805 • Publication Date (Web): 12 Jul 2016 Downloaded from http://pubs.acs.org on July 16, 2016

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The Journal of Physical Chemistry

Thickness and Stacking Dependent Polarizability and Dielectric Constant of Graphene-hBN Composite Stacks Piyush Kumar,† Yogesh Singh Chauhan,† Amit Agarwal,∗,‡ and Somnath Bhowmick∗,¶ Dept. of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India, Dept. of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India , and Dept. of Materials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India E-mail: [email protected]; [email protected]

∗ To

whom correspondence should be addressed of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India ‡ Dept. of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India ¶ Dept. of Materials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India † Dept.

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Abstract Giant carrier mobility of graphene is significantly reduced due to external perturbations, such as substrate based charge impurities, and their impact can be minimized by encapsulating graphene between hexagonal boron nitride (hBN) layers. Using density functional theory (DFT) based ab initio calculations, we study the static response of such a composite by placing it in a vertical electric field. We find that at relatively low electric field (∼ 0.1 V/Å), although the relative permittivity (εr ) of a composite stack increases with the number of layers, but εr for a fixed stack thickness is independent of the field strength. However, at higher electric field strength, εr increases monotonically with the applied field strength even for a fixed stack thickness, signifying non-linear response. The relative permittivity changes more readily for graphene rich stacks as compared to hBN rich stacks, which is consistent with the property of the pristine phases. We also present an empirical formulation to calculate the thickness and stacking dependent effective dielectric constant of any arbitrary stack of graphene-hBN layers, which fits very well with the ab inito calculations. Our empirical formulation will also be applicable for van der Waals stacks of other two dimensional materials and will be useful for designing and interpreting transport experiments, where electrostatic effects such as capacitance and charge screening are important.

1

Introduction

Emergence of graphene 1–7 and other layered materials have paved the way for a new generation of electronic devices based on 2D materials. In graphene, electrons behave like massless Dirac Fermions, 8,9 giving rise to very high carrier mobility, ranging upto 200,000 cm2 V−1 s−1 in suspended graphene sheets 10 and around 8,000 cm2 V−1 s−1 in field effect transistor (FET) devices. 11–13 However, properties of graphene are very sensitive to the environment and carrier mobility is significantly reduced by increased scattering due to roughness of the substrate and trapped charges present in it. 14,15 Encapsulating graphene layers by other 2D materials, like hBN (hexagonal boron nitride) layers, is one of the possible routes for eliminating the environmental


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sensitivity of graphene. 16–21 Since hBN has a very high bandgap (∼ 5.97 eV) 22 and very small lattice mismatch with graphene (∼ 1.7 %), 23 it qualifies as an ideal candidate for the purpose of encapsulation in graphene based devices. Indeed, such devices are found to have superior carrier mobility (∼ 100,000 cm2 V−1 s−1 ). 24 Confining graphene layers within hBN helps to eliminate external perturbations and thus improves it’s carrier mobility and also modifies the electronic band structure by opening a band gap which can further be tuned by applying an electric field. 23,25–37 However, the effect of hBN encapsulation is not clear in terms of dielectric properties of such composite stacks. Many important practical applications like capacitance and energy storage capacity of a material as well as fundamental properties like charge screening in a material depends on it’s polarizability and relative permittivity. Although ab initio studies of polarizability and permittivity have been done for stacks of pristine materials, 38–40 graphene-hBN composite stacks are yet to be investigated in terms of dielectric properties. This necessitates a detailed study of intrinsic dielectric properties of graphene-hBN composite stacks from ab initio calculations. In this paper we calculate the polarizability (α) and relative permittivity (εr ) for graphene, hBN and their composite stacks, using density functional theory (DFT) based ab initio simulations. Since the number of possible graphene-hBN composite stacks (in terms of number of layers and order of stacking of the individual layers) is very large, it is practically impossible to cover all of them by ab initio calculations. Thus we propose a simple analytical model to predict α and εr for any possible stacking combination, in terms of number of layers, as well as arrangement of pristine layers within a given stack. Details of the ab initio calculations, and the methodology used are presented in the supporting information. 41 Here, we directly proceed to discuss the case of pristine material (either graphene or hBN stack) in Section 2, followed by a discussion on composite stacks of graphene and hBN in Section 3, and conclude our findings in Section 4.



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2

Pristine graphene and hBN stacks

When a material is placed in a transverse external electric field, it responds via a shift in the charge distribution. This leads to the formation of local dipoles inside the material, depending on it’s microscopic polarizability. Macroscopically, effect of electric field is described in terms of dielectric constant or relative permittivity of a material. Combining the microscopic and macroscopic description, α can be calculated from εr and vice versa, for example, via the Clausius-Mossotti equation which is valid for a homogeneous and isotropic dielectric material. However, layered materials like graphene and hBN are highly anisotropic, having significant difference between in plane and out of plane properties. For transport applications, such materials are often subjected to an electric field applied in the vertical direction, for example in a FET device. Prompted by this, we apply an electric field, directed normal to the plane of the graphene-hBN composite stack and study their dielectric response. We start with a discussion of the dielectric properties of the pristine phases. As shown in Figure 1(a), the relative permittivity of monolayer, bilayer and trilayer graphene does not change as a function of the applied field strength (checked up to 0.7 V/Å), while for four or more number of layers, there is a significant enhancement at higher electric field. The deviation of the relative permittivity from it’s low field value starts beyond certain critical electric field strength, whose value is inversely proportional to the number of graphene layers. On the other hand, we find that the relative permittivity of hBN is completely independent of the electric field strength, verified for stacks containing up to 8 monolayers [see Figure 1(b)]. Comparing relative permittivity of graphene and hBN, it is anticipated that the former can screen the external field more effectively inside the stack. This is confirmed by plotting the effective electric field, normalized by the external field strength, as shown in Figure 2(a) and (b), for four layers of graphene and hBN, respectively. Clearly, beyond 0.3 V/Å, the effective electric field inside the stack decreases in graphene, while it remains unchanged for hBN. Further analysis reveals that the charges accumulated at the outermost layers create an electric field opposing the external field, such that the effective field strength inside the stack is less than the applied strength. 4 ACS Paragon Plus Environment

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Average Slab Permittivity

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2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 0.0

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Eext (V/Å) Figure 1: Relative permittivity, plotted as a function of the externally applied transverse electric field strength for multi-layered (a) graphene and (b) hBN.



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Figure 2: Ratio of the effective electric field and the externally applied field in four layered stack of (a) graphene and (b) hBN. The dashed line in both panels shows Eeff /Eext for a six layered slabs at 0.5 V/Å electric field. The vertical lines indicate the location of individual monolayers. Note that in hBN, Eeff /Eext is completely independent of the applied field strength and this leads to its relative permittivity being independent of the applied electric field. (c) Top: planar average of the induced charge density in a finite electric field, plotted along the axis vertical to the slab surface. Solid lines are for graphene and the dashed lines are for hBN. Bottom: Charge accumulation at the outermost layer (calculated by integrating the induced charge density within the shaded region), which is responsible for electric field screening inside the stack. This clearly explains the reason behind better screening inside the graphene stack than compared to that of hBN. 6 ACS Paragon Plus Environment

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10

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Number of Layers Figure 3: (a) Polarizability and (b) relative permittivity of graphene and hBN, plotted as a function of number of layers. The solid and dashed line represents a low electric field strength of 0.02 V/Å (where the dielectric constant is independent of electric field) and a high electric field strength of 0.3 V/Å, respectively. A very good match is evident in the relative permittivity calculated from DFT, and that predicted by the model – Eq. (3) when the electric field strength is low. Note that for hBN, the results for both slab polarizability and the relative permittivity are identical for low and high electric fields. 7 ACS Paragon Plus Environment


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On application of an electric field, the response of the system can be quantified in terms of the induced charge density δ ρ(r), defined as the difference between the charge density in presence of electric field ρext (r), and the charge density in absence of electric field ρ0 (r). Since the applied electric field is in the zˆ direction, we take the planar average in the x-y plane, denoted by hδ ρ(z)i [see Eq. (S1) 41 ] and calculate the induced charge density per monolayer by integrating hδ ρ(z)i from zn − d/2 to zn + d/2, where zn is the location of nth monolayer and d is the inter layer spacing. In case of a surface layer, the limit is extended in the outward direction until hδ ρ(z)i becomes negligible (away from the outermost surface). As shown in Figure 2(c), the induced charge density for the outermost layer increases linearly with the external field strength for hBN, while we observe a non-linear rise in the case of graphene. As a consequence, graphene can screen the external electric field better than hBN and the difference is very prominent at higher electric field strength. Having obtained the planar averaged charge density and the effective electric field from ab initio calculations, the spatially resolved dielectric permittivity can be easily calculated, and it can be averaged in the perpendicular direction to obtain the thickness and electric field dependent εr for a given stack. 41,42 The layer dependent relative permittivity is shown in Figure 3 for both graphene and hBN, showing an increase with increasing number of layers, a trend observed in other layered materials as well. 38–40 The calculated value of the relative permittivity for a given stack, a macroscopic parameter, can be used further to compute the value of slab polarizability, a microscopic parameter. In case of a slab made of N layers, the net polarizability [summed over all the constituent atoms], termed as slab polarizability α(N), can be related to its average relative permittivity 43–45 via the following relation, Ωcell (N) α(N) = 4π

εr (N) − 1 εr (N)

,

(1)

where Ωcell (N) is the volume of the slab containing N monolayers [see Figure S1(b) and the related text in the supporting information 41 ]. Note that Eq. (1) is a direct consequence of the average induced slab polarization (P) being proportional to the local electric field (Eloc ), which is given by Eloc = Eeff + P/ε0 in 2D materials. Slab polarizability is displayed in Figure 3, as a function of the 8 ACS Paragon Plus Environment

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number of layers as obtained from Eq. (1), for 2 to 8 layers of graphene and hBN, respectively and a linear increase is observed with increasing number of layers. Similar behavior was reported earlier for graphene, 44 benzene, 43 GaS 45 and phosphorene 42 and the linear trend of the slab polarizability simply implies that the atomic polarizability (per atom) is roughly constant. Polarizability per layer per unit cell (containing two atoms) can be obtained from the slope of the α(N) curve [see Figure 3(a)] and values are found to be αgr = 0.949 Å3 and αhBN = 0.894 Å3 for pristine graphene and hBN, respectively. The linear variation in slab polarizability can be expressed in terms of bulk dielectric constant (εbulk ) of the material as, 43–45 Ωbulk α(N) = N 4π

εbulk − 1 + 2αs , εbulk

where Ωbulk is defined as the volume occupied per layer in a bulk unit cell

(2)

1

[consisting of

two monolayers and no vacuum in the z direction] and αs is a constant, which accounts for the difference in the polarizability of surface and bulk layers. From the intercept of the linear fit [see Figure 3(a)] of the polarizability data, obtained from Eq. (1), values of αs are found to be αs−gr = −0.0463 and αs−hBN = −0.0023 for graphene and hBN surface, respectively. Higher magnitude of αs for graphene as compared to hBN implies that the surface effects are more prominent in the former. The values of εbulk obtained using Eq. (2) for graphene and hBN are 3.48 and 3.09 2 respectively, which are consistent with previously reported values. 44,46–48 Combining Eq. (1) and (2), we can predict the relative permittivity of any number of layers in the limit of low electric field 42 (where εr is independent of the applied field strength, say at Eext = 0.02 V/Å considered for the plots) as, 8παs −1 Ωbulk εbulk − 1 − . εr (N) = 1 − N Ωcell (N) εbulk Ωcell (N) 1 Note

(3)

that our Ωbulk (per monolayer) is at variance with the Ωbulk defined by Tóbik and Dal Corso 43 (for two

layers). 2 In case of hBN, a good agreement with previously reported values is observed, despite ignoring the ionic part and considering only the electronic contribution to the dielectric function.



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Predicted values of the relative permittivity (up to 20 layers) are plotted in Figure 3(b), showing a very good match with the values obtained from ab initio calculations (up to 8 layers). Note that Eq. (3) has been shown to work very well for the case of layered phosphorene as well. 42 At higher electric field (where the relative permittivity is not independent of the applied field strength), while the slab polarizability of hBN is still found to be a linear function of number of layers, non-linear behavior can clearly be observed in the polarizability of graphene [see the dashed line, Figure 3(a)]. This is a direct consequence of the non-linear increase in the surface charges in graphene at higher electric field strength, as shown in Figure 2(c).

3

Graphene-hBN heterostructures

Based on the results presented so far, it is clear that graphene has better capability to screen external electric field as compared to hBN, which explains the observed difference of their dielectric properties. Similar to any composite material, graphene-hBN heterostructures are expected to have some average property, depending on the amount of the pristine phases (number of individual layers) present in it. However, considering variable thickness and order of stacking, the number of possible combinations is very large and it is impossible to characterize all of them based on ab initio calculations. To facilitate the prediction of slab polarizability and the dielectric constant of a composite stack of an arbitrary composition and thickness, we propose an empirical model in which the parameters can be obtained with the help of ab initio calculations, performed for a few structures. We emphasize that the model, which is discussed in the context of graphene and hBN in this work, is likely to work for other composite van der Waals stacks (consisting of layers of two different materials) as well. Since the number of possible arrangement of individual monolayers in a composite slab is very large, we focus on two extreme cases: graphene encapsulated between two hBN monolayers and alternately stacked graphene and hBN monolayers. As hBN can not screen the external electric field as effectively as graphene [see Figure 2], stacks of graphene (or any other 2D materials)


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46–48 Page 11 of 22


7

Microscopic Permittivity


6 7 8 6 1 1 ⇡↵ 19 s 8⇡↵s . (3) . (3) 10 ll (N 11 )⌦cell (N ) 5 12 13 ymittivity (up to 20(up lay-to 20 lay- 4 14 15good ry match ing a very good match 16 alculations (up initio calculations (up 3 17 18 shown work to work as beentoshown 19 ene as well42as . well42 . hosphorene 2 20 21 ve permittivity he relative permittivity 22 FIG. 4. Spatially resolved resolved ‘microscopic’ permittivity of seven of seven FIG. 4. Spatially ‘microscopic’ permittivity while while drength), field strength), 1 23 graphene and hBN, compared encapsulayers of hBN -10graphene -5 and 0hBN, compared 5with hBN 10with 15 encapsu24tofound nd be a to lin-be a layers still lin- of-15 lated graphene and alternately stacked hBN-graphene hetlated graphene and alternately stacked hBN-graphene hetr 25 behaviorbehavior can non-linear can z (Å) 26 ˚ ˚ erostructures, shown atshown (a) low (0.02low V/(0.02 A) and (b) high (b) high erostructures, at (a) V/A ) and 27 f graphene [see ability of graphene [see ˚ (0.3 V/A(0.3 ) electric values. of composite slabs V/˚ A)field electric fieldSchematic values.hBN Schematic of composite slabs 28 the non-linear uence of the non-linear 29 are also shown, layers graphene (blue) sandalso having shown,five having fiveof layers of graphene (blue) sandGraphene 16 are at30higheratelecraphene higher elecwiched between hBNtwo (red) monolayers and fourGraphene hBNfour lay-hBN laywiched two between hBN (red)Encapsulated monolayers and 31 g.322(c). 14 ers alternately ers stacked with three graphene layers. stacked alternately with three graphene layers. Alternately Stacked 33 34 12 35 36 RUCTURES EROSTRUCTURES two extreme cases: graphene encapsulated betweenbetween two cases: graphene encapsulated two 10 two extreme 37 hBN monolayers vs. alternately stacked stacked graphenegraphene and 38 hBN monolayers vs. alternately and 39 8 itso isfar, clear hBN monolayers. Note that, hBN canhBN not can screen ex- the exit that is clear that hBN monolayers. Note that, notthe screen 40 xternal electric electric ternal electric as e↵ectively as graphene [see Fig.[see 2]. Fig. 2]. 41 screen external electric field as e↵ectively as graphene 6 ternal field 42 sexplains the observed Thus, stacks graphene (or any other 2Dother materials) enthe observed Thus,ofstacks of graphene (or any 2D materials) en43 16,17 16,17 Similar to any to capsulated by hBN by hBN will not will havenot anyhave significant ef44 erties. Similar any 4 capsulated any significant ef45 ostructures are fect ofterms electric screening by the outer layers. N46heterostructures are in terms fect in of field electric field screening by the outer layers. 2 depending on However, since graphene has higher permittivity 47 roperty, depending on However, since graphene hasrelative higher relative permittivity 48 -15 -10 -5 0 5 10 of a com15 of a comber of individthan hBN [seehBN Fig.3], average permittivity es (number of individthan [seethe Fig.3], the average permittivity 49 dering variable variable posite stack of stack hBN and graphene is expected to be less r,50considering posite of hBN is expected to be less zand (Å)graphene 51 ber of possible than that of athat pristine graphenegraphene slab withslab equal number the number of possible than of a pristine with equal number 52 Figure 4: Spatially resolved ‘microscopic’ permittivity of seven layers of graphene and hBN,shows comossible to charof layers.ofThis is further in Fig. 4,inwhich 53 impossible to chart is layers. This is analyzed further analyzed Fig. 4,shows which 54 pared hBNspatially graphene“microscopic and alternately stacked hBN-graphene heterostructures, alculations. To withthe resolved permittivity” [ratio of [ratio of initio calculations. To encapsulated the spatially resolved “microscopic permittivity” 55 shown at (a) low (0.02 V/Å) and (b) high (0.3 V/Å) electric field values. Schematic of composite and dielectric the spatially resolved and planar averaged polarization P 56 izability and dielectric spatially and planar averaged polarization slabs are also shown, the having five layersresolved of graphene (blue) sandwiched between two hBN (red)P 41 57 41 trary composiand ; seelayers Eq. (S4), supporting information ] for both an composiand E see Eq. (S4), with supporting information ] for both e↵hBN monolayers and E four alternately three graphene layers. e↵ ;stacked 58 arbitrary l model, where kinds of kinds composite stacks (encapsulated and alternately 59 empirical model, where of composite stacks (encapsulated and alternately 60

(a)

(b)

of ab arranged) along with a comparison with their pristine hlpthe helpinitio of ab initio arranged) along with with their pristine 11 a comparison ACS Paragon Plus Environment es. We emphacounterparts. In Fig. 4,InitFig. is evident while graphene structures. We emphacounterparts. 4, it is that evident that while graphene the context of and have thehave highest and the and lowest ussed in the context of hBN and hBN the highest thepermittivity, lowest permittivity,


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encapsulated by hBN 16,17 will not have any significant effect in terms of electric field screening by the outer layers. However, since graphene has higher relative permittivity than hBN [see Figure 3], the average permittivity of a composite stack of hBN and graphene is expected to be less than that of a pristine graphene slab with equal number of layers. This is further analyzed in Figure 4, which shows the spatially resolved “microscopic permittivity” [ratio of the spatially resolved and planar averaged polarization P and Eeff ; see Eq. (S4) 41 ] for both kinds of composite stacks (encapsulated and alternately arranged) along with a comparison with their pristine counterparts. In Figure 4, it is evident that while graphene and hBN have the highest and the lowest permittivity, respectively, their composite heterostructures have permittivity values which lie in between. The difference is even more significant at higher electric field values [for example 0.3 V/Å], where the “microscopic permittivity” of graphene is roughly three times higher than that of hBN at the center of the slab. Since screening reduces the effective electric field further as we move from the surface layer towards the center of the graphene slab, the difference between graphene and hBN is more prominent in the middle of the slab. As expected, the spatially averaged relative permittivity of the composite stack is found to be range bound by εr of the pristine phases of equal slab thickness, with the actual value depending on the ratio of the number of graphene and hBN layers present in a particular stack. This was already highlighted for the “microscopic permittivity” in Figure 4, and is further illustrated in Figure 5 for the macroscopic relative permittivity for a seven layered slab of hBN encapsulated graphene vs. alternately stacked hBN-graphene and also compared with the pristine phases. Since the encapsulated graphene heterostructure has more number of graphene layers in it as compared to the alternately stacked heterostructure, the relative permittivity in the former is higher. To go beyond the ab initio calculations, and to facilitate the prediction of the dielectric constant of an arbitrary stack of two different layers, let us consider the polarizability of a composite stack. A general model for polarizability of a composite slab, made of N1 layers of graphene and N2


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4 .8

A v e r a g e S la b P e r m it t iv it y

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G ra p h e n e h B N E n c a p s u la te d G r a p h e n e A lte r n a te s ta c k in g

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Figure 5: Comparison of the relative permittivity (calculated for a seven layered stack) as a function of the electric field strength, of pristine graphene, hBN and their encapsulated and alternately stacked heterostructures. As expected, the relative permittivity of the composite stack lies between that of pristine stacks of graphene and hBN.



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40

Total no. of Layers Figure 6: Slab polarizability [Eq. 4] and relative permittivity [Eq. (5)] calculated from the model and compared with the values obtained from DFT calculations (at 0.02 V/Å) for (a) graphene encapsulated by hBN monolayers and (b) alternately stacked hBN-graphene slabs.


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layers of hBN, arranged in any arbitrary combination, can be written as α(N1 , N2 ) = N1 αgr + N2 αhBN + (αs1 + αs2 ) {z } | {z } | Bulk

Surface

αgr + αhBN + NI αI − NI , | {z 2 }

(4)

Interface

where NI is the number of hBN-graphene interfaces in the slab. The two surfaces (s1 and s2) of the slab can be made of either graphene or hBN and the related constants (αs−hBN and αs−gr ) are already calculated in Section 2. Values of αgr and αhBN , needed to account for the bulk portion, are also given in Section 2. The only unknown term, αI (which is required to account for the interface contribution to the slab polarizability) can be calculated from ab initio simulations of a slab, containing just three monolayers (hBN-graphene-hBN or graphene-hBN-graphene) and the value is found to be 0.890 Å3 [see Section SII 41 ]. Using the values of the slab polarizability obtained from Eq. (4), we can further predict the relative permittivity of a composite slab from our model. Rearranging the terms in Eq. (1), we can express the relative permittivity of a composite slab, made of N1 layers of graphene and N2 layers of hBN as,

4πα(N1 , N2 ) −1 , εr (N1 , N2 ) = 1 − Ωcell (N1 , N2 )

(5)

where α(N1 , N2 ) is the polarizability of the composite slab [obtained from Eq. (4)] and Ωcell (N1 , N2 ) is the slab volume [see Section SIII 41 ). Using Eq. (4) and Eq. (5), we calculate the relative permittivity of (a) graphene layers encapsulated between two hBN monolayers and (b) alternately stacked graphene and hBN monolayers (the latter forming both the outer layers), and compare them with the results of the ab initio calculations. Let us consider the case of encapsulated graphene first. Given that the number of graphene layers in the slab is equal to N1 and hBN forming the two outermost layers, Eq. (4) reduces to

α(N1 , 2) = (N1 − 1)αgr + αhBN + 2αs−hBN + 2αI . 15 ACS Paragon Plus Environment

(6)


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Using the values of α(N1 , 2) obtained from Eq. (6), we can predict the relative permittivity of the slab from Eq. (5). Thickness dependence of εr (N), as predicted by the model, is plotted in Figure 6(a) and compared with the values obtained from ab initio calculations, performed for 4 to 8 layered stacks and excellent agreement is observed between the model and ab initio results. Next, we consider alternately stacked graphene and hBN slabs. Predicted values of the slab polarizability (for a heterostructure containing N1 layers of graphene and N2 = N1 + 1 layers of hBN) obtained from Eq. (4) is used in Eq. (5) to calculate the relative permittivity from the model, as illustrated in Figure 6(b). Evidently the values predicted from the model are in excellent agreement with those obtained from ab initio calculations, performed for 3 to 11 layered composite stacks of graphene and hBN. A comparison of values predicted by our model vs. values calculated using ab inito method is given in Section SIV. 41 Before concluding, we would like to point out that the model, which is discussed in the context of graphene and hBN in this work, should also work for other composite (consisting of layers of two different materials) stacks for low electric field strengths, as long as the interlayer interactions are dominated by van der Waals forces. Furthermore, we believe that this model will also work for materials like MoS2 , where more hybridization is observed among the orbitals of constituent atoms, as long as the surface polarizability is accounted for correctly. However further investigation is required to verify this.

4

Conclusion

To summarize, we investigate the thickness and electric field dependent dielectric properties of graphene, hBN and their composite stacks using ab initio DFT calculations. We find that at relatively low electric field, the relative permittivity of graphene, hBN and their composite stacks is independent of applied field strength and it increases with increasing number of layers, similar to other layered materials. To investigate the thickness and stacking dependent dielectric constant of a composite stack of any number of layers, we propose a model to predict the relative permittivity


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and polarizability of the stack (for low electric field strength), which is successfully compared with the values predicted from ab initio calculations. However, ab initio calculations are necessary at high electric field, when the relative permittivity is no longer independent of applied field strength, particularly in graphene rich composite stacks. Electric field dependent permittivity of graphenehBN composite stacks can be attributed to the superior electric field screening ability of graphene as compared to hBN. Our study will be useful to design and interpret transport experiments based on gated devices of stacked materials, to correctly account for the electrostatic effects such as charge screening and capacitance.

Supporting Information • SI. Computational details • SII. Estimation of surface polarizability, αI • SIII. Ωcell for different stacks • SIV. Comparison of α(N) of graphene-hBN stacks obtained from DFT and model

Acknowledgements AA acknowledges funding from the DST INSPIRE Faculty Award. SB acknowledges funding from SERB Fast Track Scheme for Young Scientist. YSC and AA acknowledge funding from CSIR extramural research division (project no. 22(0699)/15/EMR-II). We also thank CC IITK for providing the HPC facility.

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Encapsulated Graphene

Alternately Stacked

8 6 4 2 -15

-10

-5

hBN Graphene Encapsulated Graphene Alternately Stacked

0

5

10

z (Å)

TOC Figure


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Thickness and Stacking Dependent Polarizability and Dielectric

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