Thickness and electric field dependent polarizability and dielectric constant inphosphorene

Piyush Kumar,1 B. S. Bhadoria,2 Sanjay Kumar,3 Somnath Bhowmick,4 Yogesh Singh Chauhan,1 and Amit Agarwal3

1Dept. of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India2Dept. of Physics, Bundelkhand University, Jhansi, 284128, India

3Dept. of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India4Dept. of Materials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India

(Dated: March 24, 2016)

Based on extensive first principle calculations, we explore the thickness dependent effective di-electric constant and slab polarizability of few layer black phosphorene. We find that the dielectricconstant in ultra-thin phosphorene is thickness dependent and it can be further tuned by applyingan out of plane electric field. The decreasing dielectric constant with reducing number of layersof phosphorene, is a direct consequence of the lower permittivity of the surface layers and the in-creasing surface to volume ratio. We also show that the slab polarizability depends linearly on thenumber of layers, implying a nearly constant polarizability per phosphorus atom. Our calculationof the thickness and electric field dependent dielectric properties will be useful for designing andinterpreting transport experiments in gated phosphorene devices, wherever electrostatic effects suchas capacitance, charge screening etc. are important.

I. INTRODUCTION

Monolayer black phosphorene1 has emerged as apromising material for p-type FET (field effect transis-tor) operation, on account of it’s direct bandgap of mag-nitude 1.5 eV and reasonably high hole mobility value,theoretically predicted and experimentally measured tobe around 10,000 and 1,000 cm2V−1s−1, respectively1–3.As a consequence, transistors based on phosphorene ex-hibit excellent ION/IOFF ratio of 104 − 105, as well ascarrier mobility comparable to that of MoS2 (200 – 1,000cm2V−1s−1)1,2,4. Individual layers of black phosphorushave puckered honeycomb like structure and they areheld together by weak van-der Waals forces, facilitat-ing separation of monolayer or few layers by mechan-ical or liquid exfoliation1,5,6. Phosphorene is a highlyanisotropic material and values of elastic modulus, elec-tron and hole effective mass, calculated along mutuallyorthogonal zigzag and armchair direction, have a ratioof 3.5, 6.6 and 42.3, respectively3, which is also mani-fested in experimental observations reporting anisotropicelectronic transport and optical properties1,7–9.

It has been shown that the intrinsic bandgap of phos-phorene and other layered materials can be modulatedby applying external perturbation like strain and electricfield10–19. More interestingly, it has also been predictedthat an electric field applied out of the plane of phos-phorene can induce a tunable Dirac cone, and addition-ally induce a normal insulator to topological insulator tometal transition20,21. Such studies vividly demonstratethe tunability of electronic properties of multi-layeredphosphorene, on application of an external electric field.Motivated by this, in this article we study the thicknessand electric field dependent effective dielectric constantof N -layers of phosphorene subjected to an out of plane(transverse) electric field. The dielectric constant and po-larizability of a material, in addition to being fundamen-

tally important properties, are also very useful for char-acterizing the material’s electrostatic properties such aselectronic charge screening which determines the strengthof Coulomb interaction, capacitance, energy storage ca-pacity, and its transport characteristics22–26.

In this article we use density functional theory (DFT)based electronic structure calculations, including van-derWaals interactions, to explicitly show that the dielectricconstant of phosphorene, perpendicular to the phospho-rene plane, depends on the number of layers and thestrength of the applied transverse electric field. Usingthese calculations for a few layers (from two to eight)phosphorene, we obtain the bulk dielectric constant andthe surface polarizability of phosphorene atoms, which inturn are used to correctly extrapolate the dielectric con-stant for any number of layers, at least in the low electricfield regime. The article is organized as follows: in sec-tion II we report the structural parameters and electronicproperties of multi-layer phosphorene, followed by a dis-cussion of the impact of transverse electric field on chargescreening in section III, and it’s effect on the slab polar-izability and dielectric constant in section IV. Finally wesummarize our findings in section V.

II. STRUCTURAL AND ELECTRONICPROPERTIES OF MULTI-LAYER

PHOSPHORENE

We calculate the structural and electronic proper-ties of multilayered black phosphorene based on DFTcalculations, using a plane wave basis set (cutoff en-ergy 100 Ry) and norm conserving Troullier Mar-tins pseudopotential,27 as implemented in QuantumEspresso.28 The electron exchange-correlation is treatedwithin the framework of generalized gradient approxima-tion (GGA), as proposed by Perdrew-Burke-Ernzerhof(PBE).29 The dispersion forces among multiple phospho-

arX

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all]

23

Mar

201

6

2

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-1

0

1

2

Ener

gy (e

V)

M X Y M

EF

G

a)

b)

a

b

R1

R2θ2

θ1

d

c) d)

a = 4.5178 Åb = 3.3268 Åd = 3.2047 ÅR1 = 2.2376 ÅR2 = 2.2720 Åθ1 = 96.042oθ2 = 103.033o

e) f)

a)

b)

a

b

R1

R2θ2

θ1

d

c) d)

a = 4.5178 Åb = 3.3268 Åd = 3.2047 ÅR1 = 2.2376 ÅR2 = 2.2720 Åθ1 = 96.042oθ2 = 103.033o

e) f)a = 4.52 A

b = 3.33 A

d = 3.20 A

R1 = 2.24 A

R2 = 2.27 A1 = 96.04

2 = 103.03

0 2 4 6 8 10-2

0

2

Eext = 0

Density of States

Eext = 1.36 V/Å

(a)

(b) (c)

Eext = 1 V/A

Eext = 0

t

FIG. 1. (a) Top view and side view of bilayer phosphorene(BP) for AB stacking of layers, calculated using optB88-vdW.(b) Band structure and (c) Density of States of BP, in theabsence (red curve) and presence (blue curve) of a trans-verse electric field of 1.0V/A. The red curve clearly showsan intrinsic direct bandgap of 0.48 eV at the Γ point. Notethat in presence of a large transverse electric field (bluecurve), the bandgap reduces to zero signifying a semiconduc-tor to metal transition which is also known to be a topo-logical phase transition.21 Structural properties for bilayerAB stacked phosphorene are consistent with those reportedearlier.10

rene layers are taken into account by using non-localvan-der Waals exchange correlation functional optB88-vdW.30–34 Brillouin Zone integrations are performed us-ing a k-point grid of 16× 20× 1. Further we have used avacuum region of 20 A in the direction perpendicular tothe phosphorene plane to prevent any interaction amongthe spurious replica images. All the structures are fullyoptimized until the forces on each atom are less than 0.01eV/A. The effect of transverse electric field is simulatedusing a sawtooth potential, while taking dipole correctioninto consideration.35

In Fig 1(a), we illustrate the crystal structure and theelectronic band structure of AB stacked bilayer phos-phorene. Comparing with their bulk values, we findthat in bilayer phosphorene the lattice parameter a is0.9% larger, while b and inter-layer distance d remainsthe same [see Table I]. With increasing number of lay-ers, magnitude of a decreases towards it’s bulk value of4.48 A [see Table I]. The GGA-PBE estimated bandgapin bilayer phosphorene has a value of 0.48 eV, which de-creases towards it’s bulk value of 0.04 eV, the magnitudebeing inversely proportional to the number of layers [seeTable I]. Although GGA-PBE can qualitatively predictthe electronic band structure of a material correctly, it’slimitations are well known regarding bandgap underesti-mation. For example, a value of 1.04 eV (almost twicethan our GGA-PBE estimate) is reported for bilayer

NL a (A) b (A) d (A) Bandgap (eV)

2 4.52 3.33 3.21 0.48

3 4.51 3.33 3.21 0.28

4 4.51 3.33 3.21 0.19

5 4.50 3.33 3.21 0.13

6 4.50 3.33 3.21 0.09

8 4.49 3.33 3.21 0.07

Bulk 4.48 3.33 3.21 0.04

TABLE I. Structural parameters and bandgap of multi-layered and bulk black phosphorus calculated using GGA-PBE.

phosphorene, based on hybrid functional (HSE06) basedcalculations, which is known to predict bandgap moreaccurately.10 Evidently the band structure is anisotropic(see curvature along Γ-X and Γ-Y directions), which isalso going to be reflected in effective mass, mobility etc.and this qualitatively explains the experimentally ob-served strong directionality of electronic, thermal andoptical properties of multilayered black phosphorene.1,7,8

III. IMPACT OF A TRANSVERSE ELECTRICFIELD AND CHARGE SCREENING

Having described the structural and electronic prop-erties of multi-layered phosphorene, we now proceed tostudy the impact of a transverse electric field on it. Asreported in the literature10,20,21, with increasing exter-nal transverse electric field (Eext) in multilayered phos-phorene, its bandgap decreases and ultimately reduces tozero [see Fig. 2 (a)], leading to a topological phase transi-tion with a phase characterized by the Z2 = 1 index36–39.However, in this article we will focus on the evolution ofthe dielectric properties of multi-layered phosphorene innormal insulator state, i.e., before it turns into a topo-logical insulator and eventually into a metal at relativelyhigh transverse electric field.21 According to our GGA-PBE calculation, for bilayer phosphorene bandgap clos-ing occurs at Eext = 0.93V/A [see Fig. 1(b)-(c)]. Theexplicit dependence of the bandgap on the transverseelectric field is shown in Fig. 2(a). Evidently the crit-ical electric field needed to close the bandgap and inducea topological phase transition decreases with increasingnumber of layers, which is consistent with the fact thatthe actual bandgap also varies inversely with number oflayers.

There is a limit to the electric field strength that canbe applied to few layers of phosphorene without compro-mising its structural stability. Binding among adjacentlayers [because of van-der Waals (vdW) interaction] areexpected to weaken in presence of externally applied elec-tric field, making the structure unstable beyond certainfield strength. This is verified by computing the total en-ergy of bilayer phosphorene (measured with reference toit’s energy at equilibrium interlayer spacing of 3.2 A and

3

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5-0.4

-0.2

0.0

0.2

0.4

Ener

gy (e

V)

Interlayer Distance (Å)

Eext = 0.0 V/Å Eext = 0.8 V/Å Eext = 1.2 V/Å Eext = 1.4 V/Å Eext = 1.8 V/Å Eext = 2.1 V/Å Eext = 2.3 V/Å Eext = 2.7 V/Å Eext = 3.2 V/Å

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

Band

gap

(eV)

Eext (V/Å)

2L 3L 4L 5L 6L

(a)

(b)

FIG. 2. (a) Variation of the (GGA calculated) bandgap withthe applied electric field for multi-layered (two to six layers)phosphorene. (b) Total energy of bilayer phosphorene, cal-culated with reference to it’s energy at equilibrium interlayerspacing of 3.2 A and zero external field, plotted as a functionof interlayer distance for different Eext. Reduced stability ofbilayer AB stacked phosphorene suggests that it can be moreeasily exfoliated at higher electric fields.

zero external field) by changing the interlayer distance,for different values of Eext [see Fig. 2(b)]. The vdW bind-ing energy per phosphorus atom in absence of externalelectric field is about 48.1 meV, while that at Eext = 1.4V/A, it is 21 meV and it further decreases to zero atEext = 1.8 V/A.

Qualitatively, a vertical electric field pushes the elec-tronic cloud from the bottom surface towards the topsurface. This in turn results in induced charge densityas depicted in Fig. 3(a), which is qualitatively similar tothe induced charge density reported in other 2D materi-als such as graphene22,25, MoS2

23, multilayer GaS films40

etc. To calculate the effective induced charge density inthe z direction (perpendicular to the phosphorene lay-ers), denoted by ρind(r), we take the difference betweenρext(r) (charge density in presence of external field) andρ0(r) (charge density in absence of external field), andthen average it over the plane of phosphorene layer (x-yplane). In order to smoothen out the variations in thecharge density over the inter-atomic distances, we fur-ther use a Gaussian (filter) smoothing function, havingthe width of the order of inter-atomic distance and thus

-10 -8 -6 -4 -2 0 2 4 6 8 10-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Indu

ced

char

ge d

ensi

ty (1

0-3 e

/Å3 )

Distance along supercell height (Å)

Eext = 0.1 V/Å Eext = 0.4 V/Å

-8 -6 -4 -2 0 2 4 6 80.0

0.4

0.8

1.2

Effe

ctiv

e El

ectri

c Fi

eld

(V/Å

)

Distance along supercell height (Å)

Eext = 0.1 V/Å Eext = 0.3 V/Å Eext = 0.5 V/Å Eext = 0.8 V/Å

-20 -15 -10 -5 0 5 10 15 200.000

0.005

0.010

0.015

Effe

ctiv

e El

ectri

c Fi

eld

(V/Å

)

z (Å)

-20 -15 -10 -5 0 5 10 15 200.001

0.002

0.003

0.004

0.005

Effe

ctiv

e El

ectri

c Fi

eld

(V/Å

)

Distance along supercell height (Å)

2L 4L 6L 8L

(a)

(b)

(c)

FIG. 3. (a) The induced charge density for bilayer phospho-rene, within the supercell, at two different Eext. The two ex-treme peaks become more asymmetric at higher electric fieldas depicted in the figure. (b) Variation of the effective electricfield inside bilayer phosphorene at different Eext. (c) The ef-fective electric field inside few layer phosphorene as functionof distance along z axis for 2 to 8 layers at a fixed Eext = 0.08V/A.

we have

〈ρind(z)〉 =1

S

∫x

∫y

[ρext(r)− ρ0(r)] dxdy (1)

4

where S denotes the area of the supercell in the x-y planeand the 〈..〉 indicates the spatial in-plane average with theGaussian filter in the z direction. This induced ‘macro-scopic’ charge density typically increases with increasingelectric field.

The induced charge can then be used to calculate theplanar averaged and Gaussian filtered effective polariza-tion in the z direction using,

∂〈pind(z)〉∂z

= −〈ρind(z)〉 , (2)

along with the boundary condition that the polarizationvanishes in the region of vanishing charge density.

The induced charges in turn screens the external elec-tric field by generating a screening electric field (Eρ),and consequently the effective electric field inside phos-phorene is reduced to Eeff(z) = Eext −Eρ(z). The effec-tive electric field in the z direction can be calculated bysolving the Poisson equation for the screening potentialwhich arises form the induced charges,

∂〈Eρ(z)〉∂z

= −〈ρind(z)〉ε0

, (3)

along with the boundary condition that the induced elec-tric field vanishes far away from the region of inducedcharges.

From an ab-initio perspective, Eeff(z) can also be di-rectly calculated by taking the planar average of the dif-ference between Hartree potential (obtained from DFTcalculations) at finite electric field and that at zero elec-tric field and differentiating with respect to the verticaldistance z 22,23: Eeff(z) = −∂zVH(z), where VH(z) ≡V EH (z)− V 0

H(z). We have checked that the effective elec-tric fields, calculated using both the methods are consis-tent with each other (within 1% or each other), in theregion where there are finite charges, and consequentlywe will just use Eq. (3) to report the effective electricfield used in the rest of the manuscript.

The variation of Eeff inside bilayer phosphorene, asfunction of position along z-axis at different electric fieldstrengths is shown in panel (b) of Fig. 3. Evidently bothEeff , as well as the difference of Eeff felt by the two layers,increases with increasing applied electric field strength.Aa a consistency check we note that the Eeff → Eext,at the slab boundaries where the induced charges van-ish. In panel (c) of Fig. 3, we study the variation of Eeff

for different number of layers of phosphorene, as a func-tion of the vertical distance, for a constant Eext = 0.08V/A. Note that as the number of layers increase, the ef-fective electric field decreases on account of the increasedscreening with thickness. We further note that with in-creasing number of layers, the effective electric field ishigher at the surface and decreases gradually inside thematerial. This is a direct consequence of the fact that atthe surface the field is screened only by the charges at thesurface but as we move inside the material, the chargeson the surface as well as those inside the material screenthe external field.

-20 -10 0 10 200.00

0.05

0.10

Distance along supercell height(Å)

E eff (

V/Å

)

-20 -10 0 10 20

369

121518

Distance along supercell height(Å)

Rel

ativ

e Pe

rmitt

ivity

-20 -10 0 10 20

-4-2024 Induced Charge

ind(1

0-4 e/Å

)

z(Å)

-1

0

1 Potential Difference

V H (V

)

in

d(1

0

4e/

A)

-20 -10 0 10 200

4

8

Distance along supercell height(Å)

Pola

rizat

ion

(10-3 C

m-2)

P(1

03C

/m2)

z (A) z (A)

z (A) z (A)

Ee↵

(VA

)

VH

(V)

z1 z2

(c)

(b)(a)

(d)

FIG. 4. (a) Induced charge density (without macroscopicsmoothing out) and induced Hartree potential, VH(z), dueto the external electric field, (b) polarization with the verti-cal lines marking the slab boundaries where the polarizationfalls to 1% of the nearest peak values [see Eq. (6)], (c) effectiveelectric field and (d) the relative permittivity, as a functionof the distance along z axis. All panels are for six layer phos-phorene for a transverse applied field of Eext = 0.1 V/A andexcept for panel (a) all other panels show gaussian filteredquantities.

IV. POLARIZABILITY AND THE DIELECTRICFUNCTION

Having discussed the induced charge and the screenedeffective field, we now focus on the dielectric responseof N -layer phosphorene. Following Ref. [41], the micro-scopic static permittivity (planer averaged with gaussianfilter) can be defined as

εr(z) = 1 +〈pind(z)〉ε0〈Eeff(z)〉

. (4)

Using Eqs. (2)-(4), we can easily show that

∂z [ε(z)〈Eeff(z)〉] = 0 , (5)

which is analogous to the conservation of the displace-ment field component perpendicular to the interfaces inelectrostatics42. Equation (5) is also the basis for defin-ing the inverse of the permittivity of a slab (of heightz2−z1) as the average of the inverse of the height depen-dent permittivity41

1

εr=

1

z2 − z1

∫ z2

z1

1

εr(z)dz . (6)

For this manuscript, we consider the slab thickness to bebetween the points where the polarization drops to 1%of the nearest peak value associated with the topmost orbottommost phosphorene layers, as shown in Fig. 4(b).

5

0.0 0.2 0.4 0.62.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Rel

ativ

e Pe

rmitt

ivity

Eext (V/Å)

2L 3L 4L 5L 6L

0 2 4 6 8 10 12 14 16 18 20 22

3

4

5

6

7

8

Rel

ativ

e Pe

rmitt

ivity

Number of Layers

Model Calculated from DFTDFT calculatedEq. (9)

(a)

(b)

FIG. 5. (a) Variation of relative permittivity with appliedelectric field for varying number of layers. (b) Variation ofrelative permittivity with number of layers at fixed externalelectric field (Eext = 0.02V/A). The red circles represent theDFT calculated values, and the blue line represents the pre-dicted values for any number of layers as per Eq. (9), whichshould yield good results for low electric fields where the elec-tronic polarization is expected to vary linearly with the field,and the ionic contribution to the effective field is negligible.

Using the framework described above in Eqs. (1)-(4),we calculate and display the variation of the inducedcharge ρind and the induced effective potential (VH) as afunction of the vertical distance in six layer phosphorenefor a fixed external field of Eext = 0.1 V/A in Fig. 4(a).In panels (b) and (c) of Fig. 4, we study the variation ofthe (Gaussian filtered) polarization and the effective in-duced field, respectively, in the vertical direction. Panel(d) of Fig. 4 shows the variation of relative permittivityεr(z) within the supercell in the vertical direction.

The dependence of the relative permittivity of multi-layered black phosphorene, defined by Eq. (6) as a func-tion of the number of layers is displayed in Fig. 5(a).Note that the relative permittivity for bi- and tri-layerphosphorene is almost constant and independent of thetransverse electric field strength (εr = 2.9, and 3.5, re-spectively for bi- and tri-layer phosphorene) as long asphosphorene is in the insulating state. Further, even for

2 4 6 81 01 52 02 53 03 54 04 55 0

Polar

izibility

(Å3 )

N u m b e r o f L a y e r s

FIG. 6. Slab polarizability as a function of the number oflayers (red squared), which fits very well with a straight line(black). The linear behaviour of the slab polarizability withnumber of layers for low electric fields, implies that the polar-izability per phosphorus atom is nearly constant and it has avalue of 1.42 × 4πε0 A3. For a comparison, the polarizabilityof a carbon atom in multilayered graphene sheet has a value25

of 0.5 × 4πε0 A3 . Here Eext = 0.02V/A.

four to six layered phosphorene, the relative permittivityis almost constant for small electric fields and varies withthe transverse electric field only at larger field strengths(roughly Eext ≥ 0.1 V/A). In Fig. 5(b) we study thevariation of εr with number of layers for smaller electricfields. Evidently, the relative permittivity increases withthe number of layers, and seems to be saturating slowlywith increasing number of layers towards its bulk valueof εr = 8.3 for black phosphorus43,44.

Besides the relative permittivity, atomic/molecular po-larizability which relates the induced charge polarizationor alternately dipole moment (in the i direction) to thelocal effective electric field (in the j direction) via therelation, pind,i =

∑j αijEloc,j is also of great interest.

Note that here, Eloc is the local electric field within theslab, which is in turn related to the macroscopic field Eeff

in the slab, via the relation Eloc = Eeff + pind/ε0, wherepind is the average induced polarization in the slab. Themolecular polarizability αij is typically a tensor, howeverfor this manuscript, we will focus on the polarizability inthe z direction for a transverse electric field in the z di-rection, i.e., α = αzz. Now it is straightforward to relatethe slab polarizability (in the z direction) α of an isolatedphosphorene stack of N layers, to the respective relativepermittivity24–26, as

α(N) = ε0Ωcell

(εr(N)− 1

εr(N)

), (7)

where Ωcell is the extended supercell volume to includethe surface charges which extend beyond the top andbottom atomic phosphorene layers by δ [see Fig. 4(b),and Eq. (10)]. In Eq. (7), εr → 1 implies α → 0 whichrepresents a slab of vacuum. Further in Eq. (7), εr →∞

6

implies α→ ε0Ωcell, which is the polarizability of a metalslab. In general for an insulator, finite polarizability isproportional to the fraction of the supercell volume thatacts like a metal under a transverse electric field. Thus,if the polarizability per atom in phosphorene does notvary much, then the slab polarizability is expected tobe a linear function of the number of layers. Note thatEq. (7) is valid only under the assumption that polariza-tion is linearly proportional to the applied electric field,and further that there is no ionic contribution to the localelectric field, both of which are likely to be valid only forsmall values of the applied electric field. We plot the po-larizability as a function of the number of layers in Fig. 6and find that indeed the slab polarizability is a straightline, as expected. Similar linear behaviour of the slab po-larizability with the number of layers, has been demon-strated earlier for other two dimensional materials, suchas benzene slabs24, graphene25, GaS11 etc.

Following Refs. [24–26], the slab polarizability can alsobe expressed in terms of the bulk dielectric permittivityof black phosphorus εbulk via the relation

α(N) = NΩbulkε02

(εbulk − 1

εbulk

)+ 8πε0αs , (8)

where Ωbulk is the volume of the bulk unit cell with twolayers, εbulk is the dielectric constant of bulk black phos-phorus, and αs is surface polarizability, which capturesthe difference in the dielectric properties of a slab andbulk. Based on the straight line fit to the slab polariz-ability data in Fig. 6, we obtain the value of εbulk = 9.3and αs = 0.09 A3. Evidently, the surfaces are less polar-izable than the bulk. Here we emphasize that the bulkvalue of the relative permittivity of black phosphorenefrom the polarizability, is more than that of the actualvalue of 8.343,44. This discrepancy of less that 15% incovalent solids, is known to be a direct consequence ofsimplistic assumption, of a solid being a collection of in-dependently polarizable bi-layers which do not interactwith each other, that is used to derive Eq. (8). Similardiscrepancy has also been reported earlier for graphene25

and Si41.Finally we note that Eqs. (7)-(8) can also be combined

to predict the dielectric permittivity for any number oflayers, once we have the results for a few layers, at leastin the low electric field regime where the polarization isexpected to be linearly dependent on the external field.Eliminating α(N) form Eqs. (7)-(8), we obtain

εr(N) =

[1− N

2

Ωbulk

Ωcell(N)

εbulk − 1

εbulk− 8παs

Ωcell(N)

]−1

,

(9)where

Ωcell(N) = ΩbulkN − 1

2+

Ωbulk

2(d+ t)(t+ 2δ) . (10)

In Eq. (10), t is the is the thickness of a single phospho-rene layer due to its puckered nature and d+ t indicates

the interlayer spacing [see Fig. 1], and δ is the distancebetween z1 (or z2) and the exact supercell boundary i.e.where the bottommost (or topmost) atoms are located,as depicted in Fig. 4(b). Physically δ indicates the regionof charge spillover beyond the outermost atomic layers,and it does not vary significantly with varying numberof layers for low electric fields. In Fig. 5(b), we displaythe predicted effective dielectric constant from Eq. (9)for up-to 20 layers, and evidently it shows a very goodmatch with the actual DFT calculated values.

V. CONCLUSION

To summarize, we have studied the dielectric proper-ties of few layer black phosphorene, using first princi-ple electronic structure calculations and find that (a) ingeneral the relative permittivity increases with increas-ing number of layers, ultimately saturating to the bulkvalue, and (b) it can be tuned to a certain extent by anexternal electric field. The decreasing dielectric constantwith reducing number of layers of phosphorene in the lowfield regime, is a direct consequence of the lower polariz-ability of the surface layers and the increasing surface tovolume ratio.

In addition to the effective dielectric constant, we alsocalculate the slab polarizability of multilayered blackphosphorene, and find that for small electric fields itdisplays a linear relationship with the number of layers,implying a nearly constant polarization per phosphorusatom as expected. However for large electric fields, whichdecreases with increasing number of layers, for exampleEext > 0.1 V/A for 6 layers, and Eext > 0.2 V/A for4 layers, as per GGA calculations — see Fig. 5(a), thissimple relationship breaks down, on account of possiblebreakdown of the linear relationship between polariza-tion and the applied field, and increasing impact of ioniccontribution to the effective field. In the low field regime,one can model the system as a collection of well spacedand independent bi-layer units, and obtain the slab po-larizability in terms of the bulk dielectric constant andsurface polarizability on one hand, and on the other theslab polarizability can be related to the layer dependentdielectric constant. This allows us to extrapolate our cal-culations for few layers, to much larger number of layersand obtain an empirical relation for the thickness depen-dent dielectric constant for any number of layers – seeEq. (9).

Finally we note that all the calculations presented inthis article, are valid only for the insulating regime ofphosphorene. Further in a realistic experimental sce-nario, the effective dielectric constant is also likely tobe affected by the environment, for example the sub-strate used, passivation method, etc. and incorporat-ing these effects would require a more detailed numericalstudy. However our study offers a good starting point,and demonstrates the thickness dependent and electricfield tunability of the dielectric constant in phosphorene,

7

which provides an additional handle for optimally design- ing, and interpreting phosphorene based devices.

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