Strain engineering the charged-impurity-limited carrier mobility in phosphorene

Strain engineering the charged-impurity-limited carrier mobility in phosphorene

Yawar Mohammadi , Borhan Arghavani Nia Corresponding author. Tel./fax: +98 831 427 4569, Tel: +98 831 427 4569. E-mail address: yawar.mohammadi@gmail.com
Abstract

We investigate, based on the tight-binding model and in the linear deformation regime, the strain dependence of the electronic band structure of phosphorene, exposed to a uniaxial strain in one of its principle directions, the normal, the armchair and the zigzag directions. We show that the electronic band structure of strained phosphorene, for the experimentally accessible carrier densities and the uniaxial strains, is well described by a strain-dependent decoupled electron-hole Hamiltonian. Then, employing the decoupled Hamiltonian, we consider the strain dependence of the charged-impurity-limited carrier mobility in phosphorene, for both types of carriers, arbitrary carrier densities and in both armchair and zigzag directions. We show that a uniaxial tensile (compressive) strain in the normal direction enhances (weakens) the anisotropy of the carrier mobility, while a uniaxial strain in the zigzag direction acts inversely. Moreover applying a uniaxial strain in the armchair direction is shown to be ineffective on the anisotropy of the carrier mobility. These will be explained based on the effects of the strains on the carrier effective masses.

Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran

Department of Physics, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran

Keywords: Phosphorene; Tight-binding model; Band Structure; Strain; Carrier Mobility.

1 Introduction

Since successful isolation of a single layer of graphite[1] called graphene, as the first real two-dimensional lattice structure which shows novel appealing properties[2, 3], many researchers tried to synthesis or isolate new two-dimensional materials. These efforts resulted in finding other two dimensional materials such as BN[4], transition metal dichalcogenides[5], silicene[6, 7, 8, 9] and recently phosphorene. Phosphorene is a single layer of black phosphorus, which can be isolated by mechanical exfoliation[10, 11] of black phosphorus. In a single layer of black phosphorus, each phosphorus atom covalently couples to three nearest neighbors. This configuration of phosphorus atoms results in a honeycomb-like lattice structure. However, due to the hybridization of and atomic orbitals, it forms a puckered surface. The electronic band structure of phophorene has been studied using different methods such density functional theory calculations[12, 13], method[14, 15] and tight-binding model[13, 16, 17]. These considerations show phophorene is a direct-band-gap insulator, but with an anisotopic band structure. This novel band structure leads to many attractive properties[18, 19, 20, 21].

Strain tuning is an effective means to tune the physical properties of two dimensional materials (for a review, see e.g. Ref.[2, 22]). Puckered structure of phosphorene makes this easier, so one can tune and control its electronic and mechanical properties by strain, confirmed by recent studies on the effects of uniaxial and biaxial strains in phosphorene [12, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. These works examined effects of strains applied along three principle directions which preserve group point symmetry of phosphorene[16], its zigzag and armchair edges and the direction normal to its plane. It has been shown that a uniaxial strain in the direction normal can decrease its band gap and even leads to an insulator to metal transition[12, 27, 34]. Moreover, the effects of an in-plane uniaxial strains along zigzag and armchair edges[28, 29, 30] on the band gap of phosphorene has been studied. Some other researchers has studied the effects of uniaxial and biaxial strains on the band structure[30, 31, 32, 33] and the optical properties[31] of phosphorene, confirming the capability of stain as an effective means to tune the properties of phosphorene. These works showed, when the uniaxial strain is applied along the armchair direction, the properties of phophorene change further. But, recently, it has been shown[35] that the most effective direction to apply a strain and tune the band gap of phosphorene is an in-plane direction (not being along armchair nor along zigzag) with a direction angle about counted from the armchair edge.

According to the high-potential capability of strains to tune the properties of phosphorene, driving an analytical relation for the Hamiltonian of strained phosphorene is very desirable, and can be used to examine the effects of the strains on the electric, optical and magnetic properties of phosphorene. In this paper, starting from the well-known 4-band tight-binging Hamiltonian of phophorene[13, 16], we obtain a strain-dependent tight-binding Hamiltonian for phosphorene. In this paper we work in the linear deformation regime and only consider uniaxial strains. To benefit from the group point symmetry of phosphorene and reduce the 4-band Hamiltonian to a 2-band Hamiltonian and achieve an analytical result, we restrict our consideration to the uniaxial strains applied along three principle directions of phosphorene which preserve its group point symmetry. Thanks to this symmetry, we can obtain analytical relations for its band energies which can be used to explore easily the effects of the uniaxial strains on properties of phosphorene. Searching for low-energy structures in strained phosphorene, we use continuum approximation and derive the corresponding Hamiltonian dominating low energy excitations. Then, by taking into account the weak interband coupling of conduction and valance bands, we project the low-energy Hamiltonian into a decoupled Hamiltonian[36, 37, 38] and show that for the experimentally accessible carrier density, the decoupled bands agree well the bands obtained from the tight-binding Hamiltonian. Motivated by this fact and the recent studies on the carrier mobility in phopsphorene[10, 20, 37, 30], then we apply our decoupled Hamiltonian to consider the strain dependence of the charged-impurity-limited carrier mobility in phosphorene. Our result shows that one can tune the amount and the anisotropy of the mobility in phosphorene, by making use of a uniaxial strain in the normal or zigzag direction.

The rest of this paper is organized as follows. In Sec. II we reproduce the known 4-band Hamiltonian of phosphorene. In Sec. III we explain how one can, in general, insert the effects of the strains in the Hamiltonian and obtain a general formalism for the strain-dependent Hamiltonian of phosphorene. Sec. IV devoted to consider the strain dependence of the charged-impurity-limited carrier mobility in phosphorene. We end the paper by summarizing our results in Sec. V.

2 Structure and tight-binding Hamiltonian of phosphorene

The lattice structure of phosphorene and the necessary lattice parameters to construct the tight-binding Hamiltonian of phosphorene, including the lattice constant, the bond angles and the transfer energies, have been introduced in Fig. 1. The unit cell of phosphorene (solid-line rectangle in Fig. 1) consists of four phophorus atoms, two atoms in the lower layer represented by the grey circles (called and ) and two atoms in upper layer represented by the red circles (called and ). Hence, the tight-binding Hamiltonian of phophorene can be written in terms of a matrix as

(1)

acting in with being the two-dimensional momentum. Notice that . Moreover, it has been shown[13, 16] that if we only retain the transfer energies up to the fifth nearest neighbors, the tight-binding approximated band structure agrees well with its density functional theory band structure. These transfer energies are[13] , , , and . So we can rewrite the Hamiltonian matrix as

(2)

where the matrix elements are given by , , , and . Here is a vector which is drawn from (The origin of the cartesian coordinate system) to one of the ith nearest neighbors (See Fig. 1). They are , , , and where . One can take into account the group point symmetry in phosphorene and project the four-band Hamiltonian into a reduced two-band Hamiltonian as[16]

(3)

acting in . The corresponding energy bands, obtained by diagonalizing the Hamiltonian matrix, are given by

(4)

where denotes to the conduction(valance) band. We have shown the energy spectrum of phosphorene obtained from two-band Hamiltonian in Fig. 2. It is evident that minimum (maximum) of the conduction (valance) energy band is at point. If we apply continuum approximation to the obtained two-band Hamiltonian and retain the terms up to the second order in k, we can reproduce the known Hamiltonian of phosphorene[16, 17],

(5)

where , , , , , and which agree well with the other calculations [17](Notice that in our calculations the zigzag edge lies along the x-axis.). The corresponding energy spectrums are given by

(6)

where denotes to the conduction(valance) band. It is evident that the energy spectrum is linear in the direction while in the direction it is parabolic. Due to the large band gap, which leads to a weak interbans coupling, one can decouple the electron and the hole bands into a low energy regime. In this approximation, Eq. 6 can be written as[17, 36, 37, 38]

(7)

In this approximation the electron and hole effective masses in the and directions are given by, , , and , which is the mass of a free electron, in good agreement with recent result[13]. To see that in what region this approximated energy bands agree well with the other results, we have shown all three set energy bands obtained from the tight-binding, the low energy and the decoupled Hamiltonians, in Fig. 2. One can see that in the direction all three set energy bands agree well in a wide range of the energy and the momentum. Moreover, in the direction the tight-binding energy bands and the low-energy bands agree well too, but the decoupled bands overlap with them only up to () with respect to the bottom (top) of the conduction (valance) bands. This corresponds to and for the electron and hole densities. These indicate that the low energy excitations in phosphorene are well described by the decoupled Hamiltonian[17, 36, 37, 38].

3 Strain-dependent tight-binding Hamiltonian

In this section we rederive the tight-binding Hamiltonian of phosphorene in the presence of the uniaxial strains applied along the principle directions of phosphorene. To insert the effects of the applied strain in the tight-binding Hamiltonian of phosphorene, first we must determine the effects of the strain on the transfer energies and the bond lengths. It has been shown[39] that the transfer energies between and orbitals, which construct the electronic bands of phosphorene, depend on the bond length as . To obtain this relation, it has been supposed that the applied strain doesn’t change the bond angles and only change the bond lengths. Within the linear deformation regime, this is a reasonable assumption. Since the change in the bond angles in a strained lattice, at least, includes the terms of second order in terms of the applied strain. So they can be ignored in the linear deformation regime. Hence, we only need to determine the strain dependence of the bond lengths and the other inter-atomic distances.

Let us construct our formalism in a general case in which phosphorene is exposed to strains applied along all three principle directions of phosphorene, the armchair (x-direction) and the zigzag (y-direction) edges and the the direction normal to the phosphorene plane (z-direction). So, the deformed coordinates are given by

(8)

where , and are the normal strains applied along the x-, y- and z-directions respectively. In this paper, we restrict our considerations to the linear deformation regime, so the bond lengths and the other atomic distances, in general, can be expanded in terms of all components, , and as

(9)

where and are the unstrained and strained bond lengths respectively and , and are the strain-related geometrical coefficients, given by , and . If we insert Eq. 9 into the relation , and expand it in terms of the strains and only retain the terms up to the first order in we get

(10)

where and are the unstrained and the strained transfer energies respectively.

As mentioned above, the strains applied along all three principle directions of phosphorene don’t break symmetry. So the electronic excitations in strained phosphorene are dominated by the reduced two-band Hamiltonian, Eq. 3, but after substituting the deformed transfer energies and bond lengths into it. Recalling the relations obtained for the strained bond lengths and the transfer energies, and substituting them into the tow-band Hamiltonian, Eq. 3, we get

(11)

where , , , and . The corresponding electron and hole energy bands are given by

(12)

with denoting to the electron (hole) band. It easy to show that strained phopsphorene has a direct band gap at point, in agreement with the recent density functional theory [12, 27, 34, 28] and tight-binding[40] calculations done in the linear deformation regime. Similar to the unstrained case, to capture the low energy physics of strained phosphorene, one can expand the matrix elements around point and only retain the terms up to second order in and first order in . Hence, the low energy Hamiltonian becomes

(13)

The values of the matrix elements depends on the directions in which the strains are applied (See appendix). The applied strains can affect, by changing the energy gap and , on the interband coupling. When the interband coupling is weak, one can project the low energy two-band Hamiltonian into a decoupled Hamiltonian which is given by

(14)

where , and

(15)

In the reminder of this section we consider the effects of the strains applied along all three principle directions of phosphorene on its electronic band structure, as a key feature of crystalline materials to explore their other physical properties. We have two aims. One is to see whether our tight-binding Hamiltonian reproduces previous results[12, 27, 34, 28, 40] for the energy gap of strained phosphorene. The other is to show that in what energy region the decoupled energy bands agree well with the others, obtained from the low-energy and the tight-binding Hamiltonian.

Uniaxial strain along the normal direction (z-axis)- Let us first explore effects of a uniaxial strain in the normal direction (z-direction), and . If we recall the relations obtained for the the strained bond lengths and the transfer energies, and substitute them into Eq. 12, we get

(16)

for the strain-induced modulation in the energy gap. This shows that the energy gap of phosphorenre decreases (increases) linearly when it is exposed to a uniaxial tensile (compressive) strain in the normal direction. This is in agreement with the previous first-principle[12, 27, 34] and tight-binding [40] studies on the strain-induced modulation in the energy gap of strained phosphorene, done in the linear deformation regime. This can also be seen in Fig. 3 where we have shown the energy bands of strained phosphorene for different values of obtained by diagonalizing Eq. 11 (black curves), Eq. 13 (red dashed curves) and Eq. 14 (green dotted-dashed curves). In this figure right (left) panels show the energy bands of phosphorene in the presence of a uniaxial tensile (compressive) strain applied in the normal direction, and in each panel the energy bands have been drown in both and directions.

This figure also shows that a uniaxial tensile (compressive) strain in the normal direction enhances (weakens) the anisotropy of the both electron and hole energy bands slightly (See black cures in Fig. 3). This becomes more clear in the next section, where we consider the strain dependence of the carrier mobility in strained phosphorene. Moreover one can see that in the presence of a uniaxial tensile (compressive) strain, the energy range in which the decoupled bands agree well with the other bands becomes limited (extended). This is mainly due to the effect of the stain on the energy gap (See Eq. 16). When the energy gap increases (decreases), the coupling of the conduction and the valance band is enhanced (weakened) and the decoupling-band approximation becomes more (less) accurate. For a uniaxial tensile (compressive) strain about (), the decoupled conduction band overlap well with the tight-banding conduction band up to () with respect to the bottom of the conduction band. By making use of , where is counted from the bottom (top) of the conduction (valance) band, one can show this agreement corresponds to () electron density. This agreement for the valance band is up to () with respect to the top of the valance band, corresponding to () hole density.

Uniaxial strain along the zigzag edge (y-axis)- When phosphorene is exposed to a uniaxial strain along its zigzag edge, the strain-induced modulation in its energy gap is given by

(17)

which shows that a uniaxial tensile (compressive) strain in the zigzag edge increases (decreases) linearly the energy gap. This agrees well with the recent studies[10, 28, 40]. Figure 4 shows that in the presence of a uniaxial tensile (compressive) strain along the zigzag edge, the anisotropy of the band structure is weakened (enhanced). Moreover it is evident that, in the presence of a uniaxial tensile (compressive) strain about (), there is good agreement between the decoupled and the tight-binding conduction bands up to () with respect to the bottom of the conduction band. In the valance band the overlapping is up to () with respect to the top of the valance band.

Uniaxial strain along the armchair edge (x-axis)- In the presence of a uniaxial along the armchair edge of phosphorene, the strain-induced modulation in its energy gap is given by

(18)

which shows that the energy gap is a linear function of the applied strain, and increases (decreases) when phosphorene is exposed to a uniaxial tensile (compressive) strain in agreement with the recent studies[10, 40]. Comparison of Eqs. 17 and 18 shows that, for same uniaxial strains along the zigzag and armchair edges, the uniaxial strain along the zigzag edge induces a larger band gap variation. Effects of the applied strain on the anisotropy of the band structure can be seen in Fig. 5 which, as it is expected, is unlike the effects of the uniaxial strain along the armchair edges. In the presence of a uniaxial tensile strain about along the zigzag edge, the overlapping of the decoupled band with the tight-binding is up and for the conduction and the valance bands respectively, while for a compressive strain about they agree only up for both conduction and valance bands.

We end this section by this conclusion that the electronic band structure of strained phosphorene, for the experimentally accessible carrier densities and the uniaxial strains applied along all three principle directions of phosphorene, is well described by the decoupled Hamiltonian. Motivated by this fact, we apply it to consider strain engineering the charged-impurity-limited carrier mobility in phosphorene.

4 Strain engineering the charged-impurity-limited carrier mobility

In this section, employing our strain-dependent decoupled Hamiltonian, we investigate the strain dependence of the impurity-limited carrier mobility in phosphorene for both types of carriers, electron and hole, and along both armchair and zigzag edges. The carrier mobility, , is defined as where is the electrical conductivity, is the carrier density and is the electron charge. To calculate the electrical conductivity we use the semi-classical Boltzmann transport theory combined with the relaxation time approximation. Moreover we restrict our calculation to the steady state and suppose that the two-dimensional electron gas in phosphorene is homogenous, so the electrical conductivity is given by

(19)

where is , is the spin degeneracy, is the two-dimensional momentum and is the electron velocity in the direction with and being the corresponding electron or hole momentum and mass. is the energy band obtained from the strained-dependent decoupled Hamiltonian (Notice we have omitted the electron and hole indexes in and .), is the Fermi-Dirac distribution function and is the relaxation time. Let us suppose that the impurities are static, of symmetric potential and have no internal excitations. So the relaxation time is given by

(20)

where is the number of impurities per unit area, and is the scattering angle between and . is the Fourier transform of the potential of the charge impurity and is the effective dielectric constant with and being the dielectric constant of the substrate ( for substrate[19]) and the encapsulating layer respectively which for vacuum is zero. is the dielectric function which within the random phase approximation is given by , where is the polarizability function. The polarizability function can be written[19] as

(21)

where is the Fermi energy of strained phosphorene for a fixed carrier concentration.

If we introduce new variables as and , we have for the energy bands with . In the new momentum, space the electrical conductivity is given by

(22)

leading to for the electrical conductivity of strained phosphorene at zero temperature, where with being Fermi momentum in the new momentum space. is given by

(23)

where is the carrier density of states at the Fermi energy and

(24)

is the anisotropic Fermi momentum. In Eq. 24, is counted from the x-axis, and is the carrier density in strained phosphorene, being a linear function of Fermi energy as same as the carrier density in the ordinary two-dimensional electron gas. Hence the zero-temperature carrier mobility in strained phosphorene is given by.

In Fig. 6 we have shown our numerical results for the strain dependence of the charged-impurity-limited electron (left panels) and hole (right panels) in phosphorene exposed to the uniaxial strains in the normal direction (z-axis). The upper (lower) panels shows the carrier mobility along its armchair (zigzag) edge, and orange () to black () curves correspond to different carrier densities with . The density of the charged impurities is supposed to be which is typical of the substrate. Figure 6 shows that the carrier mobility along the armchair direction is higher than that along the zigzag direction, as same as that in unstrained phosphorene[10, 20, 37, 30]. This can be understood by this fact that, in the presence of both uniaxial tensile and compressive strains, the carrier effective mass along the armchair edge is always smaller than that along the zigzag edge. This can be tested by making use of the Eqs. 15, 25. and 26. Moreover one can see that the carrier mobility along both armchair and zigzag directions increases by increasing the carrier density. This is the familiar feature of the ordinary two-dimensional electron gas[41], arising from the linear dependence of its carrier density on the Fermi energy (In phosphorene the carrier density depends on the Fermi energy as ). Figure 6 also shows that in the presence of a tensile (compressive) strain in the normal direction, the carrier mobility along the armchair edge increases (decreases), while the carrier mobility along the zigzag edge decreases (increases). This property originates from the effect of the strain on the anisotropy (and consequently the carrier effective mass) in phosphorene, as explained in the previous section. To explain this property further, we rewrite the relation of the carrier mobility as . It is easy to show that the effect of the strain on part is weak and it mainly affects on part. By making use of the Eqs. 15, 25. and 26, one can show that applying a uniaxial tensile (compressive) strain in the normal direction decreases (increases) both electron and hole effective masses in the armchair (zigzag) direction, and consequently their mobilities in the armchair (zigzag) direction increase (decrease).

In Fig. 7 we have compared the effect of the direction of the applied straina on the carrier mobility in phosphorene. Figure 7 shows that, unlike the strains in the normal direction, applying a uniaxial tensile (compressive) strain in the zigzag direction decreases (increases) both electron and hole mobilities in the armchair (zigzag) direction. This originates from their different effects on the anisotropy (and consequently the carrier effective mass) in phosphorene, as explained in the previous section and above. This figure also shows that applying a uniaxial strain in the armchair direction weakly change the carrier mobility in phosphorene. Moreover Figs. 6 and 7 show that applying a uniaxial tensile (compressive) strain in the normal (zigzag) direction enhances (weakens) the anisotropy of the carrier mobility in phosphorene. While in the presence of a uniaxial compressive (tensile) strain in the normal (zigzag) direction, the carrier mobility is weakened (enhanced).

5 Summary and conclusions

In Summary, we investigated the electronic band structure of strained phosphorene within the linear deformation regime and based on the tight-binding model. We restricted our consideration to the uniaxial strains applied along one of the principle directions of phosphorene, the normal, the armchair and the zigzag directions. We showed that the derived strain-dependent energy spectrums reproduce the previous results for the energy gap of strained phosphorene. Then we applied the continuum approximation to derive the corresponding low-energy Hamiltonian. Moreover we showed, when the interband coupling is weak, the low-energy Hamiltonian can project into a decoupled electron-hole Hamiltonian. We found that the electronic band structure of the strained phosphorene, for the experimentally accessible carrier densities and the mechanical strains, is well described by the decoupled Hamiltonian. Motivated by this fact we used our strain-dependent decoupled Hamiltonian to investigate the strain dependence of the charged-impurity-limited carrier mobility in phosphorene. We examined the dependence of carrier mobility on the direction of mobility, the carrier type, the carrier density and the direction of the applied strain. We showed the dependence of the carrier mobility on the direction of mobility, the carrier type and the carrier density is same as that in unstrained phosphorene. Moreover, as a point worthy of mention, we found that applying a uniaxial tensile (compressive) strain in the normal direction decreases (increases) carrier mobility in the armchair (zigzag) direction. While in the presence of a uniaxial tensile (compressive) strain in the zigzag direction the carrier mobility is decreased (increased). We also showed that a uniaxial strain in the armchair direction don’t changed the carrier mobility approximately. These properties were explained based on the effect of the applied strain on the anisotropy of the carrier effective mass in phosphorene.

Appendix A Appendix: Calculating the elements of the strain-dependent low-energy Hamiltonian matrix

The matrix element in Eq. 13, in the linear deformation regime, depend in general on the applied strain as

(25)

where the coefficients of the applied strain, for a uniaxial strain in the normal direction, are given by

(26)

These coefficients, when the strain is applied in the zigzag edge (y-axis), become

(27)

while for a uniaxial strain along the armchair edge, they are