Fermi Level Determination for Charged Systems via Recursive Density

Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 4014−4019

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Fermi Level Determination for Charged Systems via Recursive Density of States Integration H. A. Tahini, X. Tan, and S. C. Smith* Department of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra 2601, Australia

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S Supporting Information *

ABSTRACT: Determining the Fermi level position for a given material is important to understand many of its electronic and chemical properties. Ab initio methods are effective in computing Fermi levels when using charge-neutral supercells. However, in the case where charges are explicitly included, the compensating homogeneous background charge, which is necessary to maintain charge neutrality in periodic models, causes the vacuum potential to be ill-defined  which would otherwise have been a reliable reference potential. Here, we develop a method based on recursively integrating the density of states to determine shifts in the Fermi level upon charging. By introducing incremental charges, one can compute the density of states profile and determine the shift in the Fermi level that corresponds to adding or removing a given increment of charge δq, which allows the evaluation of the Fermi level for any arbitrary charge q. We test this method for a range of materials (graphene, h-BN, C3N4, Cu, and MoS2) and demonstrate that this method can produce a reasonable agreement with models that rely on localized compensating background charges.

C

charge−voltage relationships, which has so far limited our ability to include charging energies when assessing chargeresponsive materials. The current difficulty in calculating work functions of charged systems using most density functional theory (DFT) codes is the fact that upon the introduction of excess charge into the supercell a compensating background charge needs to be added to avoid divergence in the periodically repeated cells. This is normally done by removing the G = 0 component from the Fourier expansion of the electrostatic potential.13,14 However, this uniform background charge makes it impossible to correctly calculate a vacuum electrostatic potential, as no matter how large the supercell might be there will always be present a certain amount of charge density in the vacuum region, and the electronic potential energy will grow linearly with the vacuum spacing.15 Attempts have been made to address this issue, offering a range of post processing, as well as fully ab initio schemes. Such attempts appear most notably in the context of surface charged defects as suggested by Komsa and Pasquarello16 or Vinichenko et al.,17 where they used classical electrostatics and dielectric profiles to correct for finite size supercell errors. Lozovoi et al.18 presented a scheme making use of a reference electrode at a finite distance from the charged slab. This grounded reference electrode set to 0 V at the supercell boundaries is obtained by solving the Poisson equation and choosing adequate boundary conditions, which allows the determination of the electrostatic contribution of the compensating background charge. This method was extended

harge modulation and injection is a promising avenue for tuning the binding and adsorption properties of surfaces and has garnered a wide interest in the past few years.1−7 This charge modulation route has allowed a number of systems to exhibit an enhanced capacity toward molecular adsorption, such as H2 storage and CO2 capture, when their surfaces are charged with electrons or holes, allowing a switchable adsorbate capture−release based on charge−discharge mechanisms.1,2,8 For example, it was found that while CO2 weakly physisorbs on the surface of boron nitride (h-BN) under ambient neutral conditions,1 when a certain amount of charge is introduced into the system, CO2 binding is greatly enhanced. Similarly, the capacity of graphitic carbon nitrides (g-C4N3 and g-C3N4) for H2 storage was dramatically enhanced upon electron injection, reaching storage capacities of up to 7 wt %.8 Many other materials have been identified to exhibit this type of behavior.2,3,6,9,10 In these studies, the approach has been to compare binding free energy changes for the neutral and charged cases, omitting the fact that charge needs to be added into the sorbent material first, which requires an energy input. This energy input or cost needs to be taken into account for an accurate prediction whether or not a material is truly chargeresponsive. The reasoning behind this is that materials that are poor electrical conductors such as h-BN, with a band gap of ∼6 eV,11 would consume significant energy during charging that will offset any exergonic binding of a CO2 molecule, as was predicted.12 To include such charging energies, one needs to obtain charge−voltage relations by taking into account the evolution of the work function or the Fermi level with the amount of charges added or removed to a given material. Current ab initio methods are mostly limited to computing neutral work functions and thus do not permit the evaluation of such © 2018 American Chemical Society

Received: May 25, 2018 Accepted: July 3, 2018 Published: July 3, 2018 4014

DOI: 10.1021/acs.jpclett.8b01631 J. Phys. Chem. Lett. 2018, 9, 4014−4019

Letter

The Journal of Physical Chemistry Letters

Figure 1. Compensating background charge to maintain overall supercell charge neutrality is applied via two schemes: (a) a localized version where the charge is confined to atomic layers leaving the vacuum region charge neutral and (b) by applying the charge homogeneously through the entire volume of the supercell. (c) The variation in the averaged electrostatic potential of graphene with applied charge and a homogeneous background charge. The neutral Fermi level was set to 0 eV.

Figure 2. (a) DOS profiles of a neutral graphene layer and one with 4.77 × 1014 e−/cm2. The 0 eV mark represents the Dirac cone, and shaded regions indicate occupied states. As charge is added, clear perturbations are induced in the conduction band as visible in the 0.6−1.4 eV range. (b) Integrated total DOS of a neutral graphene system showing the shift in the Fermi level, δFL, when a charge δq = ±0.2e− is applied.

by Krishnaswamy et al.19 and was applied to study charged interfaces arising from the formation of a two-dimensional (2D) electron gas at oxide surfaces. On the other hand, approaches that are fully ab initio include methods that tend to localize the compensating charge within the bulk region of the supercell, thus leaving the vacuum region essentially neutral, which include the use of Gaussian compensating background charges at atomic sites20 or the use of virtual crystal approximation and a dipole correction in vacuum.21 Another approach to study the variation of work functions with charge relies on using H atoms in the vicinity of a surface.22,23 Here, the idea is that H will donate an electron becoming H+, with the donated electron transferred to the sorbent material thus charging it and changing its work function. Difficulties with this approach are that the presence of H+ will introduce additional dipoles that will affect the work function; moreover, with this method, one can only negatively charge the material. In this work, we present a new technique which relies on computing densities of states (DOS) and their integrals to track the changes in the Fermi level as incremental charges are added or removed, which is then translated to charge−voltage curves and eventually allows the calculation of charging

energies. We tested the method on four different types of materials using two popular DFT codes for comparison. In DFT codes, energies are always defined relative to some arbitrary potential, and hence, values of total energies are by themselves meaningless. For the same reason, energy eigenvalues defining the top and the bottom of the valence and conduction bands (in the case of semiconductors or insulators or the positions of Fermi levels in metals) are not well-described by definition. To obtain physically meaningful band edge positions and Fermi levels, the arbitrary potential energy present in a given calculation needs to be eliminated. This is normally done by calculating the local potential energy and averaging it in the plane perpendicular to the vacuum direction (conventionally ⊥ xy-plane) and setting the potential to 0 eV in the vacuum layer. This leads to a definition of the work function as Φ = Evac − E F

(1)

In real systems, Φ is termination-dependent and highly sensitive to details of the surface (reconstructions, defects, etc.). Yet eq 1 gives a reasonable agreement with experiments and has been widely used in the literature.24−29 As can be seen, the ability to calculate work functions rests entirely on being 4015


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Figure 3. Evolution of the Fermi level or voltage (expressed relative to SHE) with charge for (a) graphene, (b) h-BN, (c) C3N4, (d) Cu(111), and (e) MoS2. The RDI is calculated with a homogeneous background charge using DMol3 and VASP and compared to the model that employs localized compensating charge to calculate the FL shifts directly via eq 1.

renders the use of the neutral integrated DOS to estimate Fermi level shifts for high charge states questionable, as these states are not accounted for in the neutral integrated DOS. To overcome this, we introduce a scheme based on recursive DOS integration (RDI), which allows us to calculate the FL shift for any charge. Suppose we wish to calculate the work function for a given charge q. We begin by partitioning q into N increments such that q = ∑N δqi . The total number of electrons in a supercell for any given number of increments is

able to evaluate the vacuum potential unambiguously. However, this picture fails dramatically when charge is introduced into the supercell. Figure 1c shows the changes in the local potential as charge is added to a graphene layer. This necessary compensating background charge introduces spurious interactions and electric fields that invalidate the calculated local potential. Alternatively, some codes offer a novel way to circumvent some of the issues associated with a compensating background charge. This is done by applying a small Gaussian background charge on atomic positions and therefore keeping the vacuum region effectively charge neutral (Figure 1a). This method is widely used in DMol3 and has been successfully applied, especially when calculating charge−voltage relations for a range of 2D materials.12 We will use this approach to compare with our developed methodology. The integrated DOS up to the Fermi level (Figure 2b) sums to the total number of valence electrons (as defined in the pseudopotential for the particular plane-wave code used, 4e− for carbon in our case). By using a 4 × 4 cell of graphene with a total of 32 atoms, there are 128 electrons populating the valence band up to the Fermi level. As a first-order approximation, the neutral integrated DOS gives an indication of how much the Fermi level will shift in response to added charge (Figure 2b). However, this simple approach will tend to fail as added charge will strongly perturb the DOS profile, as shown in Figure 2a. Here 4 electrons have been added which correspond to a charge density, σ = 4.77 × 1014 e−/cm2 (σ = q/A), where q and A are the number of electrons/holes and the surface area, respectively, which introduces new states around 0.6−1.4 eV above the Dirac point (Figure 2a). This

q(n) = q0 + nδq ,

n = 0,1, ..., N

(2)

where q0 is the number of electrons in a neutral supercell. The DOS is computed for each step q(n) which forms the basis of the RDI scheme. Each DOS profile computed at q(n) is used to estimate the shift in Fermi level relative to its intrinsic Fermi level when a charge δq is added, as shown in Figure 2b. Numerically, the increment charge is obtained by integrating the DOS with the Fermi level (EF(q(n)) and a corresponding shift (EF(q(n) + δq)) as given below: δq =

E F(q(n) + δq)

∫E (q(n))

ρ(q(n))dE

(3)

F

Because δq is known a priori, one can simply determine EF(q(n) + δq) by reading the corresponding shift from the integrated DOS profile at a given charge q(n) (Figure 2b). From this it follows that the FL shift is δ FL = E F(q(n) + δq) − E F(q(n))

(4)

Hence, the evolution of FL with charge can be obtained via 4016


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Figure 4. Charging energy per supercell to impart a certain amount of electrons to (a) graphene and (b) h-BN. In agreement with expectations, it is significantly more costly to achieve the same level of charge on insulating h-BN compared to semimetallic graphene.

N

FL(q) = FL(q0) +

∑ δ FLi

point and therefore, the ease by which the material could be charged. 2D h-BN possesses a wide band gap (∼4.6 eV with DFT− PBE). Assuming that the FL is in the middle of the band gap, we obtain a work function of 3.5 eV (or a FL = −0.94 eV vs SHE). Hole charging allows the FL to access an abundance of states, which explains the steep rise in hole density for small changes in the charging voltage, and a similar behavior is observed for electron charging (Figure 3b). It is interesting to note the similarity between the RDI curves computed with VASP and DMol3, which is the result of having very similar DOS profiles. On the other hand, FL shift computed via eq 1 shows a stronger deviation from the RDI scheme as more charge is added or removed from the system beyond ±2 × 1014 e−/cm2. However, the overall trend and shape of the charge− voltage plots obtained via RDI and eq 1 remain consistent. Similarly, for C3N4, RDI curves are in very close agreement to one another with deviations of less than 0.1 eV at ±4 × 1014 e−/cm2. In the case of electron charging, there is a step in the RDI calculated with DMol3 because of the presence of an energy gap in the calculated DOS which is also mirrored by the FL shift calculated using eq 1 (Figure 3c). In the case of a highly conductive metallic system such as Cu, the charge− voltage curves possess steep gradients indicating how facile it is for charge to be induced for a small voltage change. The neutral work functions obtained with DMol3 and VASP are 4.5 and 4.6 eV, respectively, which gives rise to the small offset in the calculated charge−voltage curves obtained with the two codes as shown in Figure 3d but without affecting their gradients especially under electron charging conditions. For hole charging, the offset and the nearly constant gradients are also maintained. Figure 3e shows the behavior of a single layer MoS2, which is semiconducting with band gap of 1.9 eV.31,32 This gap (1.73 eV with DFT-PBE, in good comparison to other theoretical results32,33) is clearly visible in Figure 3e when charge is added. Following a similar trend as shown above for h-BN, MoS2 displays steep changes of electron/hole densities for relatively small applied voltages. The charge−voltage relations are useful in determining the cost of charging a material. This charging energy Ec(q) is evaluated as12

(5)

while keeping in mind that Φ(q) = −FL(q). The approach presented here relies on the ability to compute the local potential under neutral supercell conditions to obtain a referenced position of the Fermi level first and the ability to compute the density of states profiles under charged conditions. We want to stress that the quantities in eq 4 are obtained from the same calculation with charge q(n) and therefore are not subject to band misalignments. Added electrons will populate bands above the neutral Fermi level forming a new Fermi level. Likewise, removing electrons (hole charging) lowers the Fermi level and creates a new level in a region where bands were previously occupied. Therefore, the density of states in the vicinity of the Fermi level plays a major role in determining the ease by which electrons can be added or removed from a system and hence the shift of the Fermi level. In order to test and exemplify our proposed scheme, we chose a number of systems that are of practical use in gas capture and catalytic reactions, namely, graphene, h-BN, C3N4, Cu(111), and MoS2. These systems encompass metallic and insulating surfaces and will therefore be expected to show markedly different response to charging. As mentioned before, DMol3 offers the option to constrain background charges to the atomic layers of the material, therefore leaving the vacuum region neutral, which allows the direct computation of the work function (Fermi level) as given by eq 1. The RDI scheme is computed using a homogeneous background charge (jellium) as implemented in VASP and DMol3, which is then compared to FL shifts computed directly with DMol3 using the localized background charge via eq 1. Figure 3 shows the FL shifts calculated using these three approaches. FLs were expressed relative to the standard hydrogen electrode (SHE, −4.44 eV relative to vacuum30). Graphene being a semimetal does not have a band gap and exhibits a negligible DOS around the Dirac point. The n- or pdoping of graphene allows accessing states that are otherwise empty or occupied. The work function is ∼4.6 eV, and therefore its FL is ∼0.16 eV relative to SHE under neutral conditions (Figure 3a). Anodic or cathodic voltages of the order 0.5 V are needed to impart ∼5 × 1013/cm2 electrons or holes. Beyond these voltages, smaller changes in the voltage are required to add significantly more charge to graphene, reflecting the growing density of states far from the Dirac

Ec(q) =

∫0

q

[E F(q′) − E F(0)] dq′

(6)

Figure 4 shows the cost of charging graphene and h-BN from neutral conditions to charge densities of ∼6 × 1014 e/cm2 4017


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with an all-electron double numerical basis set with polarization function (DNP) and a real space cutoff radius set to 4.1 Å. Similar energy and forces convergence criteria were used as above. In addition, all core and valence states were included when evaluating the DOS profiles. The models were constructed using 4 × 4 supercells for graphene, h-BN, and MoS2 and a 2 × 1 and 2 × 2 for C3N4 and Cu(111), respectively.. For Cu(111), four atomic layers were used. A vacuum region of ∼20 Å was used to minimize interactions along the z-direction.

computed using FL shifts obtained directly via eq 1 and our RDI scheme. Two features are quite noticeable here: (i) When using the same code, eq 1 and RDI yield almost identical charging energies as shown for graphene and h-BN (panels a and b of Figure 4, respectively). The deviation between the RDI obtained using VASP and those using DMol3 are the results of deviations in the charge−voltage plots, as shown in Figure 3, which is due to variations in the DOS profiles for each charge generated by the different codes. (ii) The charging curves are in agreement with expectation that it is more facile to charge a material that is either conductive or with a small gap compared to a wide-gap material. In the case of graphene, 8.5 eV (6 eV using VASP) (∼0.1 eV/Å2) is needed to achieve a charge density of ∼6 × 1014 e/cm2, whereas ∼13 eV (∼0.15 eV/Å2) per supercell is needed for h-BN to achieve the same charge density. While the RDI scheme is robust and self-consistent when using the same computational codes and setup, numerical variations will arise because of different approximations and implementations (for example: plane waves vs localized orbitals) in different codes, leading to differing DOS profiles and hence different charge−voltage plots. In addition, the RDI scheme in a jellium model is perhaps more physically intuitive compared to the compensating localized background charge, which is normally confined to a small atomic region and could lead to spurious interactions in the local potentials. In conclusion, we have demonstrated a new method to evaluate work functions and Fermi level shifts upon charging. Traditional DFT methods that rely on the computation of a plane averaged electrostatic potential in the vacuum region to determine work functions will fail when a homogeneous background charge is present in the supercell. To overcome this we introduce a method based on recursive DOS integration. The DOS is computed at small charge increments, and the shifts in the Fermi level are determined by integrating the DOS to determine the shifts in the Fermi levels (δFL) corresponding to adding or subtracting a charge increment δq. The method is tested for a number of materials and shown to yield consistent results among the codes and implementations used (localized vs homogeneous compensating background charge). The method also has the potential to be extended to investigate charged point defects in solids where the Fermi level is usually adopted as that in a neutral system but which should be treated as a charge-dependent variable instead. Overall, we anticipate that this work will contribute to better understanding and quantifying charging effects in chargeresponsive materials.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.8b01631. Additional details about the effect of vacuum on computed charge−voltage curves as well as the choice of charge increments δq (Figures S1 and S2, respectively) (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

H. A. Tahini: 0000-0001-5454-0983 X. Tan: 0000-0002-6352-8828 S. C. Smith: 0000-0002-5679-8205 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was undertaken with the assistance of resources provided by the National Computing Infrastructure (NCI) facility at the Australian National University.

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REFERENCES

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COMPUTATIONAL METHODS Our DFT calculations were performed using two popular codes that differ in their implementations of basis sets: VASP34,35 and DMol3.20 With VASP, the projector augmented wave (PAW) method36 was used in treating core and valence electrons. Exchange−correlation was parametrized using the PBE37 functional with a plane-wave cutoff energy set to 500 eV. Self-consistent field iterations were considered converged when the energy change was less than 1 × 10−5 eV and while residual forces on atoms were no greater than 1 × 10−3 eV/Å. The Brillouin zone was sampled using dense meshes with grid spacing set to 0.00975 Å−1. For the total DOS integration to work effectively, we included all valence states in our evaluation by considering states that are up to 50 eV below the Fermi level. In DMol3, we used the PBE functional along 4018


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Fermi Level Determination for Charged Systems via Recursive Density

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