Surface-Bulk Model for d0 Ferromagnetism in ZnS Quantum Dots

Article pubs.acs.org/JPCC

Surface-Bulk Model for d0 Ferromagnetism in ZnS Quantum Dots and Wires Vitaly Proshchenko, Anri Karanovich, and Yuri Dahnovsky* Department of Physics and Astronomy/3905, University of Wyoming, 1000 E. University Avenue, Laramie, Wyoming 82071, United States ABSTRACT: We study d0 ferromagnetism in ZnS quantum dots (QDs) and nanowires (NW). To find the magnetization of the medium and large size nanocrystals (NC), we introduce the surface-bulk (SB) model where the separately calculated surface and bulk contributions to the total magnetic moment allow us to find the magnetization for a large nanocrystal (NC). For nanowire calculations the accuracy of the SB model varies from 0.2% to 21% depending on the Zn vacancy concentrations on the NC surface and in the NC core. We find that the higher the concentration of the Zn vacancies, the larger the total magnetic moment in the nanocrystal. However, the magnetic moment increases faster for quantum dots rather than for nanowires. We also study the cases where the concentrations of Zn vacancies can be different on the NC surface and in the core. From the comparison of the experimental and theoretical NW magnetic moments, we find that the experimental magnetization is 1.4 × 103 smaller than the calculated one. Such a huge discrepancy can be explained from the assumption that not all magnetic moments due to Zn vacancy participate in the ferromagnetism, and there are some regions with zero magnetism and uncoupled (paramagnetic) spins. The random orientation of nanoowires could be another reason for the weak ferromagnetism.

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INTRODUCTION Magnetic nanomaterials with high Curie temperature (Tc) are essential in the fast growing fields of nanomagnetism, spintronics,1,2 and biomedicine.3 The important class of magnetic nanomaterials is transition metal (TM) doped semiconductor nanocrystals (NCs), which simultaneously exhibit both a magnetic order and unusual optical properties.4−20 The TM nanocrystals, however, can lead to the formation of hazard free radicals that make them unsuitable in medicine. There is another type of ferromagnetic material with Tc much above the room temperature and with no TM doping.21−28 Such nanocrystals exhibit d0 (d or f electrons are not involved) ferromagnetism associated with unpaired electron spins due to the intrinsic defects, vacancies. One of the most attractive NC with d0 ferromagnetism is ZnS. A ZnS crystal is found to be in both stable cubic zinc-blende and hexagonal würtzite crystal structures with the bulk band gap of 3.7−3.8 eV.29−31 ZnS compounds are widely used as materials for cathode ray tube, field emission display phosphors, electroluminescent devices, and infrared windows32−39 as well as sensitizers for quantum dots sensitized solar cells.40 The theoretical description of such materials becomes important in the explanation of the existing experiments and prediction new properties. It is very difficult to computationally study NCs of medium or large sizes, i.e., large quantum dots or thick nanowires, because of a large computational time and insufficient computer memory. Therefore, it is desirable to describe NCs within some approximate and at the same time reliable models, which allow for faster computations. To facilitate large scale calculations, we propose the new approach based on the surface-bulk model (SB), which considers the NC © XXXX American Chemical Society

magnetization as the sum of surface and core contributions. We verify the SB model from the direct electronic structure calculations for medium size NCs and then use it for the large sizes of quantum dots and nanowires. The SB model is suitable for the description of nanoferromagnetism in ZnS würtzite nanocrystals, specifically quantum dots28 and nanowires,27 and therefore can predict magnetization for any shape and size nanoparticle with different Zn vacancy concentrations.

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COMPUTATIONAL DETAILS Electronic-structure calculations have been performed within the density functional theory (DFT) using the Vienna ab initio simulation package (VASP).41−44 The Pedrew−Burke−Ernzerhof (PBE) exchange-correlation functional45,46 within the generalized gradient approximation (GGA) and the projectoraugmented plane-wave (PAW) pseudopotential47,48 with the cutoff energy of 400 eV are employed for all calculations. The PBE functional has been chosen because it provides the fastest computational time for the electronic structure calculations. The Γ-centered k-point grid has been generated from the Monkhorst−Pack scheme.49 In this work we conduct the electronic structure calculations of the würtzite ZnS bulk, (101̅0) surfaces, and nanowires with and without Zn (or S) vacancies. To make the smooth transition from a surface to bulk, we use the scheme where the surface contains the three or five layers; the first and last layers (we consider them as the Received: March 2, 2016 Revised: May 11, 2016

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DOI: 10.1021/acs.jpcc.6b02191 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

magnetic moments of the surface we choose the 8 × 1 × 8 kpoint grid. The bulk density of states have been calculated with the 15 × 15 × 15 k-point mesh. The vacuum spaces of 15 and 20 Å are chosen to ensure that the periodic images are well separated for the surface and nanowire calculations, respectively. To study the Zn vacancy concentration dependencies of the bulk and surface magnetic moments, we choose V0Zn as 100%, 25%, 12.5%, 1.5%, and 0% for the bulk and 100%, 25%, 6.25%, and 0% for the surface. In the exact DFT nanowire calculations we consider the 6.6% and 26.6% Zn vacancy concentrations only on the surface (there are no vacancies in the core). We also conduct the studies where the vacancy concentrations on the surface and in the core are different with 6.6% on the surface and 12.5% in the core and 26.6% on the surface and 25% in the core.

surface layers) are optimized having symmetrically located vacancies while the middle layer(s) representing the core has (have) the fixed bulk geometry. The conjugate gradient optimization method with 4 × 4 × 4, 4 × 1 × 4, and 1 × 1 × 4 k-point meshes has been employed for the bulk, surface, and nanowire structure relaxation, respectively. For efficient magnetic moment calculations we choose a k-mesh, which provides us the most reliable and fast computations. For this purpose we perform the calculations of the magnetic moment with the different k-meshes and compare the obtained results, which are demonstrated in Tables 1 and 2 for the bulk and surface, respectively. Table 1. Comparison of the Magnetic Moment for the Different k-Meshes for the Bulk with V0Zn = 25% k-mesh

magnetic moment/cell, bulk V0Zn = 25%

4×4×4 8×8×8 11 × 11 × 11

2.84 μB 2.32 μB 2.37 μB

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ZnS BULK CALCULATIONS WITH WÜ RZITE CRYSTAL STRUCTURE A ZnS crystal with no vacancies is diamagnetic and therefore does not exhibit any ferromagnetism. The simple explanation is presented in Figure 1a where the two levels, nondegenerate a1 and triply degenerate, t2, split in the tetrahedral crystal field, are fully occupied. The vacancies can be generated by the absence of sulfur or zinc atoms in a ZnS nanocrystal. In our computations we first study the magnetic properties in the presence of S vacancies in the würzite crystal. We choose V0S = 12.5% and find that there is no ferromagnetic order in this case. Indeed, in the absence of six sulfur sp electrons removed from the triply degenerate t2 state only the two Zn 4s electrons with the opposite spins remain in the a1 crystal-field state, resulting in the zero total magnetic moment as shown in Figure 1b. If we consider Zn vacancies, the two Zn 4s electrons are removed from the t2 state, and according to the Hund’s rule, the configuration with the highest spin has to be chosen as depicted in Figure 1c. For the spin density of states calculations we study α (spin ↑) and β (spin ↓) spin density of states for (a) the bulk crystal with no vacancies, (b) 12.5% of the S vacancies, and (c) 12.5% of the Zn vacancies as shown in Figure 2. Figure 2a demonstrates that the partial spin density of states for α and β electrons are completely identical, resulting in the zero total

Table 2. Comparison of Magnetic Moment for Different kMeshes for the Surface with V0Zn = 25% k-mesh

magnetic moment/cell, surface V0Zn = 25%

4×1×4 8×1×8 11 × 1 × 11

1.62 μB 1.53 μB 1.53 μB

In Table 1 we present the magnetic moment calculations in the ZnS bulk with V0Zn = 25% (a Zn vacancy concentration). The results reveal that there is no significant difference between the calculations with the k-meshes of 8 × 8 × 8 and 11 × 11 × 11. Thus, we conclude that computations with the k-mesh 8 × 8 × 8 are reliable and can be used in the further calculations. Then we run the surface calculations for the magnetization with V0Zn = 25%. The computational results are shown in Table 2. As we can see from this table, there is no difference between the calculations with the k-meshes of 8 × 1 × 8 and 11 × 1 × 11. Moreover, the calculations with the lowest k-mesh of 4 × 1 × 4 can be also considered as reliable. For the further calculations of

Figure 1. Single electron configuration schemes for (a) zero vacancy, (b) S vacancy, and (c) Zn vacancy structures. In the lower part of all figures we present the tetrahedral structures in a crystal field for (a), (b), and (c) cases. B


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Figure 3. Zn vacancy concentration dependence of the total magnetic moment per unit cell for the 3D crystal.

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ZnS WÜ RZITE SURFACE ELECTRONIC STRUCTURE CALCULATIONS As for the bulk crystal we have found that the presence of the S vacancies leads to zero magnetization for the 1010̅ surface plane. The monolayer magnetic moment already includes the surface-bulk interface effect that is discussed in more detail below. For the Zn vacancies we calculate the V0Zn dependence of the surface magnetic moment/unit cell, mS (see Figure 4),

Figure 2. α (spin ↑) and β (spin ↓) spin density of states for (a) 3D crystal with no vacancies, (b) 12.5% of S vacancies, (c) 12.5% of Zn vacancies, and (d) 12.5% of Zn vacancies only for the sulfur sp electrons.

Figure 4. Zn vacancy concentration dependence of the total magnetic moment per unit cell in the 2D würzite surface structure.

magnetic moment. For the S vacancies (see Figure 2b) the densities for the α and β states are also identical, resulting in the zero magnetization. In case (c) the contribution of one type of electrons (α) is higher than for β, resulting in the nonvanishing total magnetic moment that leads to the ferromagnetism. According to the mechanism introduced above (see Figure 1), the origin of the ferromagnetism is due to sulfur sp electrons. The calculations presented in Figure 2d confirm this conclusiononly the sulfur s and p electrons contribute to the nonvanishing total magnetic moment. In Figure 3 we study how the bulk magnetic moment/unit cell, mB, depends on the Zn vacancy concentration. As depicted in this figure, mB increases with the concentration of Zn vacancies, reaches the maximum, and then drops to zero at V0Zn = 100%. We have not checked the higher (>25%) Zn vacancy concentrations because in this case the magnetic moment depends on a specific configuration of adjacent Zn vacancies in a crystal. Such a physical situation will be investigated in separate reseach. The results from Figure 3 indicate that the magnetic value is not always proportional to the Zn vacancy concentration. The monotonic behavior is true only for low V0Zn.

which always increases with V0Zn. Contrary to the bulk, the magnetic moment is nonvanishing even if the concentration of Zn vacancies reaches 100%. Indeed the Td crystal symmetry is not conserved for the monolayer anymore, and therefore the physical picture based on the Td symmetry and described by Figure 1 is invalid. For the surface each sulfur atom has nonzero spin due to the four p electrons resulting in some nonzero total magnetic moment for the whole monolayer.

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SURFACE-BULK MODEL FOR ZnS NANOCRYSTALS: JUSTIFICATION AND VERIFICATION In general, it is very difficult to find electronic structures for medium and large quantum dots and nanowires because of the large computation time and computer memory limitations. Therefore, it becomes necessary to develop some approximate schemes where computations are significantly facilitated. For this reason we introduce the surface-bulk (SB) model where all physical properties can be presented as the direct sum from the surface and bulk contributions. In such a model we neglect all interactions caused by the surface-bulk interface (however we explain below how partially to take them into account) and also C


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The Journal of Physical Chemistry C the curvature of the surface taking place in quantum dots and wires. Apparently, this model should work well for larger systems where the interface contribution is negligible compare to the bulk and surface ones. For the same reason a surface curvature distortion becomes minimal for large quantum dots and wires. Thus, within the SB model the total NC magnetic moment is defined as the sum of the surface and bulk magnetic moments (see eq 1). MNC = NSmS + NBmB

Table 3. Comparison between the Exact DFT and SB Calculations for the Nanowire Magnetic Moment per Unit Length MNC/unit length (μB) V0Zn surface V0Zn = 6.6% surface V0Zn = 26.6% surface V0Zn = 6.6% + core V0Zn = 12.5% surface V0Zn = 26.6% + core V0Zn = 25.5%

(1)

where mS and mB are the surface and bulk magnetic moments per unit cell. NS and NB are the surface and bulk unit cell numbers, respectively. From the NS and NB numbers and the volumes of surface and bulk unit cells, VS and VB, we can find an NC volume and therefore evaluate an NC size, VNC defined by eq 2.

VNC = NSVS + NVVB

exact DFT calculations

SB calculations

discrepancy (%)

3.47 11.73

3.86 11.76

10 0.2

4.52

5.7

18.63

16.31

21 12.5

the conclusion that the SB model is very reliable and can be used for further magnetic moment calculations. Then we verify the magnetization calculations for spherical quantum dots with the surface V0Zn = 4.5%. The quantum dot diameter is approximately equal to 1.2 nm (it is a medium size QD). For such a small quantum dot the core does not exist for the geometrical reason and therefore does not contribute to the magnetization. For the verification of the SB model we have conducted only one SB calculation and compared the obtained magnetization with the DFT magnetic moment already found in ref 28. In this calculation we choose the zinc blende crystal structure. As shown in Table 4, the discrepancy between the SB model and exact DFT calculations is only 10%. Thus, the SB model works very well for quantum dots as well.

(2)

For the quantum dot volume, VNC = πD /6 (here D is a QD diameter), and for the nanowire volume, VNC = πD2L/4 (here D and L are a nanowire diameter and length, respectively). The SB model is very convenient because we can widely employ solid state codes for magnetization calculations. The schematic picture of the model for quantum dots and wires is presented in Figure 5a,b. 3

Table 4. Comparison between the Exact DFT and SB Calculations for the Quantum Dot Magnetic Moment vacancy concn, V0Zn surface V0Zn = 4.5%

Figure 5. Schematic picture of the surface and core layers in (a) spherical quantum dot and (b) quantum wire.

MNC, exact DFT calculations28

MNC, SB calculations

discrepancy

1.8 μB

2.0 μB

10%

To verify the influence of the interface between the surface and bulk layers, we preform the calculations for the two different cases (see Figure 6): (a) one surface monolayer + one

Besides the NC size and Zn vacancy concentration dependence of the NC magnetic moment, it is important to determine the magnetic moment of the whole nanocrystal per unit cell: mNC = MNC/NNC

(3)

Here mNC stands for the total NC magnetic moment pre unit cell, MNC is the total NC magnetic moment, and NNC = NS + NB is the total number of the surface and core unit cells. Such a quantity describes the ferromagnetic “strength” of the NC with Zn vacancies. Although we intuitively understand that the SB model can be applied for large systems, we nevertheless would like to verify its validity for the intermediate size nanocrystals where the standard DFT calculations are possible. The comprehensive verification of the SB model is made for different types on nanostructures: nanowires, quantum dots, and layers. In Table 3 we present the calculations of the nanowire magnetic moment per unit length from the exact DFT and SB-model calculations for the different surface and core Zn vacancy concentrations. We choose the nanowire with the diameter 1.9 nm (it is a medium size NW) and unit cell length of 6.25 Å. As follows from the calculations (see Table 3), the discrepancy varies from 0.2% to 21%. Such a small discrepancy leads us to

Figure 6. Schematic presentation of the surface-bulk calculations where the surface monolayers are optimized and the bulk monolayer(s) is (are) not optimized having the bulk unit cell values The calculated structures have (a) the two surface monolayers and the single bulk monolayer in the middle and (b) the two surface monolayers and the three bulk monolayer in the middle. The bubbles represent the Zn vacancies.

bulk monolayer + one surface monolayer and (b) one surface monolayer + three bulk monolayers + one surface monolayer. These two cases with one and three bulk monolayers (in the middle) include the effect of the interaction between the surface and bulk layers. In case b, the bulk layers are thicker and the interaction between the two surface monolayers on the both sides of the system is weakened. For the two surface monolayer calculations we have conducted the geometry optimization while for the bulk monolayer(s) no geometry optimization has been performed. D


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magnetic moment for the four different cases: (a) all vacancies are placed on the QD surface (the core does not have any vacancies), (c) all vacancies are located in the QD core (the QD surface does not have any vacancies), (b) all vacancies are placed on the NW surface (the core has no vacancies), and (d) all vacancies are located in the NW core (the NW surface has no vacancies). If the vacancies are on the NW surface, the magnetic moment is a linear function with the NW size as shown in Figure 8b. The similar behavior but for the quantum dots is described by the nonlinear increasing function (see Figure 8a). These curves demonstrate that the size dependence for the quantum dots is faster than that of the nanowires. If we compare the magnetizations for the cases where the Zn vacancies are only in the core of QDs and NWs, we see from Figures 8c and 8d that the MNC size dependencies are similar. The total magnetizations for quantum dots and nanowires have nevertheless different size dependencies. Indeed for a quantum dot, the core magnetization is proportional to the cube of a QD diameter while the surface magnetic moment is proportional to the square of a QD diameter. For nanowires the size dependence is weaker; i.e., the core magnetization is proportional to the square of a NW diameter while the surface magnetic moment is the linear function of a NW diameter. For the smaller NC sizes the surface vacancies contributions dominate over the bulk ones. All NW calculations presented in Figures 8, 9, and 10 are performed for the 1 μm long nanowire. The next step is to combine surface and bulk magnetic moments to make up the total nanocrystal magnetization depending on the NC size and vacancy concentration. In Figures 9a and 9b we present the results of the size-dependent calculations for the total NC magnetic moment with the 12.5% core and 6.25% and 25% Zn surface vacancy concentrations. All curves demonstrate the monotonic increase of magnetic moment with the both nanocrystal size and V0Zn. As mentioned in the discussion for Figure 8, the total NC magnetization grows faster for quantum dots than nanowires. Besides the calculations of the total NC magnetic moment, it is important to determine an NC magnetic moment per unit cell defined by eq 3. Such a quantity describes the ferromagnetic “strength” of an NC with Zn vacancies. In Figure 10 we present the NC size dependence on the NC magnetic moment per unit cell with the 25% Zn vacancy concentration. All studied cases demonstrate the different NC size dependencies. Indeed, if all vacancies are placed on the surface, mNC monotonically decreases, reaching the zero value for very large NC sizes. This dependence can be explained from the following considerations: the total number of the NC unit cells contains the number of the surface unit cells plus the bulk unit cells with no magnetic moments. With the increase the NC size the fraction of core unit cells dominates over the fraction of the surface unit cells. Thus, mNC vanishes because there are more empty core cells in a nanocrystal. If the vacancies are in the NC core only, the dependence of mNC on a nanocrystal size is rather different (the red curve in Figure 10a,b). To choose the smallest size nanocrystal, we have started the calculations from the 1 nm NC diameter because NCs with diameters below this value are not well-defined within the SB model (no bulk contribution). This is the reason why the magnetization begins from zero value at DNC = 1 nm. The magnetic moment increases with the NC size and reaches the constant value, which corresponds to the bulk magnetic moment per unit cell at V0Zn = 25%. If Zn vacancies are in both NC core and on the

Cases a and b describe the interface effect between the surface and bulk. As shown in Table 5 the effect of the interface is also Table 5. Surface-Bulk Interface Effect: (a) Optimized Monolayer, (b) Two Optimized Monolayers with the One Unoptimized Bulk Monolayer, and (c) Two Optimized Monolayers with the Three Unoptimized Bulk Monolayersa no. of surface layers, SL (a) one SL (b) one SL + one bulk SL + one SL (c) one SL + three bulk SL + one SL a

magnetic moment/cell (μB)

magnetic moment/ layer/cell (μB)

discrepancy (%)

1.63

1.63

0.0

5.13

1.71

4.7

7.07

1.41

13.5

The discrepancy is calculated with respect to case a.

small. It is only ∼13.5%. Thus, we conclude that the SB model is reliable and correctly describes the magnetization within the 10%−20% accuracy. In order to include the interface effect, we present the procedure of choosing a monolayer by detaching it from the bulk (see Figure 7). In Figure 7a, we schematically draw the

Figure 7. Schematic procedure of the detaching of the surface monolayer from (a) the one-monolayer + one-bulk-layer + onemonolayer system, (b) the single monolayer is detaching from the system, and (c) the detached monolayer where the surface-bulk interface interaction is included. The bubbles represent the Zn vacancies.

one-monolayer + one-bulk-layer + one-monolayer system. In such a system we symmetrically include the surface effects for both sides to exclude the bulk monolayer surface states if only a single surface monolayer is considered. The surface monolayer is optimized in the system depicted in Figure 7a. The detached monolayer is shown in Figure 7c. Figure 7b demonstrates a detaching process. The optimized monolayer with some Zn vacancies (the bubbles in Figure 7) is used for the further SB calculations. The SB model can be very successful for large size nanocrystals because the fraction of the interface contribution between the surface and bulk becomes negligible compare to the bulk and surface contributions. The other reason (which is in favor of the SB model) is the surface and core curvature distortion that becomes negligible for large nanocrystals. For small size QDs the model could be inaccurate because a surface and bulk are ill-defined. This model is very convenient if one employs solid state codes. By using the SB model, we can also describe nanocrystals with different surface and bulk Zn vacancy concentrations.

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COMPUTATIONAL RESULTS In this section we discuss the calculations of nanowire and quantum dot magnetic moments depending on an NC size and Zn vacancy concentration within the SB model. In Figure 8 we present the QD and NW size dependences of the total E


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Figure 8. NC size dependence of the total magnetic moment for the two different cases: (a) all the vacancies are placed on the QD surface, (b) all the vacancies are located in the NW surface, (c) all the vacancies are placed in the QD core, and (d) all the vacancies are located inside the NW core.

Figure 10. Size dependence on the magnetic moment per unit cell at the 25% Zn vacancy concentration for (a) the quantum dot and (b) nanowire.

Figure 9. Total NC magnetic moment dependence on the nanocrystal size for different partial 12.5% core vacancy concentration and surface vacancy concentrations, 6.25% and 25% for (a) the quantum dot and (b) nanowire.

reaches the plato for the larger NCs. The dependencies are very similar for the both quantum dots and nanowires. In Figure 11 we demonstrate how the concentration of Zn vacancies changes the value of a magnetic moment at the fixed NW size corresponding to the experimental diameter of 2.0 nm and length of 1.0 μm.27 We find that the magnetic moment linearly increases with the concentration of Zn vacancies. Then

surface, the size dependence of mQD and mNW is presented by the black curves in Figures 10a and 10b for quantum dot and nanowire, respectively. These curves contain the features from both cases discussed above. The black curves start from the nonzero value as for the blue curves. Then it goes down and F


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have also found that the higher the concentration of the Zn vacancies, the larger the total magnetic moment in the nanocrystal. If the both core and surface contain Zn vacancies, the magnetic moment increases with a quantum dot or nanowire diameter as depicted in Figure 9. The “strength” of NC ferromagnetism can be described by a magnetization per unit cell, mNC, defined by eq 3. If all vacancies are on the surface, mNC monotonically decreases reaching the zero value for very large NC sizes as shown in Figure 10. Such a behavior can be explained due to the prevailing zero magnetization contribution of the core cells vs surface nonvanishing magnetism. For the quantum dot with core vacancy defects, mNC increases with the QD or NW sizes (see the red curve in Figure 10a). The similar behavior is true for the nanowires (see Figure 10b). If QD and NW contain the vacancies in the core and on the surface simultaneously, the magnetic moment goes down with the NC size but not so strongly as in the case of the pure surface Zn vacancies. Thus, we conclude that the magnetization per unit cell becomes less efficient for larger nanocrystals. Figure 11 demonstrates how the NW magnetization depends on the concentration of the Zn vacancies. We have found that it is a linear function with V0Zn. The experimental magnetization has been compared with the theoretical estimation for the nanowire with D = 2.0 nm, L = 1 μm, and V0Zn = 23%.27 The calculations presented in Figure 11 provide the magnetization value of 2.7 × 104 μB while the experimental value is 0.05 emu/ g = 3.7 × 10 μB, which is 1.4 × 103 smaller than the calculated magnetic moment. Such a huge discrepancy could be explained from the assumption that not all magnetic moments participate in the ferromagnetism, and there are some regions with zero magnetism and uncoupled (paramagnetic) spins. This hypothesis has to be thoroughly investigated and proved in future research. The other reason for such a large discrepancy could be because the experimental Zn vacancy concentration in the nanowire is much smaller as suggested by the authors of ref 27 (certainly not the 3 orders of magnitude). There is another reason for small ferromagnetism in the experiments. As discussed by the authors in refs 50 and 51, the random orientation of GaN nanowires doped by Mn impurities50 or graphene nanoflakes51 could be the main reason for the small magnetization in these systems.

Figure 11. Zn vacancy concentration dependence of a total NW magnetization for a 2.0 nm in diameter and 1.0 μm length.

we compare the experimental and theoretical magnetizations in the nanowire with D = 2.0 nm and L = 1 μm and V0Zn = 23%.27 The calculations presented in Figure 11 provide the magnetization value of 2.7 × 104 μB while the experimental value is 1.4 × 103 smaller. Such a significant discrepancy could be due to the presence of some regions with zero magnetic moments and uncoupled spins.

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CONCLUSIONS We have theoretically studied from the electronic structure calculations at zero temperatures the possibility of d 0 nanoferromagnetism due to vacancies in ZnS quantum dots and nanowires. We have found that the S vacancies do not cause the ferromagnetism. For Zn vacancies we believe that the mechanism of the d0 ferromagnetism is due to the unpaired sp sulfur electrons in the tetrahedral crystal field as depicted in Figure 1. To calculate the magnetization induced by Zn vacancies for medium and large nanocrystals, we have introduced the surface-bulk model where the surface and bulk contributions to the total magnetic moment allow us to determine the magnetization in the most efficient way. We have thoroughly verified and justified the SB model by the comparison of the exact DFT with SB calculations for nanowires and quantum dots for the medium size nanocrystals. We have found that for the nanowires the accuracy of the SB model varies from 0.2% to 21% depending on the Zn vacancy concentrations (see Table 3). For a spherical quantum dot of the intermediate size of 1.2 nm we have found that the discrepancy between the SB and exact DFT calculations reported in ref 28 is only 10% (see Table 4). We have also provided the procedure how to choose a monolayer where the effect of the interface interaction between the monolayer and bulk is taken into account and then use this monolayer for further surface magnetic moment calculations for quantum dots and nanowires. We have found that the interface effect for the layers is also small. It is about 13.5% (see Table 5). We have demonstrated that the total magnetic moment per unit cell increases with the bulk Zn vacancy concentration at small concentrations and then goes down at the larger values as shown in Figure 3. The surface magnetic moment, however, behaves differently with the concentration. It is always a monotonically rising function (see Figure 4). Such an interesting behavior, we believe, is due to the absence of the tetrahedral crystal field in the surface monolayer. If we consider quantum dots and nanowires, the total magnetization is a monotonically increasing function as shown in Figure 8. We

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (Y.D.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by a grant (No. DEFG0210ER46728) from the Department of Energy to the University of Wyoming and the University of Wyoming School of Energy Resources through its Graduate Assistantship program.

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REFERENCES

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Surface-Bulk Model for d0 Ferromagnetism in ZnS Quantum Dots

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