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ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2015, Vol. 121, No. 3, pp. 465–476. © Pleiades Publishing, Inc., 2015. Original Russian Text © A.G. Ryabishchenkova, M.M. Otrokov, V.M. Kuznetsov, E.V. Chulkov, 2015, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2015, Vol. 148, No. 3, pp. 535–548.

ELECTRONIC PROPERTIES OF SOLID

Ab Initio Study of the Adsorption, Diffusion, and Intercalation of Alkali Metal Atoms on the (0001) Surface of the Topological Insulator Bi2Se3 A. G. Ryabishchenkovaa,*, M. M. Otrokova,c, V. M. Kuznetsova, and E. V. Chulkova,b,c a Tomsk

State University, Tomsk, 634050 Russia de Física de Materiales UPV/EHU, Centro de Física de Materiales CFM–MPC and Centro Mixto CSIC-UPV/EHU, 20080 San Sebastián/Donostia, Basque Country, Spain c St. Petersburg State University, St. Petersburg, 198504 Russia * e-mail: [email protected]

b Departamento

Received February 27, 2015

Abstract—Ab initio study of the adsorption, diffusion, and intercalation of alkali metal adatoms on the (0001) step surface of the topological insulator Bi2Se3 has been performed for the case of low coverage. The calculations of the activation energies of diffusion of adatoms on the surface and in van der Waals gaps near steps, as well as the estimate of diffusion lengths, have shown that efficient intercalation through steps is possible only for Li and Na. Data obtained for K, Rb, and Cs atoms indicate that their thermal desorption at high temperatures can occur before intercalation. The results have been discussed in the context of existing experimental data. DOI: 10.1134/S1063776115090186

1. INTRODUCTION The field of solid state physics concerning the study of topological insulators, which are materials where spin–orbit effects are of primary importance, appeared in the middle of the last decade and then developed rapidly [1–6]. A topological insulator, as well as a normal insulator, has a bulk band gap, but, in contrast to the latter, it is inverted in a certain region of the Brillouin zone because of the strong spin–orbit coupling. As a result, a gapless state with a linear dispersion relation appears on the surface of the topological insulator in the form of a conical surface in the reciprocal space (Dirac cone). Electrons in such a state are spin polarized and are topologically protected because of time reversal symmetry from backscattering from defects, which promotes the flow of an electric current almost without loss of energy. For this reason, topological insulators are very promising for spintronics devices and quantum computers. The topological protection effect is confirmed both by ab initio calculations of the electronic structure and by experiments on studying possible channels of scattering of electrons in the topological surface state. In particular, it was shown in [7] that the topological surface state of binary tetradymite-like chalcogenides “survives” even after the complete removal of the surface Se (Te) monolayer. The topological protection effect was experimentally confirmed by scanning tunneling microscopy measurements for the surface of the

Bi0.92Sb0.08 solid solution [8], which demonstrated disorder at the atomic scale, and for surfaces of Bi2Se3 and Bi2Te3 containing defects [9–11]. However, the topological protection effect also holds after the deposition of foreign nonmagnetic atoms on the surface of a topological insulator, because such a modification of the surface does not break the time reversal symmetry. It has recently been shown [12] that the topological state of Bi2Se3 remains coherent up to high concentrations of an adsorbate at which the Fermi surface undergoes significant hexagonal distortion. However, the investigation of topological insulators with the use of foreign atoms includes not only experiments on studying electron scattering channels in the topological state. For example, the intercalation of copper into van der Waals gaps (in other words, the introduction of a foreign atom into interblock gaps) in the topological insulator Bi2Se3 results in the appearance of superconductivity [13] and the intercalation of silver [14, 15] can be used to control exchange interactions in blocklayered topological insulators doped with magnetic atoms [16]. Numerous experiments indicate that the deposition of various atoms and molecules on the surface of topological insulators [12, 17–22] leads to the appearance of Rashba states in their electronic spectrum, which coexist near the band gap with the Dirac cone [23, 24]. In particular, it was shown [19] that the Rashba parameter in such systems can reach 1.3 eV Å, which is much larger than that in normal semiconduc-

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tors. Furthermore, the possibility of controlling the position of the Dirac point in the bulk gap in Bi2Se3 by means of doping its surface with carbon atoms [25, 26], i.e., by the method significantly different from those proposed previously [27–29], was recently demonstrated. Finally, when studying the thermoelectric properties of the Bi2Se3 and Bi2Te3 compounds, the intercalation of Li and Cu atoms [30–35] is used to increase the thermoelectric Q factor or as an intermediate stage in the process of the fabrication of nanostructured thermoelectric materials.

and photoemission intensity of core 3d Rb (0.23 ML) levels at room temperature after a series of annealings at temperatures of 384, 394, and 417 K (for 5 min at each temperature) showed that the intensity of these levels did not change, excluding thermal desorption. At the same time, the intensity of the doublet of 3d Rb levels with a lower binding energy increased, whereas the intensity of the doublet with a higher binding energy decreased. For this reason, the authors concluded that Rb atoms changed their positions and were intercalated into van der Waals gaps in Bi2Se3.

Chemically active alkali metal atoms are widely used to dope the surface of topological insulators [12, 17–22, 36, 37], topological Dirac semimetals [38], and semiconductors with strong Rashba splitting [39, 40]. Their deposition makes it possible to significantly reduce the chemical activity of the surface of topological insulators [12, 22], to “bare” the Dirac point in photoemission experiments with p-doped topological insulators [36], and to control the electrostatic potential of the surface of topological insulators at the atomic scale [37]. It is noteworthy that the implementation of such effects depends on the coating of an adsorbate, temperature of its deposition, and duration and temperature of annealing. In particular, it was shown in [22] that the electronic structure of the rubidium-doped Bi2Se3(0001) surface becomes insensitive to the deposition of oxygen only after annealing. Thin Rb coatings (0.006–0.073 ML) and a low temperature of measurements (4.3 K) were used in [37]; as a result, the position of adatoms was controlled by the tip of a scanning tunneling microscope and, thereby, artificial structures of adatoms were created on the Bi2Se3(0001) surface. Another example is the deposition of chemically similar K [17] and Rb [22] atoms close in size on the Bi2Se3(0001) surface under various conditions. In the former case, comparison showed that the electronic spectrum obtained immediately after the deposition of potassium (in a time of 4 min at a temperature of 6 K) is significantly different from that obtained after the gradual heating of the surface to T = 220 K in 36 h. At the same time, it was revealed that the electronic structure of the surface measured at T = 220 K is almost the same as that obtained immediately after deposition in 1.5 min at T = 6 K (without subsequent heating). The authors of [17] explained this observation by the heating-induced partial desorption of K atoms from the Bi2Se3(0001) surface, which leads to a partial recovery of the spectrum of the doped surface of the topological insulator. The electronic structure of the Bi2Se3(0001) surface did not change after the deposition of 0.23 Rb ML at the temperature T = 190 K and subsequent heating to 350 K in 1 min. In this case, the subsequent scanning tunneling microscopy measurements for 0.12 (0.025) Rb ML at a temperature of 1.2 K revealed a decrease in the adsorbate coating by 60% (20%) after annealing at T = 400 K for 10 min. The measurements of the positions

In view of the experimental data presented above, it seems useful to obtain a priori information on the adsorption, diffusion, and intercalation of alkali metal atoms on the surface of topological insulators from ab initio calculations. Such a theoretical study makes it possible to obtain detailed information on localization, activation temperatures of diffusion, and diffusion lengths of adsorbates, which can be used to plan experiments and to interpret their results. For this reason, we performed the ab initio study of adsorption, diffusion, and intercalation of Li, Na, K, Rb, and Cs atoms on the (0001) surface of the topological insulator Bi2Se3. 2. MODEL AND METHOD OF CALCULATION The Bi2Se3 compound is crystallized in the tetradymite structure (space group R 3 m). This structure is formed by the periodic packing of quintuple-layer blocks alternating along the hexagonal axis of the crystal in the sequence such that xy coordinates of atoms in each layer are repeated (Fig. 1a). Atomic layers in a quintuple layer follow as Se–Bi–Se–Bi–Se and atomic bonds are ionic covalent, whereas the neighboring blocks are connected by van der Waals forces. As a result, so-called van der Waal gaps, which are quite large interlayer spaces consisting of distorted octahedra and tetrahedra in a ratio of 1 : 2, are formed (Fig. 1a). All tetrahedra in the bulk of the material are equivalent, whereas these tetrahedra near the surface are not equivalent because of the difference between the interplanar distances in the surface and subsurface quintuple layers, which is due to the relaxation of the surface. As a result, two groups of tetrahedra appear and the energies of systems with intercalated foreign atoms are expected to be different. Below, we specify nonequivalent tetrahedra by the notation of atomic layers that form the van der Waals gap nearest to the surface and to which the vertices of tetrahedral belong. To this end, we denote the sequence of atomic layers as ABCABC, beginning with the surface selenium layer (Fig. 1a). The xy coordinates of a foreign atom placed in a tetrahedron coincide with the coordinates of a Se atom located at the vertex of the tetrahedron. Thus, two tetrahedral sites–B and C–are possible for the atom intercalated into a van der Waals gap. The octahedral site of an intercalated atom is denoted as A. The

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AB INITIO STUDY OF THE ADSORPTION, DIFFUSION (a)

467

(b) A B

f h

C 0 1

VDW

2 4 5

3

b

A B C A B C

QL

Bulk

(c)

0] [011

7 8

9

f h b f h b f h b f ' h' b' B' A' C' E, eV

[0001] [2110]

AC

A C

0.8 0.4

A

VDW

0

A B C A B 0.8 0.4 0

i

t

Ea

Ea

[1120] Bi

f = fcc hollow h = hcp hollow b = bridge t = top

ACBACBACBACBACBACBACBACBACBACB A

A

6

t [0110 II

QL

[11 20]

Vacuum

B C A

Se ACBACBACBACBACBACBACBACBACBACB A

Fig. 1. (a) Crystal structure of Bi2Se3(0001) with the sequence of atomic layers ABCABC in the first two quintuple layers. Digits 0–5 and 6–9 mark the octahedron and tetrahedron of Se atoms in the van der Waals gap, respectively. (b) Plan view of the Bi2Se3(0001) surface containing [01 1 0]- and [11 2 0]-oriented steps. The upper terrace “is coated” by a light semitransparent layer for clarity. Symmetric sites are marked on the lower terrace: (t) top, (b) bridge, (f) fcc, and (h) hcp. Thin dashed lines show 1 × 1 and 3 × 3 hexagonal cells. The thick dotted, solid, and dashed broken lines are the diffusion paths of alkali metal atoms on the surface, near the step, and in the van der Waal gap, respectively. (c) Side view of the Bi2Se3(0001) surface containing a [01 1 0]oriented step and type-II termination. The dash-dotted broken line shows the type-I atomic termination. The diffusion energy profile of an isolated Li atom on the surface, near the [01 1 0]-II step, and in the van der Waals gap corresponds to the diffusion path shown in panel (b); E ai and E at are the activation energies of intercalation and return to the terrace, respectively.

atomic environment at the octahedral site is less dense than that at the tetrahedral site. In particular, the unoptimized (i.e., before relaxation) lengths of the Li–Se (Li–Bi) bond for Li in an octahedron and a tetrahedron are 2.71 Å (2.83 Å) and 2.39 Å (2.94 Å), respectively.

used for all alkali metal atoms under consideration. Furthermore, the diffusion of Li and Rb atoms in the van der Waals gap was studied with a 4 × 4 cell because nudged elastic band method calculations used to study diffusion are significantly complicated because of a large number (251) of atoms in the 5 × 5 cell.

To study adsorption and diffusion of isolated alkali metal adatoms on the Bi2Se3(0001) surface, we used 3 × 3 supercell in the xy plane containing one quintuple layer and a vacuum layer with a thickness of no less than 20 Å. To determine the equilibrium positions of alkali metal atoms intercalated into the van der Waals gap, we used a supercell containing two quintuple layers. It was revealed that isolated Cs atoms in the van der Waals gap cannot be simulated with 3 × 3 and 4 × 4 cells because the optimization of atomic positions in these cases resulted in the homogeneous broadening of the van der Waals gap both near the intercalated atom and far from it. The reason is that the atomic radius of a Cs atom (2.65 Å [41]) is larger than the size of the van der Waals gap in Bi2Se3 (2.49 Å [26]). This problem was resolved with a 5 × 5 supercell, which was

The diffusion of alkali metal atoms near a [01 1 0]oriented ([11 2 0]-oriented) step was studied with the use of a 3 × 9 (2 3 × 8) quintuple layer below a truncated 3 × 4 1 (2 3 × 4) quintuple layer simulating the 3 step. Two different atomic truncations, which can be implemented in the same calculation cell, were considered for the former orientation. The use of such cells allows the adequate simulation of various positions of the foreign atom with respect to the step: on the terrace and in the van der Waals gap far from the step and near it. However, as was mentioned above, nudged elastic band method calculations are complicated because of a large number (211 or 251, depending on the orientation of the step) of atoms in the cell.


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For this reason, the diffusion of alkali metal atoms near steps was studied by standard calculations of total energies, which were calculated for 18 (15) positions of an adatom in the supercell containing the [01 1 0]-oriented ([11 2 0]-oriented) step. The optimization of atomic coordinates of the substrate and the z coordinate of the adatom itself was performed for each position of the adatom. As is shown in Figs. 1b and 1c, an f (h) site on the lower terrace, which is farthest from the [01 1 0] ([11 2 0]) step, was chosen as the initial position of the adatom. Then, the adatom was successively placed at the neighboring b and h (b and f) sites located closer to the step with the chosen termination than the initial position. Thus, the adatom “followed” the diffusion path … f → b → h → b → f … revealed for the surface (see Section 3.2). Symmetric sites on the surface (f, h, b, and t) near the step are no longer equivalent to the respective sites far from it because the energies of the adatom at them are different. These sites will be marked by a prime (f', h', …). To calculate the activation energies of intercalation (E ai ) and return to the terrace (E at ), we “connected” the f' site to the octahedral site nearest to the step in the van der Waals gap (A' in Fig. 1c) by a sequence of close positions of the adatom. After that, the atom was successively placed at symmetric sites along the path A' → B → A → B → A …, which is the minimum-energy path in the van der Waals gap [42]. It is noteworthy that this path is almost equivalent to the path … A → C → A … passing through the tetrahedral sites C (see Figs. 1b, 1c). It is also seen in Fig. 1c that the diffusion energy profile at the edge of the terrace has a step shape. The activation energy is calculated in this case as the difference between the energies of the initial equilibrium position near the edge of the step (A') and the transition position highest in energy [43]. This procedure was performed for all geometries ([01 1 0]-I, [01 1 0]-II, and [11 2 0]), thus studying the dependence of E ai and E at on the orientation of the atomic termination of the step. The activation energies E ai and E at were calculated according to the above procedure only for Li and Rb atoms. The calculations were performed within the electron density functional theory with the use of the projector augmented-wave method [44] in the VASP implementation [45, 46]. The exchange-correlation potential was described in the generalized-gradient approximation [47], which ensures good agreement of the calculated band spectra of the surface of topological insulators with the experimental spectra (see, e.g., [1, 4, 11, 48, 49]). The van der Waal interaction was taken into account within the DFT-D2 approach proposed by Grimme [50], whereas the spin–orbit interaction and spin polarization were not taken into account. A number of test calculations show that both adsorbed and intercalated metal atoms have no local

magnetic moments. To seek transition states and diffusion activation energies of adatoms on the surface and in the van der Waals gap, we used the nudged elastic band method [51, 52]. The cutoff energy in calculations within the projector augmented-wave method was 400 eV. Relaxations of atomic positions were performed with the use of a 2 × 2 × 1 grid of k points when studying adatoms on the surface and in the van der Waals gap and with the use of a 3 × 1 × 1 (2 × 1 × 1) grid when studying adatoms near the [01 1 0]-oriented ([11 2 0]-oriented) step. The total energies were calculated with a double denser grid of k points in the twodimensional Brillouin zone. The optimization of atomic positions continued until forces become weaker than 0.025 eV/Å for each atom in the supercell. The selenium layer of the lower surface of the Bi2Se3 film was fixed at relaxation in all calculations.

3. RESULTS AND DISCUSSION 3.1. Adsorption of Alkali Metal Atoms on the Bi2Se3(0001) Surface Four symmetric sites can be indicated on the Bi2Se3(0001) surface: (t) top, (b) bridge, (f) fcc, and (h) hcp (see Fig. 1b). Two interstitial sites (f and h) are due to the packing of atomic planes of Bi2Se3 in the [0001] direction, which alternate in the sequence ABC ABC ABC … (Fig. 1a). To determine the equilibrium positions of adatoms on the surface, we calculated the adsorption energies (1) E ads = E 0 − E , where E0 is the energy of the system when the adatom is spaced by 10 Å from the substrate (i.e., the interaction of the adatom with the substrate is excluded) and E is the energy of the system when the adatom is on the surface. Thus, the position that has the maximum adsorption energy (i.e., the minimum energy E) is the most favorable among the four sites mentioned above. The calculations of the adsorption energies presented in Table 1 show that the f site is the most favorable position for alkali metal atoms on the Bi2Se3(0001) surface. For most atoms (K, Rb, and Cs), the h site has a close activation energy (difference is about 30 meV) and can also be an equilibrium position at low temperatures. For Li and Na atoms, the Table 1. Activation energies Eabs (in electronvolts) of alkali metal atoms on the Bi2Se3(0001) surface calculated for the (t) top, (b) bridge, (f) fcc, and (h) hcp sites

t b h f

Li

Na

K

Rb

Cs

1.454 2.227 2.429 2.554

1.145 1.751 1.924 1.975

1.377 1.821 1.934 1.965

1.418 1.847 1.947 1.979

1.593 2.000 2.089 2.114


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AB INITIO STUDY OF THE ADSORPTION, DIFFUSION Table 2. Atomic radii r [41], equilibrium lengths d of bonds with the nearest Se atoms, heights z0 of adatoms over the surface, and the charge Δq transferred from the adatom to the surface

r, Å d, Å f h z0, Å f h Δq, e f h

Li

Na

K

Rb

Cs

1.520 2.549 2.623 0.963 1.189 0.874 0.885

1.860 2.885 2.938 1.609 1.722 0.850 0.859

2.320 3.251 3.286 2.188 2.239 0.852 0.858

2.480 3.389 3.425 2.379 2.421 0.859 0.864

2.650 3.529 3.569 2.566 2.617 0.856 0.863

difference between the activation energies at the f and h sites is 125 and 51 meV, respectively; consequently, the fcc sites are the most probable positions on the ideal surface. The top site is significantly unfavorable for all atoms under consideration, particularly for small atoms. In contrast to the top site, the bridge site has a noticeably higher adsorption energy and, as will be shown below, is a transition position at diffusion on the surface. The sensitivity of the adatom to its position on the surface decreases with an increase in the size of the adatom. Table 2 presents the equilibrium lengths of bonds d with the nearest Se atoms, the height z0 of the adatom over the surface, and charge Δq transferred by the adaE, eV 0.40 0.35 0.30

Li Na K Rb Cs

~

b

b

0.25 0.20 0.15

h

0.10 0.05 f 0

~

f

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 Reaction coordinate, Å

Fig. 2. Energy profiles of the shortest diffusion path … f → b → h → b → f … of alkali metal atoms on the Bi2Se3(0001) surface. Here, f and f (f ≠ f ) are the initial and final states (f ≠ f ), respectively; b and b (b ≠ b ) are the transition states (saddle points) for the f → b → h and h → b → f jumps, respectively; h is the intermediate state; and the direction of the h → b → f jump makes an angle of 120° with the direction of the f → b → h jump (see the thick dotted broken line in Fig. 1b).

469

tom to the surface at the f and h sites. The charge transfer was estimated by the Beider method [53–55]. It is seen in Table 2 that the length of its bond with the nearest Se atom, as well as the height of the adatom over the surface, increases with the size of the adatom. Since the charge transfer to the substrate Δq is independent of the type of an atom, this effect is only due to an increase in the atomic radius of the adsorbate (Table 2). The equilibrium length of the bond of the Na atom with the nearest Te atoms on the Bi2Te3(0001) surface calculated in [56] is 3.05 Å. The length of the bond is larger than that in the case of the Na adatom on the Bi2Se3(0001) surface (Table 2) because the Te atom is larger than the Se atom. It is also noteworthy that the coverage in [56] was 0.25 ML, whereas the coverage in our work is 0.11 ML. 3.2. Diffusion of Alkali Metal Atoms on the Bi2Se3(0001) Surface Knowing the adsorption positions of atoms on the surface, one can study their diffusion. According to the nudged elastic band method calculations, a jump from a certain f site to another position for all atoms under consideration occurs through b and h sites in the sequence … f → b → h → b → f …. Figure 2 shows the energy profiles of a half of the above diffusion path, i.e., f → b → h. It was found that the bridge site b has the highest energy on the energy profile along the diffusion path and is a transition state for all adatoms under consideration. It is seen in this figure that the activation energy significantly depends on the size of the adatom. In particular, an energy of 110–140 meV is sufficient for Cs, Rb, and K atoms to jump from the f to h site, whereas this energy for Na and Li is 217 and 322 meV, respectively. It is remarkable that the activation energy of the “inverse” h → f jump is slightly lower than the activation energy of the “direct” f → h jump because of a lower activation energy at the h site. We also note that the activation energies of the f → b → h (h → b → f) jump obtained within the nudged elastic band method are in agreement with the differences between the activation energies Eads(f) – Eads(b) (Eads(h) – Eads(b)) within an accuracy of 1.2–4.3%, depending on the type of atoms and the direction of the jump, see Table 1 and Fig. 2. This circumstance indicates the reliability of the performed calculations of activation energies. Knowing activation energies, one can estimate the diffusion length of adatoms as a function of the temperature [42]:

Λ = 2α Dt,

(2)

where α is the dimension (α = 2 in the case of diffusion in two dimensions, e.g., on the surface or in the van der Waals gap), t is the time, and D is the diffusion coefficient depending on the temperature, which was


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Λ, μm 103

Tmax = E amax /4kB, where E amax is the maximum barrier for the corresponding adatom. For example, a Li atom migrating on the surface meets two barriers Efh =

101 10−1 10−3 10−5 10−7 10−9

0

100

200

300

400

500

Li Na K Rb Cs 600 700 T, K

Fig. 3. Temperature dependences of the calculated diffusion lengths of alkali metal adatoms on the Bi2Se3(0001) surface in the logarithmic scale for the diffusion time t = 1 min.

calculated using the approach proposed in [42]. According to that work, the diffusion coefficient of the adatom with the energy profile shown in Fig. 2 can be calculated by the formula

ν ν l2 D = 6 fh hf , 2α ν fh + ν hf

(3)

where νfh(νhf) is the jump frequency from the f(h) site to the h(f) site, l = a0/ 3 is the distance between the f and h sites, and a0 = 4.132 Å is the optimized lattice parameter of the substrate (the optimized lattice parameters and interplanar distances of bulk Bi2Se3 differ from the respective values recently reported in experimental work [26] by less than 0.35%). The jump frequencies were estimated by the formula

ν = ν 0e −E a /kBT ,

0.322 eV and Ehf = 0.2 eV. Thus, E amax = 0.322 eV and Tmax = 940 K, which are larger than the respective values for, e.g., Rb by a factor of 3. The temperature scale is limited by 700 K because the thermal desorption of quintuple layer blocks of the Bi2Se3(0001) surface occurs above this temperature [26]. It is seen in the figure that the diffusion lengths of larger atoms (K, Rb, and Cs) at temperatures below room temperature are at least an order of magnitude larger than those of smaller atoms (Na and Li). The diffusion activation temperatures are also very different. In particular, Li and Na atoms “are frozen” in adsorption positions up to temperatures of 60 and 80 K, respectively, when their diffusion length is no more than 0.1 Å within 1 min. This temperature for K, Rb, and Cs atoms is approximately 40 K. Thus, experiments on the control of the position of alkali metal adatoms on the Bi2Se3(0001) substrate with the use of the tip of a scanning tunneling microscope are hardly possible above the indicated temperature because the positions of adatoms vary continuously. At higher temperatures, atoms begin to move on the surface; i.e., diffusion is activated. The diffusion lengths of K, Rb, and Cs atoms on the Bi2Se3(0001) surface at temperatures of about 80 K reach 1 μm/min, whereas diffusion lengths of Li and Na atoms reach this value at 200 and 130 K, respectively. The indicated length is about an order of magnitude larger than the typical distance between steps on the surface of epitaxially grown Bi2Se3(0001) thin films [58–60]. The width of terraces on the (0001) surface of Bi2Se3 single crystals grown by the Bridgeman method can reach 1.5 μm [61]. For this reason, to study the intercalation of adatoms into van der Waals gaps [14, 22, 60], where efficient intercalation occurs through steps [15], it is necessary to heat a system up to temperatures above the indicated values.

(4)

where e −E a /kBT is the Boltzmann coefficient determining the jump probability at given activation energy Ea and temperature T. The vibrational frequency ν0 = E a /2m/2δ reflects the dynamic coupling between phonons of the substrate and vibrations of the atom at the adsorption position, where δ is the distance between adsorption and transition positions and m is the atomic mass. Formula (3) is valid only for the case of low adsorbate coverage and under the condition Ea > 4kBT, which guarantees that jump frequencies of the adatom are much lower than its vibrational frequencies near the equilibrium positions [57]. Figure 3 shows the temperature dependences of the calculated diffusion lengths of alkali metal adatoms on the Bi2Se3(0001) surface (t = 1 min). According to the condition Ea > 4kBT, each line in the figure finishes at

3.3. Intercalation of Alkali Metal Atoms on the Bi2Se3(0001) Step Surface As was mentioned in the Introduction, the desorption of potassium atoms from the Bi2Se3(0001) surface and the intercalation of rubidium atoms into van der Waals gaps of Bi2Se3 owing to the heating of the system were reported in [17] and [22], respectively. In the former case, the surface with the adsorbate was heated from 6 to 220 K in 36 h, whereas in the latter case, it was heated to higher temperatures of 350–417 K in shorter times (1–10 min). The conclusion on the desorption of potassium atoms in [17] was based on the analysis of the band structure of the surface, which exhibited partial recovery and, finally, had the form almost coinciding with the spectrum obtained for lower coverages (shorter deposition time). The con-


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AB INITIO STUDY OF THE ADSORPTION, DIFFUSION Table 3. Energies (in electronvolts) of alkali metal atoms at the (A) octahedral and (B and C) tetrahedral sites in the van der Waals gap of Bi2Se3(0001) measured from their energies at the (f) fcc site on the surface

A B C

Li

Na

K

Rb

Cs

–0.463 –0.031 –0.053

0.078 0.867 0.863

1.708 2.545 2.596

2.482 3.178 3.259

3.304 3.564 3.712

clusion on intercalation in [22] was based on the measurements of the intensity of the main 3d levels of rubidium. First, the intensities of the indicated levels did not change after annealing; i.e., thermal desorption was excluded. Second, the intensity of the doublet of 3d levels at a lower binding energy increased owing to the intensity of the doublet at a higher binding energy, which indicates a change in the state of adatoms attributed in [22] to their intercalation. Thus, the conclusions on the reasons for the observed phenomena were based in the cited works on indirect data, rather than on direct evidence. It is known that a BiSe (BiTe) antisite defect located in the lower selenium (tellurium) layer of the subsurface quintuple layer (i.e., the layer forming the upper edge of the van der Waals gap nearest to the surface) can be detected by scanning tunneling microscopy [62–64]. This feature has a three-leaf shape and a transverse size of about 3 nm. An example of evidence of intercalation into the van der Waals gap in Bi2Se3 is the appearance of convex triangular features with a transverse size of about 3 nm that were observed in [14] on the scanning tunneling microscopy image of the topography of the Bi2Se3(0001) surface after the deposition of silver at room temperature. Such an evidence of intercalation was not presented in [22], although features corresponding to intercalated rubidium atoms could be detected by means of scanning tunneling microscopy. Nevertheless, since the annealing-induced decrease in the number of rubidium atoms on the surface was observed in [22] in particular for low coating (0.025 ML), this situation can be analyzed (we recall that the results of our calculations are valid only for the case of low coating). Unfortunately, potassium coating was not reported in [17] and any scanning tunneling microscopy image of the doped surface was not presented. This situation will also be analyzed for the case of low coverage. It was recently shown [15] that the intercalation of silver atoms into van der Waals gaps of Bi2Se3 is due to the presence of steps on the surface of Bi2Se3: adatoms enter into interblock spaces, diffusing on the surface and reaching steps. At the same time, the penetration of silver atoms into van der Waals gaps through interstices and/or vacancies of the surface quintuple layer of Bi2Se3 is much less probable because the activation energies of such processes are high. Since the atomic

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radii of alkali metal atoms (see Table 2) are larger than the radius of silver atoms (1.44 Å [41]), their penetration through such a mechanism is even less probable. Indeed, the calculated activation energy of the penetration of the Li atom (smallest among the alkali metal atoms under consideration) into the van der Waals gap of Bi2Se3 from its surface through interstices of the upper quintuple layer is approximately 2 eV. As will be shown below, the activation energy of the intercalation of the Li atom through a step is much lower. According to the calculations of diffusion lengths presented in Fig. 3, each of the alkali metal atoms under consideration can cover a path of at least 1 μm in minute at temperatures above 200 K. Thus, adatoms quite rapidly reach steps the distance between which on the Bi2Se3(0001) surface varies from 0.1 to ~1.5 μm, depending on the sample growth method [58–61], and can penetrate into van der Waals gaps if the intercalation activation energies are not too high. It is reasonable to ask whether intercalation is energetically favorable. The calculations of the total energies show that penetration into van der Waals gaps of Bi2Se3 reduces the energy of only Li atoms, whereas the energies of other alkali metal atoms increase in this process (Table 3). This increase is very large for K, Rb, and Cs atoms: the fcc site on a terrace is more favorable by 1.708, 2.482, ad 3.304 eV than the octahedral site (A), which is the most favorable position in the van der Waals gap for all atoms under consideration. The intercalation of K, Rb, and Cs atoms is apparently possible only at very high temperatures. As is seen in Table 3 and Fig. 4a, the difference between the energies of B and A positions is a nonmonotonic function of the size of an alkali metal atom. The reason for such a behavior is as follows. Being at the octahedral site, Li and Na atoms quite weakly distort the structure of the matrix. The displacement is maximal for Bi atoms with the same xy coordinates as the intercalated atom, but located above and below it (Figs. 4b, 4d). In-layer distances between Se atoms forming an octahedron differ only slightly from the lattice parameter a0 = 4.132 Å in both cases of Li and Na (Figs. 4c, 4d). The distortion of the matrix at the tetrahedral site B becomes much more noticeable because of a denser atomic environment (see Section 2). This distortion can be described as an increase in the volume of the tetrahedron B that is formed by Se atoms and in which a Li (Na) atom is placed, because the side of a triangle in its base increases at relaxation from a0 to 4.23 Å (4.34 Å) and its vertex is shifted upward by 0.27 Å (0.5 Å). This shift leads to both vertical and horizontal displacements of Bi atoms that are the nearest neighbors of the Se atom located at the vertex of the tetrahedron B (Fig. 4d). Such a distortion of the structure of the matrix results in an increase in the energy by 0.423 eV (0.789 eV) as compared to the situation where the Li (Na) atom is located at the octahedral site. According to the pre-


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Fig. 4. (a) Differences between the total energies of alkali metal atoms at the tetrahedral (B or C) and octahedral (A) sites in the van der Waals gap of Bi2Se3. (b) Absolute values of the vertical displacement of (circles connected by the solid line) Se atoms forming the octahedron A, (circles connected by the dotted line) the Se atom at the vertex of the tetrahedron B, see panel (d); (squares connected by the solid line) Bi atoms located above and below the atom intercalated into the octahedron A, and (squares connected by the dotted line) Bi atoms being the nearest neighbors of the Se atom located at the vertex of the tetrahedron B, see panel (d). (c) In-layer distance between (circles connected by the solid line) Se atoms forming the octahedron A, (circles connected by the dotted line) Se atoms located in the base of the tetrahedron B, and (squares connected by the dotted line) Bi atoms that are the nearest neighbors of the Se atom located at the vertex of the tetrahedron B; a0 = 4.132 Å is the lattice parameter of Bi2Se3, i.e., the equilibrium in-plane Se–Se or Bi–Bi distance in the pure matrix. (d) Alkali metal atom (black ball) and its environment in the (A) octahedral and (B) tetrahedral sites.

sented vertical displacements and in-plane distances, Na atoms, whose atomic radius is larger than that of Li atoms, are responsible for stronger distortions of the matrix. This tendency mainly holds for the K atom. However, this tendency is violated for Rb and Cs atoms: the side of the triangle in the base of the tetrahedron B decreases with an increase in the size of the foreign atom. In this case, the upward displacement of the vertex of the tetrahedron corresponding to the indentation of the Se atom into the quintuple layer reaches giant values of 1.67 and 2.02 Å for Rb and Cs atoms, respectively. This means that intercalated Rb and Cs atoms stronger distort the upper quintuple layer and weaker distort the lower quintuple layer. At the same time, being placed at the octahedral site, they almost identically distort the structures of both blocks and the degree of this distortion increases with an increase in the size of the foreign atom. Correspondingly, the nonmonotonic behavior of dBi–Bi (Fig. 4c) correlates with the nonmonotonic behavior of the difference between the energies of alkali metal atoms at the tetrahedral (B) and octahedral (A) sites in the van der Waals gap of Bi2Se3 (Fig. 4a). A similar situation was observed in the case of the tetrahedral site C, which differs from the site B: the base and vertex of the tetrahedron formed by selenium atoms belong to the upper and lower selenium layers, respectively, forming the van der Waals gap. The difference between the total energies for the sites B and C is 81 and 148 meV for Rb and Cs, respectively (Table 3). This can be due to the exclusion of the displacement of Se atoms of the lower surface of the Bi2Se3 film at relaxation. Indeed, the absolute value of the vertical displacement of the vertex of the tetrahedron C for Rb and Cs is smaller than the absolute value of the vertical displacement of the vertex of the tetrahedron B by 0.22 and 0.18 Å, respectively. A more correct analysis of this situation for Rb and Cs atoms requires 5 × 5 supercell contain-

ing three quintuple layers, thereby, requires very large computational consumptions, and has not been performed in this work. For a more detailed description of the possible intercalation of alkali metal atoms into van der Waals gaps of Bi2Se3, we calculated the corresponding activation energies with cells containing a step. Figure 5 summarizes the results of studying the diffusion of Li and Rb atoms on the terrace and in the van der Waals gap both far from the steps and near them. To simplify discussion, a schematic diffusion energy profile is presented along with the activation energies of intercalation and return to the terrace averaged over three cases under consideration, [01 1 0]-I, [01 1 0]-II, and [11 2 0]. We analyze the motion of Li and Rb atoms at temperatures above T = 200 K. At such temperatures, Li atoms rapidly reach steps, where they acquire a significant energy gain (about 0.5 eV on average) as compared to the fcc site on the terrace far from the step. The subsequent motion of the adatom is determined by the ratio of the activation energies of intercalation and return to the terrace whose average values are about 0.58 and 0.82 eV, respectively (we do not consider the motion of the adatom along the step). This ratio is E ai / E at ≈ 0.71. Thus, it can be expected that Li atoms be intercalated into van der Waals gaps at temperatures sufficient for overcoming the barrier E ai ≈ 0.58 eV. It is noteworthy that a close value E ai / E at ≈ 0.65 was obtained in [15, 42] for the Ag atom. Since the intercalation of Ag atoms was observed in the experiment [14] at room temperature, the same temperature is expected to be sufficient for the intercalation of Li atoms. As the activation energy of return to the terrace E at for Li is 240 meV higher than the intercalation


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Fig. 5. Schematic of the diffusion energy profiles of (lower) Li and (upper) Rb atoms near a step on the Bi2Se3(0001) surface. The numbers are the (left) activation energies of diffusion on the surface, (to the left near the step) activation energies of return to the terrace E at , (to the right near the step) intercalation activation energies E ai , and (right) activation energies of diffusion in the van der Waals gap. All energies are given in electronvolts. The energies E ai and

E at are given for [01 1 0]- and [11 2 0]-oriented steps and two atomic terminations in the former case, see Section 2 and Fig. 1. The values E ai and E at averaged for E ai and t E a over three indicated cases are marked as Average.

activation energy E ai , temperatures can be chosen such that Li atoms will be almost completely intercalated and partially deintercalated. The activation energy of diffusion in the van der Waals gap for Li atoms is 0.62 eV and transition states are located approximately at 2/3 of the path between octahedral and tetrahedral sites (see the energy profile in Fig. 1c). Thus, the diffusion of Li atoms in the van der Waals gap of Bi2Se3 is much slower than that on its (0001) surface. In contrast to Li atoms, Rb atoms, reaching steps, have a much smaller probability of penetration into the van der Waals gap because the average intercalation activation energy is very high (2.89 eV) and E ai / E at ≈ 8.8 (Fig. 5). Thus, the intercalation of Rb atoms in the low-coverage regime maybe possible only partially and only at very high temperatures. The question of the possibility of the intercalation of Rb atoms can be answered by means of the kinetic Monte Carlo calculations, but such a study is beyond the scope of this work. Nevertheless, we calculated the activation energy for the diffusion of Rb atoms in the van der Waals gap of Bi2Se3; this energy appears to be 0.72 eV, which is higher than the activation energy for the diffusion of Li surprisingly only by a factor of 1.16, taking into account the difference between their atomic radii. To determine the reason, we analyzed the atomic

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structure of the matrix for the cases where Li and Rb atoms are located at the most favorable transition position. As was discussed above, the Li atom at the octahedral site quite slightly distorts the structure of the matrix, whereas the distortion of the matrix at the transition position becomes much more significant. This distortion involves the vertical (approximately 0.13–0.2 Å) and horizontal (about 0.25 Å) displacements of Se atoms located immediately near the intercalated Li atom, as well as the vertical displacement (about 0.13 Å) of Bi atoms being nearest neighbors of the indicated Se atoms. Such a distortion of the structure increases the energy by 0.62 eV as compared to the octahedral site. In the case of Rb whose atomic radius is larger than the atomic radius of Li by a factor of 1.63 (Table 2), the distortion of the structure of the matrix is significant both for the tetrahedral (transition position) and octahedral (most stable position) sites. As a result, the activation energy for the diffusion of Rb in the van der Waals gap is quite low, although it could be significantly higher in the case of a weaker distortion of the matrix by the Rb atom at the octahedral site. As is seen in Fig. 5, the activation energy of the return of the Rb atom to the terrace for a [01 1 0]-oriented step is significantly higher than the activation energy of diffusion on the surface, leading to the formation of an energy well near the step. This means that, if a significant number of [01 1 0]-oriented steps are presented on the Bi2Se3(0001) surface, a certain number of adatoms are always confined in the energy well near them. In view of this circumstance, we recall that scanning tunneling microscopy measurements for 0.025 Rb ML on the Bi2Se3(0001) surface at a temperature of 1.2 K revealed a 20% decrease in the adsorbate coating after 10-min annealing at T = 400 K [22]. The presence of the energy well with a depth of 0.4– 0.47 eV near [01 1 0]-oriented steps can explain the partial “disappearance” of Rb atoms from the surface because some adatoms can be confined near steps at the reduction of the temperature to 1.2 K. In addition, the clustering of adatoms near steps is not excluded, as was observed in the case of the deposition of Cu on the Bi2Se3(0001) surface at room temperature [60]. Unfortunately, any scanning tunneling microscopy images of the step regions were not presented in [22]. As was mentioned above, the activation energies of intercalation and return to the terrace were calculated only for Li and Rb. Nevertheless, some qualitative conclusions can be made on the behavior of isolated K, Rb, and Cs atoms near steps on the Bi2Se3(0001) surface. In particular, since the atomic radii of Rb and K are close to each other and the presence of K atoms in the van der Waals gap is very unfavorable (Table 3), it can be assumed that barriers for the penetration of these atoms under the step are also very high. This is also likely valid for Cs atoms. Since the presence of Na atoms on the terrace is only slightly more favorable


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than the presence in the van der Waals gap, it can be expected that they will be intercalated at a certain critical temperature. It is assumed in [17] that potassium atoms are desorbed from the Bi2Se3(0001) surface in the process of heating of the system from 6 to 220 K in 36 h, whereas the authors of [22] excluded the desorption of rubidium atoms at heating to higher temperatures. Our studies do not answer the question: Are these processes possible under the indicated conditions? However, our calculations indicate that the average intercalation activation energy for Rb atoms is almost one and a half higher than their adsorption energy, which can also be considered as the desorption energy (see Table 1 and Fig. 5). This means that the desorption of isolated rubidium atoms can be activated at lower temperatures than intercalation. This is also likely valid for K and Cs atoms. We emphasize that these qualitative conclusions were obtained for low adsorbate coating. 4. CONCLUSIONS Ab initio study of the adsorption, diffusion, and intercalation of Li, Na, K, Rb, and Cs alkali metal atoms on the Bi2Se3(0001) surface has been performed for the case of low coverage. It has been shown that alkali metal atoms are adsorbed at fcc sites of Bi2Se3(0001), where they are frozen up to the diffusion activation temperatures, which are 80 and 60 K for Li and Na atoms, respectively, and is about 40 K for K, Rb, and Cs atoms. The activation energies of diffusion on the surface decrease with an increase in the size of alkali metal atoms, which increases diffusion lengths. The estimates have shown that the diffusion lengths of K, Rb, and Cs (Na and Li) atoms at temperatures above 80 K (200 and 130 K) exceed 1 μm in 1 min; as a result, they quite rapidly reach steps on the Bi2Se3(0001) surface. However, efficient intercalation into van der Waals gaps of Bi2Se3 through steps is possible only for Li and Na atoms, whereas intercalation for K, Rb, and Cs atoms is, first, very energetically unfavorable and, second, requires overcoming high energy barriers. Furthermore, it has been revealed that the desorption energy for Rb atoms is lower than the average intercalation activation energy by a factor of about 1.5, which is also apparently valid for K and Cs atoms. Thus, the desorption of K, Rb, and Cs atoms in the low-coverage regime at the heating of the system to quite high temperatures can begin before intercalation. ACKNOWLEDGMENTS We are grateful to M.A. Gonzales and Yu.M. Koroteev for useful discussions. The calculations were performed on the SKIF-Cyberia computer cluster (Tomsk).

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