ISSN 1063-7834, Physics of the Solid State, 2017, Vol. 59, No. 6, pp. 1063–1070. © Pleiades Publishing, Ltd., 2017. Original Russian Text © V.V. Pogosov, V.I. Reva, 2017, published in Fizika Tverdogo Tela, 2017, Vol. 59, No. 6, pp. 1043–1050.
METALS
Ionization Potential of a Metal Cluster Containing Vacancies V. V. Pogosov* and V. I. Reva Zaporozhye National Technical University, ul. Zhukovskogo 64, Zaporozhye, 69063 Ukraine * e-mail:
[email protected] Received August 5, 2016
Abstract—A consistent procedure for determining the ionization potential of a large metal cluster of radius RN, v, consisting of N atoms and Nv vacancies, is proposed. The perturbation theory in small parameters Rv/RN, v and Lv/Rv (Rv and Lv are average distance between vacancies and the length of electron scattering on vacancies, respectively) is constructed in the effective-medium approximation for the electron ground state energy. The effective vacancy potential profile, the electron scattering phase and length are calculated by the Kohn–Sham method for a macroscopic metal in the stable jelly model. The obtained analytical dependences can be useful to analyze the results of photoionization experiments and to determine the size dependence of the vacancy concentration, including that near the melting temperature. DOI: 10.1134/S1063783417060208
1. INTRODUCTION The solid-to-liquid phase transition in 3D samples is associated with configuration excitation, i.e., the formation of vacancy voids [1, 2]. The equilibrium vacancy concentration is calculated from thermodynamic considerations with the availability of vacancy formation energies. This value can be extracted from the annihilation spectrum of positrons localized in vacancies. In this case, the vacancy concentration remains a free parameter [3]. It was experimentally found that the melting temperature of free clusters decreases with decreasing their sizes. During melting, surface vacancy diffusion to the bulk is more advantageous for clusters with unfilled electron shells than for magic clusters [4]. There is a viewpoint that near the vacancy formation energy near the melting temperature is lower for smaller clusters, and the vacancy concentration is independent of their sizes [5, 6]. Nevertheless, despite detailed mass spectrometric and calorimetric studies of cluster premelting and melting (see, e.g., [4, 7, 8]), the question about the size dependence of the vacancy formation energy and concentration, as well as the relation to the melting process, remains open. For the first time, mass spectrometric measurements of the dissociation energy of Na +N and Al +N cluster ions were performed in [9] and [10], respectively. Conventionally, such data and the data on measured ionization potentials are used to calculate the cohesion energy of neutral clusters. Therefore, one of the urgent problems that can be formulated in the con-
text of small-sized aggregates is the determination of their ionization potential. Recently, the ionization potential AlN (N = 32–95) and its temperature dependence in the temperature range of 65–230 K were measured in [11]. As the temperature increases, an on average insignificant (~10 meV) decrease in the ionization potential is indicated. The melting temperature of such clusters is the range of 600–700 K [7, 8]. For clusters with a denumerable number of atoms, the binding energy is determined by numerical methods. For large clusters, the binding energy is usually determined by the ground state energy in an extended condensed material, taking into account the first size correction ~ RN−1 caused by the surface curvature [12, 13]. In the present study, the corrections ~RN−2 are taken into account for the first time. Furthermore, a procedure applicable to calculate the ionization potential and attachment energy of electrons of metal clusters containing point defects and impurities was developed. As an example, vacancies were chosen. This approach can be easily adapted to calculate the positron binding energy in such clusters. Methodically, the problem was solved in two stages. In the first stage, all characteristics of a semiinfinite 3D metal with flat surface (the curvature radius is zero), including the bulk vacancy formation energy, are calculated by the Kohn–Sham method. In the stable jelly model, the monovacancy is represented as a spherical cavity of atomic size in a homogeneous positively charged background caused by ions. This problem was solved in [14, 15]. As a result, the contri-
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bution of the sparse system of vacancies to the electron [14] and positron [3, 15] work functions in 3D metal was determined. In the second stage, we use these data to construct the asymptotic size dependence of the ionization potential of the spherical metal cluster of the electron attachment energy. 2. BASIC RELATIONS Let us represent the cluster as a sphere. By definition, the ionization potential and the electron attachment energy for a metal cluster with the number of atoms N, valence Z, and radius
RN = N 1/3r0
(1)
IPN = E NN e −1(RN ) − E NN e (RN ),
(2)
have the form
EA N = E NN e (RN ) − E NN e +1 (RN ),
where E NN e −1 ( E NN e +1 ) is the total energy of the sphere with the excess charge +e(–e). Here, Ne = ZN is the total number of valence electrons in the neutral cluster. This approach is based on the expansion of the chemical potential μ of valence electrons of the neutral cluster in power series of the inverse radius RN−1 and the corresponding expansion of the specific surface energy σ (liquid drop model)
μ1 μ 2 −3 + + O(RN ), RN RN2 σ σ σ(RN ) = σ 0 + 1 + 22 + O(RN−3 ), RN RN
By analogy, for the cluster with vacancies, in the asymptotic approximation, we use the expressions 2 μ1 μ − 22 + e , RN ,v RN ,v 2C N ,v 2 μ μ − 1 − 22 − e , RN ,v RN ,v 2C N ,v
IPN ,v = Wv3D − EA N ,v = Wv3D
(6)
Wv3D = W 0 + δ Wv , where W0 = –μ0 is the electron work function of the defect-free 3D metal, consisting of the bulk component and the surface dipole barrier. The quantity δWv is the correction to the bulk component of the work function, caused by the presence of vacancies [14, 15] and proportional to their relative concentration cv. Under the condition |δWv| ≪ W0, the weak dependence of the dipole barrier on the outer sample surface on the presence of vacancies in its bulk is assumed. Therefore, the further problem is reduced to the determination of the shift of the electron ground state energy εb, caused by vacancies of the neutral cluster, (7) ε b = −δ Wv . The latter term in IP in Eq. (6) is the energy of the singly charged sphere with the electrical capacitance
(
C N ,v = RN ,v ≈ C N 1 + 1 cv 3 according to the definition (5).
)
(8)
μ(RN ) = μ 0 +
(3)
where μ0 and σ0 correspond to the flat surface (RN → ∞). From the condition of mechanical equilibrium of the cluster, some rules were obtained in [16, 17], in particular
μ1 =
2σ 0 , n
(4) ⎛ σ0 ⎞ 2σ1 μ12n μ2 = − ≡ μ1 ⎜ δ1 − ⎟ , 2B0 n B0 ⎠ ⎝ where δ1 = σ1/σ0, B0 is the bulk modulus of 3D metal,
n = 3Z/(4π r03 ) is the electron density in metal, r0 is the radius of the Wigner–Seitz cell per atom. If the cluster contains Nv vacancies, then its radius
(
)
1/3 (5) RN ,v = (N + N v ) r0 ≈ RN 1 + 1 cv , 3 is the relative vacancy concentration cv = Nv/N = (r0/Rv)3, where Rv is the Wigner–Seitz supercell radius per cluster vacancy. It assumed that RN, v ≫ Rv.
3. PERTURBATION THEORY IN THE CELL APPROXIMATION Let us conditionally divide the cluster into Nv spherical Wigner–Seitz supercells of radius Rv by the number of vacancies, Nv Rv3 = RN3 ,v . The supercell muffin-tin potential is replaced by the spherically symmetric potential. The electron ground-state wave function in the cluster consisting of N atoms and Nv vacancies is defined by the wave equation 2 ⎡ − ∇ 2Ψ(r) + ⎢v eff (r ) + 2m ⎢⎣
Nv
∑ δv
eff ,v (r
i =1
⎤ − R i )⎥ Ψ(r) (9) ⎥⎦
= ε b Ψ(r), in which vacancies are centered at points Ri, Ri < RN, v. The one-electron effective potential (preliminarily calculated in the stable jelly model) is intentionally divided so that the spherically symmetric potential veff(r) forms the bottom of the conduction band and the surface barrier, and δveff, v(r – Ri) forms the ith vacancy (Fig. 1). The energy gap εb is caused by the ground state energy shift due to the presence of vacancies.
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(a)
veff(r) δveff, v(r) δWv RN, v (b)
Rv z
where T0 is the electron ground state energy due to the vacancy displacement volume, calculated in the zeroradius potential approximation; 〈δveff, v〉 Rv Rc is the cluster volume-averaged contribution of the potential energy from electron–vacancy potential δveff, v(r) ≡ veff, v(r) – v eff , where veff, v(r) is the self-consistent one-electron potential profile calculated by the Kohn–Sham method in the presence of a vacancy, v eff is the position of the bottom of the conduction band in the absence of vacancy (Fig. 1a). A measure of the effective range of the zero-radius potential is the length of electron scattering on vacancies Lv (it is also the radius of the pseudopotential for scattering s-phase) on the vacancy potential. For the electron, the vacancy represents a potential hump, therefore, Lv > 0 (in the units of Bohr radii, Lv = 1.85a0 and 1.93a0 for Na and Al, respectively). If RN, v ≫ Lv, the electron–vacancy interaction can be described using the Fermi optical approximation
T0 =
RN, v
1065
3 2Lv cv . 2mr03
(14)
3.1. Mean Field of Supercells The electron ground state energy, e.g., in the central cell (Ri = 0, Fig. 1) is found by solving the Schrödinger equation
Fig. 1. (a) Energy-level diagram of the cluster with a vacancy at the center and (b) the schematic for calculating the surface integral of the ith supercell in the perturbation theory.
Let us assume that the potential field features “spherical periodicity.” Then, as is customary in the condensed state theory, it is convenient to represent the ground-state electron wave function Ψ(r) as the product of two functions
Ψ(r) = ψ c (r)u(r).
(10)
The function ψc(r) varying on the scale of entire cluster is modulated on intervacancy distances by function u(r). The function u(r) within the supercell centered, e.g., at the point Ri, will be identical to the wave function uWS(|r – Ri|) satisfying the Schrödinger in the Wigner–Seitz supercell,
u(r) = u WS,
r ≤ Rv .
(11)
Then we write the quantity εb as the sum
ε b = ε (0) + ε (1),
(12)
⎡ 2 2 ⎤ ⎢− 2m ∇ + δ v eff ,v (r ) − T0(RN ,v )⎥ u WS(r ) = 0 ⎣ ⎦ with the boundary condition
∇ u WS(r )| r = Rv = 0,
(15)
(16)
i.e., uWS(r) behaves as a plane wave at the cell boundary. A solution to Eq. (15) can be simplified by replacing the potential profile in the cell by the second boundary condition u WS(r )| r = Lv (RN ,v ) = 0.
(17)
To this end, the inequality Lv(RN, v)/Rv ≪ 1 should be satisfied. The quantity Lv(RN, v) is the length of electron scattering on the cluster vacancy. The substitution of the expression for the electron wave function
sin{q 0[r − Lv (RN ,v )]} u WS(r ) = A q0r 4π
(18)
into the second boundary condition yields the equation
where
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tan[q 0(Rv − Lv )] − q 0Rv = 0,
(19)
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from which we find
T0 =
(15) into Eq. (9) which, after simple transformations, can be rewritten in the form
2q 02
(20) . 2m Such an approach was first used in [18]. The size dependence of T0 is naturally determined O( RN−3 )
by the dependence Lv(RN, v) = Lv + [19]. As a result, we can neglect the weak dependence of T0 and 〈δveff, v〉 Rv on RN, v. It should be noted here that the calculations by Eqs. (20) and (14) are in better agreement for lower cv. The solution to Eq. (19) in the Al supercell at cv = 0.01 and 0.05 leads to Т0 = 0.0388 and 0.2535 eV, respectively. Expression (14) gives lower values (Т0 = 0.0296 and 0.1480 eV), which indicates the lower accuracy of the optical approximation. The quantity 〈δveff, v〉 Rv is given by
∫
〈δ v eff ,v 〉 Rv = d r Ψ*(r)δ v eff ,v (r)Ψ(r) =
3cv 3 4πr0
Rv
∫
dr 4πr 2δ v eff ,v (r ),
(21)
⎡ 2 2 2 ⎢− ∇ − m ⎣⎢ 2m
⎤ ∇ u WS(| r − R i |) ∇ + v eff (r )⎥ ψ c (r) (24) u WS(| r − R i |) ⎦⎥ i =1
Nv
∑
= ε (1)ψ c (r). Equation (24) contains a “potential profile” in the form of a cross term. The cross term in Eq. (24) can be considered using the perturbation theory. Previously, a similar procedure was used to determine the ground state energy and effective masses of electrons and positrons in infinite metals. In [20, 21], the perturbation theory was constructed on the basis of cell wave functions, and ψc(r) was chosen in the form of a plane wave. In [22], the expansion of ψc(r) in plane waves in the crystal was used. In the case at hand, attention is accented on boundary conditions for ψc(r) on the surface, which results in a discrete energy spectrum of a particle. Let us determine the particle energy in the potential well veff(r) of the entire cluster, assuming that the well is deep. Then the boundary condition in Eq. (24) has the form
ψ c (r )| r = RN ,v = 0.
0
r0cv−1/3 .
into which it is convenient to substitute Rv = It is easy to qualitatively estimate the value of 〈δveff, v〉 Rv using Fig. 1a. As we found (see [15, Fig. 3]), for estimation we can take δveff, v(r) ≈ 0 even at r ≈ 3r0, despite weak Friedel oscillations; r0 = 3.99a0 and 2.99a0 for Na and Al, respectively. In large clusters, the effective potential at the cluster boundary is close to the potential of the infinite sample and represents a well of depth v eff ≈ –6 and –15.5 eV for Na and Al, respectively. For a rough estimate, the vacancy potential can be replaced by a rectangular potential hump of height U0 and of radius r0 measured from v eff , i.e., the potential bottom in the bulk (U0 ≈ +2.5 and +9 eV for Na and Al, respectively. Then
〈δ v eff ,v 〉 Rv ≈ U 0cv .
(22)
Based on numerical integration with potential profiles calculated for 3D metal with vacancy and without it, it is possible to recommend a more accurate approximation (23) ε (0) = A1cv + A2cv2 , where A1 = 4.10 eV, A2 = 3.70 eV for Na and A1 = 13.3 eV, A2 = 28.25 eV for Al. 3.2. Perturbation Theory Using the solution to the Schrödinger equation in the cell and expression (13), we can write the equation for the function ψc(r). Let us substitute Eqs. (10) and
(25)
After expanding ψc(r) in the basis of wave functions of the infinitely deep well, the energy ε(1) can be presented in the form convenient for further calculations
ε
(1)
= π2 + 〈δ V (r )〉, 2mRN ,v 2 2
(26)
in this case, as a perturbation in Eq. (24), the operator is chosen
δ V (r ) = − m
2
Nv
∇ u WS(| r − R i |) ∇. u r R (| − |) WS i i =1
∑
(27)
The diagonal matrix element represents the field δV(r) averaged over the ground state with quantum numbers nr = 1, l = 0,
〈δ V (r )〉 =
∫
d r ψ(r )δ V (r )ψ(r ),
(28)
2 sin(πr / RN ,v ) . ψ(r ) = 1 r 4π RN ,v
(29)
r < RN ,v
where
A direct estimation yields the unexpected result
⎛ ∇ u WS ⎞ 〈δ V (r )〉 ≈ ⎜ Rv (RN ,v ψ∇ψ) u WS ⎟⎠ ⎝ 2 R ≈ 2 v , mRv RN ,v
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IONIZATION POTENTIAL OF A METAL CLUSTER
where angle brackets mean integration over the cluster volume. Seemingly, such an estimate points to hierarchy fault in expansion (26), since, at first sight 〈δV(r)〉 is of the order of (RvRN, v)–1, while the order of the preceding term is RN−2 . However, angular integration in (28) introduces the new small parameter
L L 1/3 ξ = v = v cv 1, Rv r0
(31)
as a result, an additional term proportional to the ⎛ R ⎞ product ξ⎜ v ⎟ appears, i.e., the other small param⎝ RN ,v ⎠ eter. Due to the fact that the perturbation δV(r) occurs on the cell scale, it is reasonable to pass to integration over the cell in Eq. (28) and to use the Green formula
Substituting these expressions into Eq. (32), we obtain Nv ⎡d ψ 2(R ) 2 i ⎢ 〈δ V (r )〉 = − (ln u WS )| Rv 2m i =1 ⎢ dRi ⎣ (36) ⎤ 2 2 × d S r − ∇ ψ (r )| Ri d r ln u WS ⎥ . r ⎥ Si i ⎦ Using expression (18) for the wave function, we obtain
∑
∫
π
Si
∑
∫
i =1 Lv Rv , ⎪⎩3 Ri
d r ∇{ln[u WS(| r − R i |)]}∇ψ 2(r )
Nv ⎡ (32) 2 2 ⎢ {ln[u WS(| r − R i |)]}||r −R i |= Rv d S ∇ψ (r ) =− 2m i =1 ⎢ Si ⎣ ⎤ d r{ln[u WS(| r − R i |)]}∇ 2ψ 2(r )⎥ . − ⎥ Lv 4.5 nm and R > 6 nm for Na and Al, respectively. This approach seems promising for experimental determination of point defect or impurity concentrations in metal clusters. To this end, the length of electron scattering on a corresponding defect in 3D metal should be preliminarily calculated. The analytical expressions obtained are convenient to analyze results of photoionization experiments. In particular, the vacancy concentration in a cluster near to the melting temperature can be determined using these expressions. As an experimental value of the ionization potential IP (or the attachment energy EA) falls on one of calculated size and concentration dependences of IP (or EA), the cluster size and the vacancy concentration at a given temperature is automatically fixed. REFERENCES 1. R. S. Berry and B. M. Smirnov, J. Exp. Theor. Phys. 98 (2), 366 (2004). 2. R. S. Berry and B. M. Smirnov, Phys. Rep. 527, 205 (2013). 3. A. V. Babich, V. V. Pogosov, and V. I. Reva, Phys. Solid State 57 (11), 2135 (2015). 4. C. Hock, C. Bartels, S. Straßburg, M. Schmidt, H. Haberland, B. von Issendorff, and A. Aguado, Phys. Rev. Lett. 102, 043401 (2009). 5. C. C. Yang and S. Li, Phys. Rev. B: Condens. Matter 75, 165413 (2007). 6. G. Guisbiers, Nanoscale Res. Lett. 5, 1132 (2010).
7. G. A. Breaux, C. M. Neal, B. Cao, and M. F. Jarrold, Phys. Rev. Lett. 94, 173401 (2005). 8. A. K. Starace, B. Cao, O. H. Judd, I. Bhattacharyya, and M. F. Jarrold, J. Chem. Phys. 132, 034302 (2010). 9. C. Bréchignac, Ph. Cahuzac, J. Leygnier, and J. Weiner, J. Chem. Phys. 90, 1492 (1989). 10. U. R. Martin, F. Jarrold, J. E. Bower, and J. S. Kraus, J. Chem. Phys. 91, 2912 (1989). 11. A. Halder and V. V. Kresin, J. Chem. Phys. 143, 164313 (2015). 12. J. P. Perdew, M. Brajczewska, and C. Fiolhais, J. Chem. Phys. 108, 8182 (1998). 13. V. V. Pogosov, Introduction to Physics of Charged and Size Effects: Surface, Clusters, Low-Dimensional Systems (Fizmatlit, Moscow, 2006) [in Russian]. 14. A. V. Babich, P. V. Vakula, and V. V. Pogosov, Phys. Solid State 56 (5), 873 (2014). 15. A. V. Babich, P. V. Vakula, and V. V. Pogosov, Phys. Solid State 56 (9), 1726 (2014). 16. V. V. Pogosov, W. V. Pogosov, and D. P. Kotlyarov, J. Exp. Theor. Phys. 90 (5), 908 (2000). 17. V. V. Pogosov, Phys. Solid State 35 (4), 518 (1993). 18. B. E. Springett, M. H. Cohen, and J. Jortner, Phys. Rev. 159, 183 (1967). 19. I. T. Iakubov and V. V. Pogosov, J. Chem. Phys. 106, 2306 (1997). 20. J. Bardeen, J. Chem. Phys. 6, 367 (1938). 21. M. H. Cohen and F. S. Ham, J. Phys. Chem. Solids 16, 177 (1960). 22. M. J. Stott and P. Kubica, Phys. Rev. B: Solid State 11, 1 (1975). 23. T. P. Martin, Phys. Rep. 273, 199 (1996). 24. W. A. de Heer, Rev. Mod. Phys. 65, 611 (1993). 25. M. Brack, Rev. Mod. Phys. 65, 677 (1993). 26. M. A. Hoffmann, G. Wrigge, and B. von Issendorff, Phys. Rev. B: Condens. Matter 66, 014404 (2002). 27. P. Ziesche, J. P. Perdew, and C. Fiolhais, Phys. Rev. B: Condens. Matter 49, 7919 (1994). 28. J. A. Alonso and N. M. March, Surf. Sci. 160, 509 (1985).
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