ISSN 1063-7842, Technical Physics, 2008, Vol. 53, No. 5, pp. 664–667. © Pleiades Publishing, Ltd., 2008. Original Russian Text © S.Yu. Davydov, S.V. Troshin, 2008, published in Zhurnal Tekhnicheskoœ Fiziki, 2008, Vol. 78, No. 5, pp. 134–137.

SHORT COMMUNICATIONS

On Absorption on Polycrystalline Substrates S. Yu. Davydova, b and S. V. Troshinb a

Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia b St. Petersburg State Electrical Engineering University, St. Petersburg, 197376 Russia e-mail: [email protected] Received August 27, 2007

Abstract—The effect of a polycrystalline substrate on the charge of adatoms and on the work function is considered in terms of the Anderson–Newns model. Polycrystallinity is characterized by deviation δφi of the work function of the ith face from mean value 〈φi〉. It is found that polycrystallinity leads to an increase in the occupation numbers of adatoms as compared to absorption on a monocrystalline surface. This leads to a decrease in the value of charge in the case of electropositive adsorption and to an increase in its value in the case of electronegative adsorption. It is shown that the polycrystallinity effect must be manifested most strongly in the case of adsorption of gases at metals and semiconductors. PACS numbers: 73.20.Hb, 73.43.Ct, 73.61.Jc DOI: 10.1134/S106378420805023X

From the standpoint of adsorption and electron emission (both thermionic and field emission), the polycrystalline substrate is a surface containing a set of various crystal faces i, each of which is characterized by its area Si and work function φi. Sometimes, the term “spotty surface” is used [1–3], where a spot corresponds to the emergence of a certain face at the surface. The question arises: can the local value of work function φi be introduced for each ith spot (and if it can, under which condition)? It was shown in [4] that if we treat an electron as a wave packet, the uncertainity in its kinetic energy in motion in the x direction parallel to the (x, y, 0) surface is given by δE x ∼
/8m ( δx ) , 2

2

(1)

where δx is the mean square deviation of the electron coordinate. If we assume that δ E x has a value on the 2

order of
2 k Br /2m, where kBr is the value of the wavevector at the boundary of the Brillouin zone equal to π/a in order of magnitude (a is the lattice constant), we can write ( δx ) min ∼ a/2π.

(2)

Thus, work function φi of the spot can be assumed to be an informative local characteristic of a nonhomogeneous surface. Estimates of quantity δφi = |φi – 〈φi 〉| and of the ratio δφi /〈φi 〉 (〈φi 〉 is the value of work function averaged over various faces) obtained using the data from [5, 6] show that δφi amounts of tenths of an electronvolt, while δφi/〈φi 〉 amounts to one or several tenths. To describe adsorption, we will use the Anderson– Newns model [7–9], which was successfully employed for describing adsorption on metals [10–13] as well as

on semiconductors [14–16]. Let us consider adsorption of a single atom on the ith spot. In accordance with the Anderson–Newns model, the occupation number ni of such an atom can be written in the form Ωi 2 n ai = --- arctan -----, π Γ

(3)

where Ωi = φi – εa (εa is the energy of a quasi-level of the adatom, which is measured from its value in vacuum like φi). Here, we exclude the dependence of the position of quasi-level εa and its half-width Γ from a specific ith face. Such a dependence may basically emerge in view of crystallographic difference between faces and the corresponding electron densities. It should be noted that here and below we also disregard the presence of transient regions between grains primarily due to the fact that the total area of such regions is much smaller than the total area of the surface of the spots. We introduce coverage of the ith spot by adatoms, defining it as Θ i = N ai /N i ,

(4)

where Nai is the density of atoms adsorbed on the ith face and Ni is the density of adsorption centers at the ith face. Taking into account the dipole–dipole repulsion of adatoms in accordance with the theory of adsorption, we obtain the following expression for charge Zai = 1 – nai of an atom adsorbed at the ith spot and for the change ∆φi in the work function due to adsorption:

664

ON ABSORPTION ON POLYCRYSTALLINE SUBSTRATES

Ω i – ξ i Θ i Z ai ( Θ i ) 2 -, Z ai ( Θ i ) = --- arctan ------------------------------------------π Γ 3/2

ξ i = 2e λ 2

2

(5)

3/2 N i A,

∆φ i ( Θ i ) = – Φ i Θ i Z ai ( Θ i ), 2

where ξ is the constant of dipole–dipole repulsion of adatoms, 2λ is the arm of a surface dipole, and A ~ 10 is a dimensionless coefficient, which weakly depends on the geometry of the adatom lattice. Introducing relative area xi = Si/S occupied by the ith spot on the surface of a substrate of area S, we obtain the integral variation in work function ∆φ in the form M

∑ x ∆φ ,

∆φ =

i

(7)

i

i=1

where M is the number of spots on the surface; by definition, we have M

∑x

(8)

= 1.

i

We assume that adsorbed atoms are distributed over the surface at random. This is indeed observed at low temperatures, when barriers for surface diffusion are much higher than kBT and an adatom is permanently located at the same adsorption center where it arrived during the deposition of the film; in this case, we have

(6)

Φ i = 4πe N i λ,

N ai ≈ x i N a .

∆φ = 〈 ∆φ 1〉

(9)

M

Θ

∑ ( x /Θ ) = 1. i

Setting Ni ≈ NML and Θi ≈ Θ in this expression (in addition to the assumption concerning the random nature of filling the surface with adatoms), which is based on the hypothesis about approximately identical number density of adsorption centers in different spots (i.e., on the disregard of the difference in the geometry of faces), we obtain identity (8) and Φ ≈ Φi. Then equality (14) assumes the form Z a = 〈 Z ai〉 ,

Ω i – ξΘ Z ai ( Θ ) 2 〈 Z ai ( Θ )〉 = --- arctan ---------------------------------------- , π Γ 3/2

ξ = 2e λ M

∑ x ∆φ . i

(10)

i

i=1

On the other hand, it follows from expression (6) that 〈 ∆φ 1〉 = – 〈 Φ i Θ i Z ai〉 .

(11)

Let us now introduce the total number of adsorption centers at the surface, which is equal to number NML of adatoms in the monolayer on the polycrystalline surface:

∑x N . i

M

Na =

∑x N i

ai ,

(13)

i

where occupancy of the entire surface is Θ = Na/NML. Thus, introducing energy Φ and charge Za of adatoms, which characterize the state of a given surface as a whole, we obtain the relation ΦΘZ a = 〈 Φ i Θ i Z ai〉 . TECHNICAL PHYSICS

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(19)

and equals zero in remaining cases; this gives 1 〈 Z ai ( Θ, φ i )〉 = ---------πδφ

i=1

The total number Na of sites occupied by adatoms is

(18)

3/2 N ML A.

1 f ( φ i ) = --------- for φ 0 – δφ ≤ φ i ≤ φ 0 + δφ 2δφ

(12)

i

2

Here, energy Ωi = εa – φi is a random quantity and operation 〈…〉 presumes averaging over the ensemble of spots. We assume that the distribution of the values of local work functions φi obeys the equipartition law with the probability density

M

N ML =

(17)

where

2

–1

(16)

i

i=1

holds, where 〈 ∆φ 1〉 = M

(15)

Dividing both sides of this expression by Ni and taking into account relation (12), we obtain

i=1

Let us now determine the averaged value of the change 〈∆ϕ1〉 of the work function of the spot, for which the identity

665

φ 0 + δφ

×

∫

φ 0 – δφ

Ω i – ξΘ Z ai ( Θ, φ i ) arctan ------------------------------------------------ dφ i . Γ 3/2

(20)

Here, we disregard singularities associated with the emergence of grain boundaries at the surface of the substrate. For zero coverage (i.e., in the case of adsorption of 0 a single (isolated) atom), its averaged charge 〈 Z ai 〉 has the form

666

DAVYDOV, TROSHIN φ 0 + δφ

0 〈 Z ai〉

ω Ω Γ ω arctan -----i dφ i = ---------- -----+- arctan ⎛ -----+-⎞ ⎝ Γ⎠ πδφ Γ Γ φ 0 – δφ (21) 2 2 ω ω 1 ω+ + Γ – ------_ arctan ⎛ ------_⎞ – --- ln ----------------- , ⎝ Γ ⎠ 2 ω2 + Γ2 Γ –

1 = ---------πδφ

∫

∫

where ω± = Ω0 ± δφ and Ω0 = φ0 – εa. Let us consider some special cases. It follows from 0 formula (21) that averaged charge 〈 Z ai 〉 = 0 for Ω0 = 0. This result can be explained as follows: for Ω0 = 0, the center of gravity of the quasi-level of an adatom coincides with the Fermi level of the substrate; as a result, no exchange of electrons takes place between the adatom and the substrate. If Ω0 ≠ 0 and satisfies the inequality δφ  |Ω0|  Γ, we obtain 2

0

2

Ω 0 – ξΘ Z a ( Θ, φ 0 ) 2 -. Z a ( Θ, φ 0 ) = --- arctan -----------------------------------------------π Γ 3/2

4 Γ δφ 4Γ 0 〈 Z ai〉 ≈ ⎛ ± 1 – ----------⎞ – ------ ------ ------, ⎝ ⎠ πΩ 0 3π Ω 0 Ω 0

dZ a ( Θ, φ 0 ) 3 ρ a ( Θ, φ 0 )ξ ΘZ a ( Θ, φ 0 ) -. --------------------------- = – --- ---------------------------------------------------------2 1 + ρ a ( Θ, φ 0 )ξΘ 3/2 dΘ

〈 Z ai ( Θ, φ i )〉 ≈ Z a ( Θ, φ 0 ) – A ( δφ ) + 2B ( δφ ) . 2

(23)

Here, we have

Ω 0 – ξΘ Z a ( Θ, φ 0 ) -, φ 0 ) ------------------------------------------------------3/2 3 Γ [ 1 + ρ a ( Θ, φ 0 )ξΘ ] 3/2

=

(28)

Z a ( Θ, φ 0 ) ≈ [ 1 – ( 3ρ a ξ/2 )Θ ]Z a , where

ρa 0

(24)

2Γ/π( Ω 0 2

=

0

3/2

Γ2)

+

and

(29) 0

Za

=

(2/π) arctan ( Ω0/Γ). Coefficients A and B assume the form A ≈ ρ a ( 1 – ρ a ξΘ ), 0

(22)

where the plus and minus signs correspond to Ω0 > 0 and Ω0 < 0, respectively. Here, the first nonvanishing correction in δφ is indeed of the second order of smallness. Thus, corrections are small for |Ω0|  Γ and |Ω0|  Γ. Expression (22) also follows that polycrystallinity reduces the charge of a positive adatom and increases the magnitude of the negative charge of an adatom. Let us now analyze expression (20) beginning from the dependence of charge Zai on work function φi. We expand Zai into a power series in ratio δφ, retaining the terms up to the second order, inclusively:

2 ρ a ( Θ,

(27)

To define the value of charge 〈Zai(Θ, φi)〉 averaged over the polycrystalline surface, we consider some characteristic values of coatings: small coating (Θ  [|Ω0/ξZa(Θ, φ0)|]2/3), intermediate coatings (Θ ~ [|Ω0/ξZa(Θ, φ0)|]2/3), and large coatings (Θ  [|Ω0/ξZa(Θ, φ0)|]2/3). Here, for the last two cases, we assume that [|Ω0/ξZa(Θ, φ0)|]2/3 < 1. Equation (27) leads to

2

This result does not depend on polycorrect to crystallinity; i.e., the charge of the adatom is the same as on a monocrystalline substrate. If, however, |Ω0|  Γ  δφ, we have

⎛ d 2 Z ai ( Θ, φ i )⎞ -⎟ B ≡ ⎜ ---------------------------2 dφ i ⎝ ⎠ φi = φ0

(26)

0

where

(δφ)2.

ρ a ( Θ, φ 0 ) A = ---------------------------------------------, 3/2 1 + ρ a ( Θ, φ 0 )ξΘ

a

In the range of small coating, we have

〈 Z ai〉 ≈ ( ω + – ω – )/2πΓδφ = 2Ω 0 /πΓ 0

2 Γ -, ρ a ( Θ, φ 0 ) = --- ------------------------------------------------------------------π [ Ω – ξΘ 3/2 Z ( Θ, φ ) ] 2 + Γ 2

3/2

0

2

B ≈ ( ρ a /Γ ) [ Ω 0 – ξΘ ( 3ρ a Ω 0 + Z a ) ]. 0

3/2

0

0

(30)

Using relation (23), we then find 3 0 3/2 0 〈 Z ai ( Θ, φ i )〉 ≈ Z a ⎛ 1 – --- ρ a ξΘ ⎞ ⎝ ⎠ 2 –

0 δφρ a ( 1

–

(31)

3/2 0 ρ a ξΘ ).

Let us assume that δφ ρ a  | Z a |. For Θ1 < 0

0

(2/3 ρ a ξ)2/3, the charge and its dependence on δφ decreases upon an increase in Θ from 0 to Θ1. In the 0

range Θ1 ≤ Θ ≤ Θ2, where Θ2 = ( ρ a ξ)–2/3, the charge changes its sign, but its dependence on δφ decreases as before. It should be noted that the variation in work functions δφ (i.e., polycrystallinity) leads to a decrease 0 on the magnitude of charge for Z a < 0 and to its 0

increase for Z a > 0. For Θ > Θ2, the value of the charge increases with the coverage, the spread in work functions δφ contributing to this increase. 0

0

(25)

It should be noted that for Z a = 0 this takes place for Ω0 = 0 and the contribution to 〈Zai(Θ, φi)〉 comes only from the second term on the right-hand side of relation (31). This follows directly from expressions (23) and (27). Thus, the value of charge in this case is deterTECHNICAL PHYSICS

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ON ABSORPTION ON POLYCRYSTALLINE SUBSTRATES

mined exclusively by the polycrystallinity of the substrate. In the range of intermediate coverage, when Θ* ~ [|Ω0/ξZa(Θ*, φ0)|]2/3, charge Za(Θ*, φ0) ≈ 0 and A ≈ ρ *a [1 + ρ *a ξ(Θ*)3/2]–1, where ρ *a ≡ ρa(Θ*, φ0) ≈ 2/πΓ; in this case, we have 2δφ -. (32) 〈 Z ai ( Θ*, φ i )〉 ≈ – ------------------------------------3/2 πΓ + 2ξ ( Θ* ) Consequently, a nonzero value of the average charge is determined precisely by polycrystallinity in this case also. Finally, let us consider large coverages, for which Θ  [|Ω0/ξZa(Θ, φ0)|]2/3. In this range, we have Za(Θ, φ0) ≈ 0, ρa(Θ, φ0) ≈ 2Γ/πξ2Θ3 Z a (Θ, φ0), and A ≈ ξ–1, whence 2

δφ 〈 Z ai ( Θ*, φ i )〉 ≈ – ------ . ξ

(33)

Taking into account the fact that dipole–dipole interaction constant ξ is the largest energy parameter of the problem (~10 eV and higher [14–20]), corrections (32) and (33) have order δφ/ξ  1. Consequently, polycrystallinity of the substrate produces the maximal effect 0 for small coverages for adatoms whose charge Z a ~ 0 (it is well known that this characterizes adsorption of gases on metals [10] and semiconductors [17–20]. This study was performed in the framework of the target program “Development of Research Potential in Higher Education of the Russian Federation” (project no. RNP 2.1.2.1716K). REFERENCES 1. C. Herring and M. H. Nichols, Rev. Mod. Phys. 21, 185 (1949); Thermionic Emissions (Inostrannaya Literatura, Moscow, 1950). 2. L. N. Dobretsov and M. V. Gomoyunova, Emission Electronics (Nauka, Moscow, 1966) [in Russian].

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Translated by N. Wadhwa