Role of surface Cooper pair interactions on critical temperature of ultra thin film superconductors

Physics Letters A 372 (2008) 5841–5847

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Physics Letters A www.elsevier.com/locate/pla

Role of surface Cooper pair interactions on critical temperature of ultra thin film superconductors Rostam Moradian a,b,c,∗ , Mojgan Najafi a , Mohammad Elahi a a b c

Physics Department, Faculty of Science, Razi University, Kermanshah, Iran Nano Science and Nano Technology Research Center, Razi University, Kermanshah, Iran Computational Physical Science Research Laboratory, Department of Nano-Science, Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 22 November 2007 Received in revised form 19 May 2008 Accepted 15 July 2008 Available online 23 July 2008 Communicated by R. Wu PACS: 74.20.-z 74.20.Fg 74.62.-c 74.78.-w 71.18.+y

a b s t r a c t The superconductivity mechanism of Pb thin film on a Si substrate in the weak interaction regime is investigated. A discrete Fermi surface is constructed depend on the film thickness and electron density and crystallographic orientation. We consider two types of Cooper pair interactions, Cooper pair interaction at thin film surfaces and Cooper pair in the thin film volume. We have chosen the surface Cooper pair interaction, of ultra thin films superconductor proportional to the inverse of thin film thickness, while the volume Cooper pair interaction has been considered as a constant. By these assumptions, we have found oscillation feature of critical temperature T c and energy gap  in terms of thin film thickness similar to the experimental results for Pb/Si(111) thin film superconductor. However, by increasing number of Pb layers, the thin film T c goes to bulk T c . In contrast to the previous claimed constant value for the b ( T = 0)/k B T c in bulk, we have found oscillation of this parameter in terms of thin film thickness similar to the T c oscillation. © 2008 Elsevier B.V. All rights reserved.

Keywords: Superconductivity Thin film BCD theory Cooper pair interaction Critical temperature Fermi surface

1. Introduction Thin film superconductivity is a subject of great scientific and technological importance, since they are now available as epitaxial films on insulating substrate. Therefore they are the best examples for two-dimensional (2D) systems. Additionally semiconductor technology requires thinner and thinner films, so that the properties due to restriction to thicknesses of a few mono layers get also technological importance. An intriguing and unexpected feature has recently been discovered during epitaxial growth of metal thin films on semiconductors. Instead of forming three-dimensional islands of various sizes, as commonly observed for nonreactive interfaces, the metal atoms can arrange themselves into plateaus or islands of selective heights, with flat tops and steep edges, under certain growth conditions. This unusual behavior was first

*

Corresponding author at: Physics Department, Faculty of Science, Razi University, Kermanshah, Iran. Tel.: +98 918 132 4809; fax: +98 831 427 4555. E-mail addresses: [email protected], [email protected] (R. Moradian). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.07.037

observed in Ag/GaAs [1] and later in a few other systems such as Ag/Si(111) [2] and Pb/Si(111) [3–7]. It is believed that this extra stability of metal films with specific thicknesses has an electronic origin [8]. In these thin films, thickness is very much smaller than the other two dimensions; the projection of the momentum across the thickness becomes quantized. This phenomenon is known as the quantum size effect (QSE). The essential features of systems exhibiting quantum size effects follow in the simplest approaches from oscillations in the electronic density of states in terms of thickness which in turn lead to oscillations with thickness of quantities such as critical temperature T c of superconducting materials. As the thickness of a film is reduced to the nanometers scale, the film’s surface and interface confine the motions of the electron, leading to the formation of discrete electronic states known as quantum well states (QWS) [9]. This quantum size effect changes the overall electronic structure of the film. At small thickness, physical properties are thus expected to vary, often dramatically, with thickness. Experimental studies have demonstrated such variations with film thickness for properties such as the electronic density of states, electron–phonon coupling, surface energy,

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and thermal stability [10–14]. Quantum oscillations can be understood by analogy to the systematic property variations of chemical elements. The number of confined electrons in a film increases as the film gets thicker. These electrons fill quantum well states, just as the electrons in atoms fill successive shells. Quantum size effects (QSE) in superconductors have been a very attractive topic after some theoretical investigations of the superconducting critical temperature T c and energy gap for thin film [15–18]. In a metal film on a semiconductor substrate, the conduction electrons are confined by the vacuum on one side and the metal-semiconductor interface on the others [8]. QSE and the simple stability of metal thin films on a supporting substrate have been discussed in several papers [9,19,20]. The report by Guo et al. is the first definitive and quantitative demonstration of T c oscillation with Pb film thickness as well as the normal state conductance [19]. Using atomically uniform film of lead (Pb) with exactly known numbers of atomic layers deposited on a silicon (111) surface, Guo et al. observed oscillations in T c that correlated well with the confined electronic structure. Recently, Eom et al. used a scanning tunneling spectroscopy to show that both superconducting energy gap, , and critical temperature, T c , exhibit persistent oscillation without any suppression at ultra thin Pb film (5–18ML) [21]. Many physical properties modulated by QSE in the thin films have been revealed. In this Letter, we present a new model to explain these experimental results. The Letter is organized as follows. In Section 2 we introduce our model. In Section 3 we present our results and finally the last section is conclusion. In Appendix A we derive the relations for calculation of the Fermi wave vector k f of the normal thin film. 2. The T c ( K ) calculation of ultra thin film superconductor

(1)

k v kv

kv σ

where ck+ σ (ck v σ ) are the electron creation (annihilation), εkv − μ v the electron energy measured from the chemical potential μ and U is the attractive interaction between electrons with opposite spins. For weak electron interactions Eq. (1) could be written as H=

  (εk v σ − μ)ck+v σ ck v σ − ∗ c −k v ↓ ck v ↑ kv σ

−



kv

+

+ ck v ↑ c − , kv ↓

1=

S 4π

U 2 t



dk

kz  k

 × tanh

h¯ 2 (k2 2m z

1 h¯ 2 2( 2m (k2z

+ k2 ) − μ + E s (b))

+ k2 ) − μ + E s (b) 2k B T c

 (6)

,

where k z = πDn and E s (b) is the surface energy report by Czoschke z [22] depended to thickness (number of atomic layer) of thin film, b, given by E s (b) = A

Cos(2k f 0 b + B ) bχ

+ C,

(7)

where A is an amplitude parameter and B is a phase shift factor that will be dependent on the interface properties of the film and C is a constant offset. The decay exponent χ = 0.938. We separate the Cooper pair interaction, U t , into the volume term and surface term as U t = V U v + SU s ,

(8)

where U v is the Cooper pair interaction in the thin film volume and U s is the Cooper pair interaction at the thin film surfaces. Since the bulk contribution could be found in the limit of L z → ∞, where surface effects are negligible, we chose the surface Cooper pair interactions as Us =

1

αLz + β

(9)

.

Since bulk contribution is constant, volume contribution part is

To study the quantum size effects in superconducting properties of ultra thin films, we have used the BCS Hamiltonian for the effective interaction,

  + (εk v − μ)ck+v σ ck v σ − U ck+ ↑ c − c  c  , H= k v ↓ −k v ↓ k v ↑ v

have U t , then critical temperature, T c , of thin film superconductors could be found from

(2)

Uv = γ .

(10)

Therefore the thin film Cooper interaction is given by



Ut = S

γ Lz +

1

αLz + β



,

(11)

where parameters α , β , γ could be obtained by fitting calculated theoretical data with the experimental data. For the case of small L z , less than 50 layers, surface Cooper pair, U s , has main contribution. While for the case of large L z , more than 500 layers, volume Cooper pair, U v , plays main role. By solving Eq. (6), the critical temperature, T c could be calculated in terms of film thickness. 3. The numerical results and discussion

kv

where

=U



c −k v ↓ ck v ↑ 

(3)

kv

is the superconductor order parameter. For the bulk systems Eq. (3) could be converted to the following gap equation, 1=U



 1 kv

tanh

2E k v

where E k v =

Ekv 2k B T

 ,

(4)



ε2 k v + ||2 . Gap Eq. (4) for critical temperature, T c ,

of thin film superconductor is given by 1 = Ut

nf kfn   n=1 k

1 2εn,k

 tanh

εn,k  2k B T c

,

(5)

where U t is the Cooper pair interaction in the thin film. n f and k f n could be obtained from Eqs. (17) and (19) of Appendix A. If we

We start our calculation for Pb thin film on Si substrate systems. The detail of calculations of Fermi wave vector k f is given in Appendix A. Fig. 1 shows calculated k f of a normal Pb thin film in terms of layers number, b, in the thin film by Eq. (19). We found that the thin film Fermi wave vector, k f , oscillates with respect to the number of atomic layers and by increasing thin film thickness it decreases and goes to bulk Fermi wave vector. By using these calculated k f superconducting critical temperature, T c , in thin film could be calculated. To see effects of surface energy on T c , we have calculated T c for two cases, without considering surface energy, E s , and with surface energy. We fixed parameters α , β , γ , A, B, C by inserting experimental data [21] in Eq. (6). We found α = 11, β = 7.53, γ = 8.24 × 10−6 , A = 14.08, B = 0.39 and C = 0.07. Fig. 2 illustrates comparison of T c for three cases, first surface energy is considered, second without considering of surface energy and finally bulk T c . Our results show surface energy play an important role in the thin film superconductor critical temperature, T c . To clarify this, we replotted T c in terms of number

R. Moradian et al. / Physics Letters A 372 (2008) 5841–5847

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Fig. 1. Shows Fermi wave vector, k f , for the Pb(111) films vs. number of the superconducting layers (ML) in the thin film.

Fig. 2. Shows comparison of calculated T c of Pb(111) thin film in terms of layers number for three cases, finite well model, surface energy plus finite well and bulk system. Squares (triangle) illustrate variation of temperature T c vs. number of the superconducting film layers (ML) correspond to the solution with (without) surface energy. The bold line shows the bulk Pb superconductor T c = 7.2 K.

of atomic layers in the thin film by consideration of surface energy. Fig. 3 shows oscillation of T c by increasing number of atomic layers, where the envelope of oscillation has sixteen layers wave length. By increasing layer thickness the amplitude oscillation is decreased. The average T c could be considered as T c eff =

−42.13 + 7.2. 3.93b + 6.37

(12)

One can find that for 5–50 atomic layers, T c values show even–odd oscillation for either even or odd thin film thicknesses within one lobe of the envelope function and phase slip of bilayer oscillation occur [i.e., the transition from odd (even) to even (odd) oscilla-

tion]. Furthermore, T c values in our Pb/Si(111) system are lower than bulk value of 7.2 K up to 50MLs. On the other hand, increasing MLs up to 500ML goes to the bulk value for T c where it is in agreement with experimental results [21]. Fig. 4, shows comparison of our calculated T c by our theoretical method with considering surface energy with the experimental data for ML = 5–15. Since the experimental data exist just for small number of layers (small thickness), so decreasing in the amplitude oscillation is not clear for ML > 6. We calculated the superconducting gap b ( T ) in terms of temperature, T , for several thickness (b = 7, 8, 9, 10, 50, bulk) and plotted that in Fig. 5. We found 8 ( T ) < 7 ( T ) < 9 ( T ) < bulk ( T ). To clarify this, we plotted the energy

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Fig. 3. Shows calculated T c in terms of layers number in the finite well plus surface energy model. Filled diamonds illustrate a variation of critical temperature T c vs. number of the superconducting layers (5–50ML). The dashed carves are envelope functions. For the adjacent half wave lengths of envelope functions the first maximums changes from odd layers numbers to even layers numbers. The envelope amplitude damped by increasing layers number. This effect arises from the quantization of electronic states −42.13 in the Pb nanostructures due to quantum confinement. Also shown a baseline, the effective T c obtained as a T c eff = 3.93b +6.37 + 7.2 function.

Fig. 4. Shows comparison of critical temperature T c vs. number of superconducting film layers (ML) in the finite well plus surface energy model and experimental data.

gap b ( T = 0) in terms of layers number, b. Fig. 6 illustrate oscillation of b ( T = 0) in terms of b. These oscillations are similar to the T c oscillations. We found oscillations amplitude damped by increasing the film thickness. But all T c oscillation values are smaller than bulk T c . This reduction can be attributed to a suppressed baseline function with an approximation −b1 dependence, as illustrated in Fig. 3 where called T c eff . In spite of constant value for b ( T = 0)/k B T c reported by several groups, we found it oscillates with the thin film thickness that similar to the T c oscillation (Fig. 7) but in small amplitude between 1.86 and 1.96. The average of b ( T = 0)/k B T c over several layers is 1.92 while for bulk superconductors it is 1.76.

4. Conclusion A Pb/Si(111) thin film superconductor have investigated to explain features of T c in terms of thin film layers. While the T c oscillation was predicted as far back as 1963 [15] where it is due to oscillation of Fermi wave vector in the thickness dimension of layer and experimental observations have also been reported [21], our work is the first gain formula for efficient attractive interaction for 2D and 3D superconductor. We considered the surface energy and assumed the attractive interaction have two contributions, bulk and surface. The bulk interaction is assumed to be constant, while the interaction at the thin film surface is proportional to the in-

R. Moradian et al. / Physics Letters A 372 (2008) 5841–5847

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Fig. 5. Shows calculated superconducting energy gap b ( T ) vs. temperature for several number of superconducting film layers in the finite well plus surface energy model and the bulk superconductors. All energy gaps of films are smaller than bulk energy gap. Our results show that 8 ( T ) < 7 ( T ) < 9 ( T ) < bulk ( T ).

Fig. 6. Shows superconducting energy gap b ( T = 0) vs. layers number (5–23ML). The dashed carves are envelope functions. The energy gap has oscillation feature. The straight bold line is the bulk energy gap.

verse of thin film thickness. We described T c oscillation feature in ultra thin film and constant features in the bulk superconductors. These oscillations could be due to oscillation of Fermi wave vector in the thickness direction of layers and the surface cooper interaction. We found in Pb/Si(111) thin film superconductor system the existence of a 2D to 3D transition thickness somewhere between 50 and 500MLs. with this theory and that is in agreement with experimental [21]. Acknowledgements This work is supported by the Nano-Committee of Ministry of Science Research and Technology of Iran.

Appendix A. Calculation of thin film Fermi momentum, k f We consider the free electron approximation for the thin film to calculate the Fermi wave vector k f in the normal thin film systems which will be used in the calculation of superconducting thin film properties. The thin film consists of b mono atomic planes, perpendicular to the z-axis where surface of each plane is S in the x– y plane. The distance between adjacent√planes is a z . Thus, the total thickness of the film is L z = ba z  S, where b is the number of atomic planes (b = 1, 2, . . .). In the previous calculations [12] the boundary was treated by constructing an infinite well at the boundaries (z = 0, L z ) and using sine function to make the wave function zero at the boundaries. Yu et al. proposed a simple model for more realistic boundary conditions [23]. They considered

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R. Moradian et al. / Physics Letters A 372 (2008) 5841–5847

Fig. 7. Shows variation of b ( T = 0)/k B T c vs. layers number (5–23ML). Although it is claimed that b ( T = 0)/k B T c is a universal constant, but we found it has oscillations (between 1.86 to 1.96) similar to b ( T = 0) and T c vs. layers number.

Fig. 8. (a) Shows infinite well are located at z = −ε and L z + ε instead of at geometric boundaries z = 0, L z of film to allow for the leakage of electron wave functions out from the metal surface. (b) Construction of the Fermi surface in the thin film within the free-electron model. Because of confinement of the electrons to a quantum well, the Fermi sphere of allowed states is reduced to a set of subbands along the direction of confinement. The patterned area represents the occupied electronic states.

infinite well at (z = −ε , L z + ε ) instead of semi-infinite metallic conductor at (z = 0, L z ) (Fig. 8(a)) and took into account the leakage of electronic wave function across the slab geometric surface by placing the infinite wells a distance ε outside the slab, so that the film has a “physical thickness” (D z = L z + 2ε ). The wave function is assumed to be vanished at the “physical” boundary. They showed that consistent solution could be obtained if ε = 3π /8k f 0 , where k f 0 is the bulk Fermi wave vector. The electron states are described by the normalized wave function:

Φn,k (z, r ) =





2/ D z Sin

π nz Dz



1



√ e ik.r , S

(13)

where it is a plane wave in the x– y plane with 2-dimensional wave vector k = kx ˆi + k y ˆj, and a standing wave in the z-direction (0 < z < D z ), where n is a integer number n = 1, 2, . . . , where it indicates quantum states in 1-dimensional (1D) infinite quantum well in the z-direction. On the other hand, in the x– y plane the periodic boundary conditions are assumed. The solution of the

Schrödinger equation for free electron system leads to the energy eigenvalues

εk v =

h¯ 2 k2v h¯ 2  2 kx + k2y + k2z = , 2m 2m

(14)

where for thin films it is

εn,k =

h¯ 2



2m

where k =

k2x + k2y +



π 2 n2 D 2z

k2x + k2y , k v =

 =

h¯ 2 2m

 k2 +

π 2 n2 D 2z

 ,

(15)



k2x + k2y + k2z , h¯ is Planck’s constant

divided by 2π and m is the mass of the electron. In order to construct the Fermi surface (Fig. 8(b)), we will consider the electronic  which are restricted to the circles with the radii k f n states (n, k) (n = 1, 2, . . .). This circles are the cross-sections of the sphere with the radius thin film Fermi wave vector, k f , and the planes perpendicular to the z-axis, where separated by the distance π / D z . Let N e be the total numbers of the free electrons in the thin film.

R. Moradian et al. / Physics Letters A 372 (2008) 5841–5847

The occupation of the electronic states at zero temperature is performed according to the formula [24] Ne = 2

nf kfn  

1,

(16)

n=1 k

where n f is the highest occupied quantum number, for which the intersection of the plane k z = π n/ D z by the sphere with radius k f is possible k2f n = k2f −

π 2 n2 D 2z

(17)

.

For large surfaces S it is possible to replace 2-dimensional sum mation over k by the integration ( k → 4πS 2 dk), hence from Eq. (16) we get n

Ne =

f S 

2π

k2f n .

(18)

n=1

By inserting Eq. (17) into Eq. (18), we found Ne V

=

1 2π L z

n f k2f −

π2  6D 2z



2n2f − 3n f + 1 ,

(19)

where the number of electrons per unit volume of the bulk material is defined by ρ , and n f is the integer part of k f ba z /π , k ba z

n f = int( fπ ), b is the number of atomic planes. From Eq. (19) k f could be obtained if ρ and b are given. To find k f and n f −3

in the Pb/Si(111) system, we used ρPb = 0.132 Å , a z = 2.86 Å, ε = 0.7 Å and for simplicity, we take along spillage distance as the same at two interfaces of Si substrate and vacuum, in Eq. (19). The provided k f from this equation should be used in the investigation of superconducting thin film system.

5847

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