Gross R., Sidorenko A., Tagirov L. Nanoscale Devices

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Andreev Reflection in Ballistic Superconductor-Ferromagnet Contacts 273

(3)

2222

(2 ) (2 )

6(1)(1)

FFFF

Ae p

GGGG

SGGG

§·

¨¸

©¹

ppnn

¨¸



¨¸

©¹







(28)

where

,

 and 1

d. The parameter

 is the measure of the ferromagnet conduction band exchange

splitting.

Discussion of Results

Let us now compare our results with the original procedure of polarization

estimation proposed in [5]. The authors argue that the normalized

conductance measured in their work depends on polarization as

2(1 )

FS n I

GG P



  (Eqs. (4)–(6) in [5]), where ()()

II II

np np

, and

nFN



 is the conductance at high voltages across the contact ( eV ' ).

In the course of discussion, the authors silently have made the following

two assumptions: Firstly, they identified the current polarization

with

the contact polarization

()()(1)(1)

FF FF

PNvNv NvNv

nn pp nn pp

 ,

(29)

where

and v

are the density of states and the Fermi velocity in the

spin subband of the ferromagnet, respectively. We think that this

identification is not quite correct, because it tacitly implies the

independence of the total current

 through the contact of the spin

polarization of the ferromagnet in the normal state. However, it is evident

from Eqs. (22), (25) and (28) that



essentially depends on

. As a

result, the reduced conductance

(0)

GV G



 is a nonlinear function of

the contact polarization

[23]. Secondly, they do not distinguish the band

parameters of the ferromagnet from those of the superconductor in contact.

That is, they do not take into account the Fermi-momenta relations of the

contacting metals, for which we have considered three distinct cases in the

previous section. We believe that our theory allows to estimate more

accurately the polarization parameter

of the ferromagnet conduction

band by applying the proper case distinguished in the previous section to

the particular couple of materials in contact.

274 L. R. Tagirov, B. P. Vodopyanov

Conclusion

In this paper we present results of our theoretical studies of the Andreev

conductance in contacts between superconductors and ferromagnets. On the

basis of a microscopic approach we derived an original expression for the

Andreev conductance. Our expression takes into account explicitly the spin

dependence of the interface transparency and the spin-dependent

conservation laws for scattering at the F/S interface. It also distinguishes

between the metals being in contact by suitable relations between the Fermi

momentum of a superconducting metal and the Fermi momenta of the spin-

subbands of the conduction band of the ferromagnetic metal.

The work was supported by the Russian ministry of education and

science, grant CKP-KSU.

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Chemistry. Kluwer Academic Publishers v 143, pp 145-167

Superconductor-Insulator Transition in a

1-Z

Te:In Solid Solution

1 1 1 2

A. F. Ioffe Physical-Technical Inst., RAS, St.Petersburg, Russia

State Polytechnical University, St. Petersburg, Russia

Abstract: We report on the influence of the composition and In content of heav-

ily doped Pb

1-z

Te:In semiconducting compounds on their super-

conducting characteristics and low temperature conductivity.

Keywords: solid solutions, superconductivity, critical parameters, dielectric states,

magnetoresistance, low temperature

Introduction

A particular problem of superconductivity (SC) in semiconductor materials

based on the A

compounds is related to the influence of doping with

group III elements such as In and Tl, which form impurity states located

within the allowed band – in the conduction band for PbTe:In and in the

valence band for SnTe:In and PbTe:Tl [1, 2]. The doping of PbTe with In

induces an effect known as long-time photoconductivity, while the In dop-

ing of SnTe results in the transition to SC at low temperatures with a

threshold type increase of the critical temperature T

with increasing the In

content above about 1 at.% [3]. According to [2, 3], SC in SnTe:In is con-

nected with the filling of the In impurity states with a peak value of the

density of states (DOS) that exceeds the valence band DOS. The hole mo-

bility is also extremely sensitive to the Fermi level position within the im-

purity band (resonant impurity states). Furthermore, the finite energy width

of the impurity band IB is determined by hybridization of band states with

the

impurity states [4]. It should be pointed out that SnTe and solid

277

R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 277–288.

D. V. Shamshur , D. V. Shakura , R. V. Parfeniev , S. A. Nemov

278 D. V. Shamshur, D. V. Shakura, R. V. Parfeniev, S. A Nemov

1-z

Te without In doping are also superconductors, however

bulk character of the SC was determined by specific heat measurements on

SnTe

1+y

, SnTe:In and PbSnTe:In [5, 6].

The present work reports the results of a systematic study of the influ-

ence of the composition and In content on the SC characteristics as well as

on the low temperature conductivity of the In doped compounds of the tin

and lead-chalcogenides.

The Samples and Experiment

Bulk polycrystalline samples of composition (Pb

1-z

)

1-x

Te with grain

sizes d ~ 100 µm were prepared by using the metallocheramic technology

with final homogenization by annealing (600°C, 200 h) [7]. The composi-

tion of the samples varies from z = 0 to 0.9 at the fixed In content: x = 0.05,

0.16 and 0.20. The electron microprobe analysis did not reveal any trace of

a secondary phase in the samples. The samples had the p-type conductivity

with a hole concentration in the range p ≅ 5×10

– 10

-3

. The hole mo-

bility Rσ = 10 – 1 cm

/Vs was determined from the Hall and resistivity

data. The low mobility values were caused by hole scattering on the In

resonant impurity states. The SC transition was determined by four probe

measurements of the electrical resistivity ρ in the temperature range be-

tween 0.4 K and 4.2 K in

He and

He setups and applied magnetic fields up

to 12 kOe. The critical parameters T

and H

(the upper critical magnetic

field) were determined from the condition ρ = 0.5 ρ

, where ρ

is the nor-

mal-state resistivity. The SC transition width determined from the interval

between the resistivity levels 0.1 ρ

÷ 0.9 ρ

was equal to 0.1 K ÷ 0.2 K for

temperature dependences

(T) and 0.5 – 1 kOe for magnetic field depend-

ences

(H) (see Fig. 1).

Snsolutions

with SC critical parameters which are lower by an order of magnitude [5]. The

Superconductor-Insulator Transition in a Pb

1-Z

Te:In Solid Solution 279

Fig. 1. The resistance of a (Pb

0.5

)

0.8

0.2

Те solid solution as a function of tem-

perature (left) and magnetic field for T < T

(right).

Experimental Results and Discussion

Before discussing our experimental results we would like to comment on

the energy position of the impurity band (IB) in the energy band structure

of the solid solutions between SnTe:In and PbTe:In (see Fig. 2). The main

extrema in the conduction band and valence band are located at the L-points

of the Brillouin zone and exhibits an inversion of the band edge positions

and L

on changing the Pb content as indicated by the zero-gap state at

x = 0.65 (T = 4.2 K [1]). The extra maximum in the valence band is situated

at the Σ-points of the Brillouin zone. It is separated from the main extre-

mum by ∆ε

LΣ

= 0.3 eV for SnTe at 4.2 K and varies slowly in the solid

solutions [1]. The In impurity states in SnTe:In are located on the

background of the allowed valence band spectrum. They lie near the edge

of the Σ-valence band with a high DOS and shift on the energy axis towards

the conduction band edge during isovalent doping with Pb. The width of the

IB decreases from

Γ ~ 200 meV in SnTe:In to

~ 1 meV in PbTe:In due

to a reduction of the interaction between the IB states and VB (L and Σ)

states and the lack of the interaction with the conduction band in PbTe [1].

The location of the Fermi level within the IB not only determines the hole

mobility but also the SC parameters of the investigated solid solutions.

280 D. V. Shamshur, D. V. Shakura, R. V. Parfeniev, S. A Nemov

Fig. 2. The energy band spectrum of (Pb

1-z

)

1-x

Те alloys.

Figures 3 and 4 show a comparison of the SC parameters in the three se-

ries of the samples with different compositions and In doping levels over

the investigated composition range (z = 0 ÷ 0.6) when the IB shifts from the

Σ-band into the L- valence band. These figures indicate an unusual behavior

of the SC parameters in materials with impurity resonance states, which in-

teract strongly with the valence band states and pin the Fermi level of holes.

and H

(0) have bell-shaped dependences on the composition and both

the value and the position of the maximum on the z scale increase with in-

creasing In content.

Fig. 3. Dependence of the critical temperature T

(left) and the temperature deriva-

tive of H

near T

(right) on the composition.

Superconductor-Insulator Transition in a Pb

1-Z

Te:In Solid Solution 281

Fig. 4. Dependence of the upper critical magnetic field H

(0) extrapolated to T = 0

(left) and the DOS at the Fermi level N(0) (right) on the composition. The DOS

was calculated using the relation N(0) = 2.84×10

|dH

/dT |

−1

valid for “dirty”

superconductors.

The highest SC parameters are obtained for the composition

(Pb

0.5

)

0.8

0.2

Te. At low In content (5 at.%) a sharp decrease of both T

and H

(0) are observed at z > 0.2. This correlates with the total DOS at the

Fermi level N(0) estimated from our experimental data and using the rela-

tion for “dirty” superconductors (see Fig. 4).

The features in the measured z-dependencies of T

and dH

/dT can be

understood if one takes into account the rapid shift of the IB with increasing

z towards the main valence band edge. As a result, on the left-hand side of

the bell shaped curve T

(z) the IB filling by holes increases with increasing

z due to holes transferred from the valence band states to the IB states

thereby resulting in a monotonic increase of the SC parameters. On the

right-hand side of the curves the rapid decrease of the SC parameters and

the destruction of SC for T > 0.4 K are caused by the decrease of the total

DOS at the Fermi level due to the IB leaving the z-valence band with high

DOS as shown in Fig. 4. On the other hand, it is possible to restore the high

SC properties of the system at fixed z by raising the In content, that is, by

passing the Fermi level deeper into the z-valence band of the compound.

In these samples there is a strong exchange of holes between the impu-

rity states and the Σ-valence band states producing an additional smearing

of the IB. Shelankov [8] had suggested a model, which relates the high

critical SC parameters in the In doped IV-VI solid solutions to the

282 D. V. Shamshur, D. V. Shakura, R. V. Parfeniev, S. A Nemov

interaction between the quasi-local impurity states and the lattice with a high di-

electric constant.

It was shown that a mixed-valence model may be used for the description

of an energy variation of the local impurity level produced by a lattice dis-

tortion. The electron-electron interaction mediated by a virtual shift of the

localized energy level is the physical reason for the

enhancement of SC.

Besides the BCS coupling constant g

, the additional term g

imp

was intro-

duced in the weak coupling approximation:

g = (1 – a

imp

+ a

imp

(1)

Here, a

imp

= N

imp

(0)/(N

(0) + N

imp

(0)) is the fractional impurity contribu-

tion to the total DOS composed of the band value N

(0) and

imp

= 4π

imp.

/(E

)

(2)

Here, C

imp

is the concentration of impurities creating the local states,

Γ is the level width, and E is the energy level position relative to E

. Equa-

tion (1) predicts a bell-shaped dependence of T

on the Fermi level position

within the IB.

The heavily doped solid solutions with z > 0.5 show a transition from the

SC state to a dielectric state at low temperatures (Fig. 5). In the temperature

regime T = 40 ÷ 100 K the resistivity versus temperature curves show an

activated behavior with an activation energy ∆ ~ 10 meV for z = 0.8. At low

temperature in samples with the high Pb content (z = 0.9) the ρ(T) depend-

ence is weaker than an exponential one because a finite conductivity

through conduction band states (z = 0.9) appears. In the sample with

z = 0.8, for which the Fermi level coincides with the energy gap, a Mott T

–

1/4

is obtained down to 0.6 K indicating the presence of variable range hop-

ping conductivity near the Fermi level via impurity states [9]. The conduc-

tivity in the overlapping energy region of the IB and valence band states

may be less than the hopping conductivity due to the resonant scattering of

band holes.

Superconductor-Insulator Transition in a Pb

1-Z

Te:In Solid Solution 283

Fig. 5. Temperature dependence of the resistivity in (Pb

1-Z

)

0.84

0.16

Te solid so-

lutions with various Pb contents (z = 0.5 – 0.9) and fixed x = 0.16.

Due to a high In concentration in the investigated solid solutions fluctua-

tions in the composition lead to the appearance of SC regions and normal

state/SC state (N/S) interfaces in a disordered superconductor - semicon-

ductor mesoscopic system. The SC insertions in the samples reveal them-

selves in the measured temperature dependences of the resistivity especially

in the x = 0.2 series for z > 0.6 (see Fig. 6). When the SC transition in the

insertions begins, the resistivity decreases at low temperature and nonlinear

current-voltage characteristics are observed (see Fig. 7). We believe that the

resistivity decrease in Fig. 7 near the SC onset is associated with the

nonlinear current dependence of the normal exponential resistivity in the

z = 0.8 sample.

The absolute value of the low current resistivity at T = 0.4 K was one

order of magnitude less than in the N-state indicating the transition into the

SC state. The current-voltage characteristics (IVC) for the samples with

lead content z = 0.9 ÷ 0.7 correspond to a transition with decreasing z from

isolated SC regions to a structure with weak links between the SC inser-

tions. Around zero bias, the IVC deviated from the ohmic behavior. The

curves show an increase of a conductance with decreasing z what is remi-

niscent a nonzero critical current at V = 0 with a value increasing from

Gross R., Sidorenko A., Tagirov L. Nanoscale Devices - Fundamentals and Applications

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