Journal of Magnetism and Magnetic Materials 323 (2011) 2899–2902

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Journal of Magnetism and Magnetic Materials

0304-88

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jmmm

Quantum point contact conductance in ferromagneticsemiconductor/superconductor junctions

Hong Li n, Xinjian Yang

College of Science, China University of Petroleum, Dongying 257061, China

a r t i c l e i n f o

Article history:

Received 24 December 2010

Received in revised form

17 June 2011Available online 25 June 2011

Keywords:

Ferromagnetic semiconductor

Quantum point contact

53/$ - see front matter & 2011 Elsevier B.V. A

016/j.jmmm.2011.06.047

esponding author. Tel.: þ86 0546 8393771.

ail address: lihong_302@eyou.com (H. Li).

a b s t r a c t

An extended Blonder–Tinkham–Klapwijk approach is applied to study how the tunneling conductance

in ferromagnetic semiconductor/s-wave superconductor (FS/SC) junction, where the FS region is a

quantum wire, is manipulated by the mismatches of the effective mass between the FS and SC, spin

polarization in the FS, as well as the strength of potential scattering at the interface. It is demonstrated

that in the single-mode case they have different influences on the tunneling spectra.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

The advance of semiconductor spintronics has received a long-standing interest in understanding the coupling of charge and spinin semiconductors [1]. Ferromagnetic semiconductors (FS) [2,3] areof central importance to semiconductor spintronics because theyhave a conductivity compatible with that of conventional semi-conductors and the potential for a high intrinsic spin polarization,thus, providing promising conditions for efficient spin-injectioninto conventional semiconductor. (Ga,Mn)As has been successfullyincorporated into a variety of spin-injection and spin-transportdevices [4]. Authors have studied the properties of ferromagneticsemiconductor/superconductor (FS/SC) junctions [5–11] andrevealed several important features in charge transport.

On the other hand, Takagaki and Ploog [12] recently presenteda theory (TP theory) of the tunneling conductance for a quantumwire/insulator/d-wave superconductor junction in the frameworkof the Blonder–Tinkham–Klapwijk (BTK) [13] model. Since then,the TP theory has been extended to include multi-mode effects inquantum wire and p-wave junctions [14–17]. The results revealthat in the single-mode case the zero-bias conductance peak(ZBCP) appears for the p-wave, but perfectly vanishes for thed-wave because of quantum diffraction. It is naturally expectedthat a structure consisting of FS and SC will have more physicalcontents than an NM/SC structure.

In this paper, we extend the TP theory to study the tunnelingconductance for quantum point contacts in ferromagnetic semi-conductor/s-wave superconductor (FS/SC) junctions. It is demon-strated that the tunneling spectra strongly depends on the

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mismatches of the effective mass between the FS and SC, spinpolarization in the FS, as well as the strength of potentialscattering at the interface.

2. Model and formulation

The model of the FS/SC junction is shown in Fig. 1, in which theFS region is a quantum wire with width w. A very thin insulatinglayer at x¼0 is described by a d-type function V(x)¼Ud(x), whereU is dependent on the product of the barrier height and width. Werestrict our calculation to a case of single-mode occupation in theFS/SC quantum wire and neglect all the evanescent modes. Thisapproximation is justified because for powkFo2p the narrowwire has a single-mode and the energy of evanescent modes iswell above the Fermi energy [12,16].

The hole Hamiltonian in the FS is simply given by [6]

HFS ¼HFS0 �h0sþdEc ð1Þ

where HFS0 ¼�_

2r=2mnþV is the kinetic energy with mn theeffective mass of a hole plus the usual static potential, h0 is theexchange energy, s is the conventional Pauli spin operator, anddEc is the band mismatch between the FS and SC. The SC isassumed to be s-wave pairing and has the following effectivesingle-particle Hamiltonian

HSC0 ¼�_

2r2=2me�EFþV ð2Þ

where EF and me is the Fermi energy and effective mass ofthe electron, respectively. The pair potential is described byD(x)¼D0Y(x) where Y(x) is the unit step function.

We extend the Bogoliubov-de Gennes (BdG) [18] approach tostudy the transport of quasi-particles in the FS/SC structures. The

Fig. 1. The point contact FS/SC junction. The FS region is a single-mode quantum

wire with width w. Here kFw¼1.7p.

Fig. 2. Normalized conductance as a function of the bias energy for different mn/

me with z¼1.

H. Li, X. Yang / Journal of Magnetism and Magnetic Materials 323 (2011) 2899–29022900

BdG equation is given by [19–21]

HSC0 �Zsh0 DDn

�ðHSCn0 þZs

h0Þ

" #u

v

� �¼ E

u

v

� �ð3Þ

here Zs¼1 for s¼m and Zs¼�1 for s¼k.The solutions to the BdG equation in the FS and SC regions are,

respectively,

cFS ¼1

0

� �eik1sxþbs

1

0

� �e�ik1sxþas

0

1

� �eik1sx

� �f1ðyÞ ð4Þ

csc ¼

Z þ1�1

ds csuþ eifþ =2

vþ e�ifþ =2

" #eirþ xþds

v�eif�=2

u�e�if�=2

" #e�ir�x

!fsðyÞ

ð5Þ

with f1ðyÞ ¼ffiffiffiffiffiffiffiffiffiffi2=w

psin½ðp=wÞðyþðw=2ÞÞ�Yððw=2Þ�9y9Þ, fsðyÞ ¼ eisy=ffiffiffiffiffiffi

2pp

, ðu7 Þ2¼ 1�ðv7 Þ

2¼ ð1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�D2

7 =E2q

Þ=2, and eij7¼D7/9D79.

The wave numbers dependent on the wave number along the y-axis,defined by s, are given by

r7 ðsÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikðsÞ27

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2�9D92

q_2

vuutkðsÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffik2

F�s2q

k1sðsÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

FSð1þZsðsÞhÞ�p2

w2

rð6Þ

where kF ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2meEF=_

2q

, kFS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mnEF=_

2q

with h¼h0/EF.

All coefficients in Eqs. (4) and (5) can be determined by theboundary conditions at x¼0. They are cFS(0)¼cSC(0) andc0SCð0Þ�

1mc0

FSð0Þ ¼ k1zcFSð0Þ with m¼mn=me and z¼ 2meU=ð_2kF Þ.We can get

F2a�ðF1þ izþ1Þb¼ F1þ iz�1 ð7aÞ

ð1�izþF1Þa�F3b¼ F3 ð7bÞ

here

Fj ¼4m

p2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimg2

F ð1þZshÞ�1q Z gF

�gF

dq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffig2

F�q2qð1�q2Þ

2cos2 pq

2

� �fjðqÞ

q¼ws

p, gF ¼

wkF

pand j¼ 1,2,3

f1 ¼1þGþG�e�iðjþ �j�Þ

1�GþG�e�iðjþ �j�Þ, f2 ¼

2G�eij�

1�GþG�e�iðjþ �j�Þ

f3 ¼2Gþ e�ijþ

1�GþG�e�iðjþ �j�Þ, G7 ¼ v7 =u7 ð8Þ

The solutions for Eqs. (7a and 7b) are given by

a¼2F3

ð1þF1Þ2�F2F3þm2z2

ð9aÞ

b¼�F2

1þF2F3þð1�imzÞ2

ð1þF1Þ2�F2F3þm2z2

ð9bÞ

Using the BTK theory, we are able to calculate the quantumpoint contact differential conductance of the FS/SC junction attemperature T¼0 K

G¼Gs

Gn

E ¼ eV

¼ð1þF0Þ

2þ4m2z2

4F0

Xs ¼ m,k

Psð1þ9as92�9bs9

2Þ ð10Þ

where Gn is the conductance when D¼0 and a¼0, F0 is defined byEq. (8) with fj¼1 and the reflection coefficients are evaluated inE¼eV, where V is the voltage.

3. Numerical results and discussions

In the following, we take wkF/p¼1.7 [12], mn¼0.45me for the

heavy holes, and mn¼0.08me for the light holes [10,11]. The

conductance is normalized by its value at E¼2D0 beyond whichthe conductance is almost a constant.

We first study the effect of the mismatches of effective massesin the FS and SC on the quantum point contact conductance. Thecorresponding tunneling spectrum is shown in Figs. 2 and 3. It canbe seen that the ratio value m¼mn/me has a great influence on G.For the heavy holes, the normalized conductance (NC) has areduced dip structure as shown in Fig. 2a, while for the lightholes (see Fig. 2b), a V-like shape appears in the spectrum. Whenmn¼me and h0¼0, our result agrees with that in Ref. [12]. It is can

be seen that with decreased m, the conductance at the Fermi levelgradually decreases. To see more clearly these trends, the NC as afunction of m at E¼0 is plotted in Fig. 3. From it one can find thatthe value of the conductance becomes zero around m¼0.4. Thiseffect of the mismatches in the effective mass between the FSand SC is closely associated with the change of Andreev reflection(AR) [22] at the interface. Our calculation indicates that the ARprobability is greatest at m¼1 and decreases with decreased m.The AR probability would decay to zero around m¼0.38 and 0.42for spin-up and spin-down quasi-particles, respectively.

Fig. 4 shows the barrier strength effect at the FS/SC interface onthe tunneling conductance. For the heavy holes as show in Fig. 4a, atz¼0 the NC has the biggest departure from its bulk behavior.With the barrier strength increased, the NC within the energy gapgradually decreases and almost vanishes at z¼9. Fig. 4b shows the

Fig. 4. Normalized conductance as a function of the bias energy for different z

with h¼0.1.

Fig. 3. Differential conductance as a function of the ratio value m at E¼0 with

z¼1, h¼0.3.

H. Li, X. Yang / Journal of Magnetism and Magnetic Materials 323 (2011) 2899–2902 2901

NC for the light holes. A clear difference between Figs. 4a and 3b canbe seen. In this case the value of NC at zero-energy bias does notchange with z. Physically, the feature may be explained by thefollowing reason. In the FS k1s is determined by mn and h0, so withfixed h0, the mass mn for the heavy holes is bigger, which means k1sis bigger, too. Therefore the carriers can easily pass the interface andthe exchange energy is easily gained or lost by a quasi-particleAndreev reflected at the FS/SC interface, while for the light holes, theexchange energy is not easily gained or lost by a quasi-particleAndreev reflected at the FS/SC interface regardless of whether z isweak or strong.

Finally we discuss effects of the exchange energy in FS on theNC in FS/SC junctions. From Fig. 5a, one can find that for the heavyholes the amplitude of the fluctuation of the NC is greatlymodified by the magnitude of h. However, the spin polarizationeffect has little contribution to the value of the NC for the lightholes as shown in Fig. 5b. In order to check the effect more clearly,Fig. 6 shows the response of the conductance for different spins asa function of h. We can see that with increasing h, the spectra forup spins and down spins shift towards opposite directions. Itfollows that although the total NC for the light holes is insensitive

Fig. 5. Normalized conductance as a function of the bias energy for different h

with z¼1.

Fig. 6. Spin-up (a) and spin-down (b) conductance as a function of the bias energy

for different exchange energy with m¼0.08, z¼1.

H. Li, X. Yang / Journal of Magnetism and Magnetic Materials 323 (2011) 2899–29022902

to the magnitude of h, an increase of h will increase the differencein NC between spin-up and spin-down quasi-particles. Such adifference is very interesting because it is closely related toprobable coexistence of ferromagnetism and superconductivityin a small regime near the FS/SC interface. The above results alsodemonstrate that the quantum point contact conductance inFS/SC junctions strongly depends on the magnitude of mn, thestrength of potential scattering, as well as the effective exchangefield in FS.

4. Conclusions

In this paper, we study the tunneling conductance for quantumpoint contact in ferromagnetic semiconductor/ferromagnetic super-conductor junctions using the extended BTK formula. It is found that

the tunneling spectrum for heavy holes is much different from thatfor light holes. It is also demonstrated that the normalized con-ductance strongly depends on the strength of potential scatteringand the effective exchange field in the FS. An increase of theexchange energy in FS leads to an increase of the induced ferro-magnetism in SC near the interface. It is expected that the theoreticalresults obtained may be further confirmed in the future experiment.

Acknowledgments

This work is supported by the Fundamental Research Funds forthe Central Universities (09CX04068A and y0918019).

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