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Barrier-dependent switching effect in spin field-effect transistors
Solid State Communications 161 (2013) 46–49
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Barrier-dependent switching effect in spin field-effect transistors$ Jun Yang a,n, Kai-Ming Jiang b, Wen-Yuan Wu a, Lei Chen a a b
Institute of Sciences, PLA University of Science and Technology, Nanjing 211101, China Department of Physics, Shanghai Maritime University, Shanghai 201306, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 December 2012 Accepted 9 February 2013 by: A.H. MacDonald Available online 13 March 2013
We investigate the barrier-dependent conductance properties in spin field-effect transistors (SFET) by taking into account the presence of external magnetic field. It is shown that the conductance of the SFET has high peaks while the potential barriers strength varies. These peaks become more and more pronounced with the increasing potential barriers strength. Numerical results indicate that switching on or off can be realized in SFET by tuning the potential barriers strength. We also show that the conductance of the SFET exhibits an excellent double switching effect as the strength of the spin–orbit coupling varies. It is found that the application of external magnetic field will enhance the switching effect in SFET. The results may be of relevance to the implementation of SFET devices. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. Heterojunctions A. Magnetic films and multilayers A. Semiconductors D. Electronic transport
1. Introduction Spin-dependent tunneling magnetic junctions (MTJs) have recently attracted much interest due to their wide applications in spintronics. Potential applications, such as producing efficient photoemitters with a high degree of polarization of the electron beam (Spin Light-Emitting Diode, SLED), creating spin-based memory device and utilizing the properties of spin coherence for quantum computation and communication, are one of the leading reasons of such a technological challenge [1–4]. In recent years, spin Hall effect [5,6] and spin transport based on graphene [7] also attract enormous attention due to its unique electronic properties. Most of these spintronic devices are built on the base of MTJs. Magnetic double tunnel junctions with the more complicated structure FM/I/A/I/FM have more interesting physics, where FM is the ferromagnetic metal and I the insulating barrier. A stands for a semiconducting [8], or a normal metal [4,9], or superconducting layer [10]. Among these spintronic tunnel junctions, the spin-polarized field transistor (SFET) proposed by Datta and Das [11] is one of the most attractive spin-based devices. The idea is based on the Rashba spin–orbit interaction that causes spin to precess as it moves along its propagating direction. The SFET can be regarded as a ferromagnet/semiconductor/ferromagnet (F/S/F) double junctions with S as a quasi-one-dimensional electron gas (Q1DEG) or a two-dimensional electron gas (2DEG).
$ Project supported by the Innovation Program of Shanghai Municipal Education Commission (Grant no. 11YZ138), the Pre-Research Foundation of PLA University of Science and Technology (Grant no. 20110502). n Corresponding author. Tel.: þ86 2580831671. E-mail address: [email protected] (J. Yang).
0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.02.022
In recent years, several new types of SFET proposed by some research groups [12,13] attracted much interest. Recently, some theoretical work and experimentations focused exclusively on the influence of potential barrier strength in MTJs [4,14]. In this paper, taking into account the presence of external magnetic field, the Rashba spin–orbit coupling (SOC) and spin polarization in the ferromagnetic electrodes, we studied how the spin-dependent conductance in SFET is modulated by the potential barrier strength, Rashba SOC and external magnetic field. It is shown that the tunneling conductance in SFET exhibits excellent switching effect with the varying potential barrier strength, and the double switching effect appears by increasing the strength of the Rashba SOC. The switching effect becomes more and more pronounced while external magnetic field between the contacts and channel is applied.The results show that we can control the SFET on or off by varying the strength of the barrier or Rashba SOC.
2. Model and theory In this paper, we consider SFET as a Q1DEG spin-modulator devices. As shown in Fig. 1, we consider a Q1D Datta–Das model with interface barriers and an in-plane magnetic field in the Q1DEG channel. In this model, we assumed that the width of the transverse confining potential well for the Q1DEG is small enough so that the intersubband mixing can be neglected [11]. Taking into account that the magnetic field lies in the Q1DEG plane, orbital magnetic effects are absent in this geometry since they arise only from perpendicular fields [15]. In the one-band effective-mass approximation, the model Hamiltonians in the F regions of x o 0 and x 4 L and in the S region of 0 ox o L can be
J. Yang et al. / Solid State Communications 161 (2013) 46–49
Gate Source
ðÞ
ðÞ
ikx1 x
Drain
þ A e Z
1
There
Y
eiyðkx1 Þ
are
two
47
!
ðÞ
ðÞ
ikx2 x
þB e
eiyðkx2 Þ
! :
ð6Þ
1
dispersion relations in the Q1DEG : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðg mB BÞ2 =4g mB BaR kx sin g þ ðaR kx Þ2
Eð 7 Þ ðkx Þ ¼ _2 kx =2 mns þ dEc 7 2
B
and their corresponding spin eigenstates are ðeiyðkx Þ , 7 1Þy , where tan yðkx Þ ¼ tan g þ 2aR kx =ðg mB B cos gÞ. Each of the equations
X
ð7Þ
E 7 ðkx Þ ¼ EF has two roots kx1ð2Þ . All the reflection and transmission coefficients are determined by the boundary conditions. The matching conditions at the interfaces between the F and Q1DEG ^ C ¼ EC from x ¼ e to x ¼ þ e can be obtained by integrating H
Fig. 1. Schematic of the Q1DEG channel.
and from x ¼ Le to x ¼ L þ e in the limit of e!0 þ . According to the two spin channels, the transmission probability is given by [17]
given by [16] 2 ^ L ¼ p^ x þh0 s^ z , H F 2mnf
ð1Þ
2
2
T Pm ¼ 9C m 9 þ ðkk =km Þ9C k 9 : Similarly, we can obtain
^R H F
2 p^ ¼ x n 7h0 s^ z , 2mf
ð2Þ
and 2 ^ S ¼ p^ x aR s^ y p^ þ dEc þ 1 g m Bs^ n , H x 2 B 2mns _
ð3Þ
respectively. The magnetization in the left contact is assumed to be along the þz direction, whereas that in the right contact is taken to be along the þz direction (parallel configuration) or along the z direction (antiparallel configuration). The mns (mnf ) in the above equations is the effective mass of carriers in the Q1DEG n (F’s), spin matrix s^ s^ x cos g þ s^ y sin g and g is the angle between the Q1D channel and the magnetic field, and s^ i (i ¼ x,y,z) are the Pauli spin operators. The second term in Eq. (1) (Eq. (2)) describes the exchange interaction in the left (right) F with h0 being the exchange splitting energy. dEc in Eq. (3) is the band mismatch between the source and drain contacts of the channel. The second term stands for the Rashba SOC and the fourth stands for the Zeeman energy in the Q1DEG, where aR and g are the Rashba SOC parameter and the Lande´ g-factor of the S channel, respectively. We consider two interface barriers as dtype at x ¼0 and x ¼L: VðxÞ ¼ U dðxÞ þ U dðxLÞ with U the potential barrier height. In what follows we study conductance in the channel. First of all we should find out the electron eigenfunctions in the various regions. In the parallel configuration, since the spin quantization axis is along the z-axis, the spin-up and spin-down eigenstates in the two F’s have the forms ð1,0Þy and ð0,1Þy , respectively. Taking into account the spin components, a spin-up electron incident from the left F on the d-type barrier at x ¼0, the eigenfunction of ^ can be described as H 0 1 CL ðxÞ ¼ ðeikm x þ Rm eikm x Þ , ð4Þ þ Rk eikk x 1 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kmðkÞ ¼ ð2mnf =_2 ÞðEF 8h0 Þ is the Fermi wavevector of the spin-up (spin-down) electron for x o 0, and Rm (Rk ) is the corresponding reflection coefficient. In the right side of channel, the transmitted wave is given by 1 0 CR ðxÞ ¼ C m eikm x þ C k eikk x , ð5Þ 0 1 with C m and C k the transmission coefficients for x 4 L. In the Q1DEG region, the wave function (0 o x oL) can be written as ! ! ðþÞ ðþÞ ðþÞ ðþÞ eiyðkx1 Þ eiyðkx2 Þ CS ðxÞ ¼ A þ eikx1 x þ B þ eikx2 x 1 1
T Pk
ð7Þ for an incident spin-down electron in
AP the parallel configuration, T AP m and T k in the antiparallel config¨ uration. Using the Laudauer–Buttiker formula [13,17], the total conductance of the F/Q1DEG/F structure in the parallel (antiparallel) configuration is given by
GPðAPÞ ¼ GPðAPÞ þ GPðAPÞ ¼ m k
e2 PðAPÞ ðT þ T PðAPÞ Þ: k h m
ð8Þ
3. Results and discussions In this paper, we consider parallel configuration in SFET. From Eq. (8) together with Eqs. (4)–(7), we have evaluated numerically the conductance properties where Z 2mnf U=_2 kF , kR mns aR =_2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and kF 2mnf EF =_2 . The parameters used in the present calculations are as follows: mns ¼ 0:036me , mnf ¼ me with me the free electron mass, EF ¼ 2:47 eV, g ¼14.7 [18], g ¼ p=2 and dEc ¼2:40 eV. The exchange splitting energy h0 in the FMs is equal to 2PEF =ðP 2 þ1Þ with P the spin polarization. Fig. 2 displays the total tunneling conductance G versus Z for P¼0, 0.6, and 0.9. In this calculation, the Rashba SOC strength kR ¼ 1:0 107 , L ¼ 0:3 mm and external magnetic field strength B¼ 0. Fig. 2 shows a rapid monotonous drop of the conductance G with Z increasing for P¼ 0, and it can be understood by the
Fig. 2. (Color online) Total conductance G (parallel configuration conductivity) against Z for P¼ 0, 0.6 and 0.9. The Rashba spin–orbit coupling strength kR ¼ 1:0 107 and L ¼ 0:3 mm. Other parameters are described in the text.
48
J. Yang et al. / Solid State Communications 161 (2013) 46–49
Fig. 3. (Color online) The total conductance G against Z by taking into account the presence of an external in-plane magnetic field for B ¼ 0:1 T. The Rashba spin– orbit coupling strength kR ¼ 1:0 107 and L ¼ 0:5 mm. Other parameters are described in the text.
standard quantum tunneling theory. But the value of G appears with oscillations when the P is not equal to zero (P ¼0.6, P¼0.9), and this can be understood from the two spin-channel current models: taking into account the spin–orbit coupling effect, the spin up and spin down electrons have different ballistic tunneling behaviors and quantum interferences in the Q1EDG, which lead to the total conductance oscillations. As shown in Fig. 2, the peaks of G become more and more pronounced with the value of P increasing. Recently spintronic devices with applied external magnetic are attracting more and more interest [19]. In what follows we consider the total tunneling conductance G against the Z by taking into account the presence of an external in-plane magnetic field (B ¼ 0:1 T and L ¼ 0:5 mm). As shown in Fig. 3, the magnitude of conductance appears as sharp peaks with the strength of Z varying while the external magnetic field is applied, and the conductance varying from 0 to 1 while the value of P is large enough (for example, P ¼0.9), and this corresponds to a barrierdependent switching effect. According to this effect, the conductance of SFET can be realized on or off by tuning the strength of the barrier potential. Fig. 3 also shows that the barrier-dependent switching effect appears more and more pronounced with the increasing P. We can now turn our attention to how the conductance of SFET is modulated by the strength of the applied external magnetic. As shown in Fig. 4, the position of conductance peak moves from right to left with the magnitude of external magnetic field increasing and the conductance switching effect decay as the magnitude of magnetic field increases (B ¼ 0:2 T,0:5 T). That is to say, the conductance switching effect becomes pronounced only with certain strength of the B. For example, as shown in Fig. 4, an excellent switching effect appears for B ¼ 0:1 T. We further investigate the influence of Rashba SOC on the conductance of SFET. Fig. 5 shows the total conductance G versus Z for the kR ¼ 1, 3, and 4. In this calculation, we set P¼0.6 and other parameters are the same as the parameters in Fig. 2. It is found that the position of conductance peaks is also modulated by the spin–orbit coupling strength. Form Fig. 5, we can see that the larger strength of SOC is enhanced for the oscillation of conductance in SFET. In Fig. 6, we plot the total conductance G against B for Z ¼0.1, 0.5, and 1.0, where kR ¼ 1:0 107 , P ¼ 0:9 and L ¼ 0:5 mm. It is found that when the strength of the barrier is appropriate, there is also conductance switching effect in SFET and it is called
Fig. 4. (Color online) The total conductance G against Z for B ¼ 0:1 T, 0:2 T and 0:5 T. The Rashba spin–orbit coupling strength kR ¼ 1:0 107 , P¼ 0.9 and L ¼ 0:5 mm. Other parameters are described in the text.
Fig. 5. (Color online) The total conductance G against Z for kR ¼ 1, 3, and 4. Other parameters are described in the text.
Fig. 6. (Color online) The total conductance G against B for Z¼0.1, 0.5, and 1.0. kR ¼ 1:0 107 , P¼ 0.9 and L ¼ 0:5 mm. Other parameters are described in the text.
J. Yang et al. / Solid State Communications 161 (2013) 46–49
49
and the channel. It is shown that the conductance of the SFET has high peaks while the strength of the potential barriers varies. These peaks become more and more pronounced with external magnetic field being applied. It is also found that the conductance shows quantum oscillating behavior by varying the spin–orbit coupling strength and double switching effect exhibits the increasing potential barriers strength. According to the theoretical analysis and the numerical calculation, switching on or off (switching effect) can be realized in SFET by varying the potential barriers or Rashba SOC strength. The results may be of relevance to the implementation of spintronic devices.
Acknowledgments
Fig. 7. (Color online) The total conductance G against kR for Z ¼0.1, 0.2, and 1.0. P ¼0.9, B ¼ 0:1 T and L ¼ 0:5 mm. Other parameters are described in the text.
magnetic switching effect. Fig. 6 also shows that the lower value of the Z is applied, the lower magnetic field strength is required to realize the same switching effect, and the low magnetic field is easier to realize in the real devices. Finally, we study the properties of Rashba SOC strength versus conductance. In this calculation, setting P¼0.9, B ¼ 0:1 T and L ¼ 0:5 mm. As shown in Fig. 7, the magnitude of G shows a decay oscillations with increasing kR while Z¼ 0.1, 0.2, and the magnitude of G decreases with increasing Z. It is easily understood that the G oscillations arise from the ballistic transport of electrons and quantum interference in the Q1DEG, while the decay comes from the diffusive transport in the real devices [4]. Fig. 7 also shows that the two sharp peaks appear while Z ¼1 corresponds to double switching effect. The sharp peaks can be understood as follows: the increasing potential barriers strength enhances the quantum interference in the Q1DEG that leads to resonance transmission in SFET.
This work is supported by the Innovation Program of Shanghai Municipal Education Commission under Contract no. 11YZ138, the Pre-Research Foundation of PLA University of Science and Technology under Contract no. 20110502, and the Special Fund for Academic Development of Young Teachers in Shanghai Colleges and Universities under Contract no. AAYQ0920.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
4. Summary In conclusion, we have studied the barrier-dependent conductance properties in SFET by taking into account the presence of external magnetic field, the Rashba spin–orbit coupling and the potential barriers strength between the ferromagnetic contacts
[15] [16] [17] [18] [19]
Igor Zuti, S. Jaroslav Fabian, Das Sarma, Rev. Mod. Phys. 76 (2004) 323. H. Akinaga, H. Ohno, IEEE Trans. Nanotechnol. 1 (2002) 19. M. Cahay, S. Bandyopadhyay, Phys. Rev. B 69 (2004) 045303. J. Yang, J. Wang, Z.M. Zheng, D.Y. Xing, C.R. Chang, Phys. Rev. B 71 (2005) 214434. Onur Hosten, Paul Kwiat, Science 319 (2008) 787. Tomas Jungwirth, Jorg Wunderlich, Kamil Olejnk, Nat. Mater. 11 (2012) 382. C. Jzsa1, M. Popinciuc, N. Tombros, H.T. Jonkman, B.J. van Wees, Phys. Rev. Lett. 100 (2008) 236603. R. Mattana, J.-M. George, H. Jaffres, F.N. van Dau, A. Fert, B. Lepine, A. Guivarch, G. Jezequel, Phys. Rev. Lett. 90 (2003) 166601. J. Mathon, A. Umerski, Phys. Rev. B 60 (1999) 1117. Z.M. Zheng, D.Y. Xing, G.Y. Sun, J.M. Dong, Phys. Rev. B 62 (2000) 14326. S. Datta, B. Das, Appl. Phys. Lett. 56 (1990) 665. Luca Chirolli, Davide Venturelli, Fabio Taddei, Rosario Fazio, Vittorio Giovannetti, Phys. Rev. B 85 (2012) 155317. M. Buttiker, J. Phys.: Condens. Matter 5 (1993) 9361. Liu Ruisheng, Yang See-Hun, Jiang Xin, Topuria Teya, Rice Philip M, Rettner Charles, Parkin Stuart, Appl. Phys. Lett. 100 (2012) 012401; Eunsongyi Lee, Minji Gwon, Dong-Wook Kim, Hogyoung Kim, Appl. Phys. Lett. 98 (2011) 132905. S. Debald, B. Kramer, Phys. Rev. B 71 (2005) 115322. Kai-Ming Jiang, Rong Zhang, Jun Yang, Chun-Xiao Yue, Zu-Yao Sun, IEEE Trans. Electron. Devices 57 (2010) 2005. J. Yang, C. Lei, C. Rong, K.M. Jiang, Eur. Phys. J. B 62 (2008) 263. S. Bandyopadhyay, M. Cahay, Physica E (Amsterdam) 25 (2005) 399. Junjun Wan, M. Cahay, S. Bandyopadhyay, IEEE Trans. Nanotechnol. 7 (2008) 34.
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Barrier-dependent switching effect in spin field-effect transistors$ Jun Yang a,n, Kai-Ming Jiang b, Wen-Yuan Wu a, Lei Chen a a b
Institute of Sciences, PLA University of Science and Technology, Nanjing 211101, China Department of Physics, Shanghai Maritime University, Shanghai 201306, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 December 2012 Accepted 9 February 2013 by: A.H. MacDonald Available online 13 March 2013
We investigate the barrier-dependent conductance properties in spin field-effect transistors (SFET) by taking into account the presence of external magnetic field. It is shown that the conductance of the SFET has high peaks while the potential barriers strength varies. These peaks become more and more pronounced with the increasing potential barriers strength. Numerical results indicate that switching on or off can be realized in SFET by tuning the potential barriers strength. We also show that the conductance of the SFET exhibits an excellent double switching effect as the strength of the spin–orbit coupling varies. It is found that the application of external magnetic field will enhance the switching effect in SFET. The results may be of relevance to the implementation of SFET devices. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. Heterojunctions A. Magnetic films and multilayers A. Semiconductors D. Electronic transport
1. Introduction Spin-dependent tunneling magnetic junctions (MTJs) have recently attracted much interest due to their wide applications in spintronics. Potential applications, such as producing efficient photoemitters with a high degree of polarization of the electron beam (Spin Light-Emitting Diode, SLED), creating spin-based memory device and utilizing the properties of spin coherence for quantum computation and communication, are one of the leading reasons of such a technological challenge [1–4]. In recent years, spin Hall effect [5,6] and spin transport based on graphene [7] also attract enormous attention due to its unique electronic properties. Most of these spintronic devices are built on the base of MTJs. Magnetic double tunnel junctions with the more complicated structure FM/I/A/I/FM have more interesting physics, where FM is the ferromagnetic metal and I the insulating barrier. A stands for a semiconducting [8], or a normal metal [4,9], or superconducting layer [10]. Among these spintronic tunnel junctions, the spin-polarized field transistor (SFET) proposed by Datta and Das [11] is one of the most attractive spin-based devices. The idea is based on the Rashba spin–orbit interaction that causes spin to precess as it moves along its propagating direction. The SFET can be regarded as a ferromagnet/semiconductor/ferromagnet (F/S/F) double junctions with S as a quasi-one-dimensional electron gas (Q1DEG) or a two-dimensional electron gas (2DEG).
$ Project supported by the Innovation Program of Shanghai Municipal Education Commission (Grant no. 11YZ138), the Pre-Research Foundation of PLA University of Science and Technology (Grant no. 20110502). n Corresponding author. Tel.: þ86 2580831671. E-mail address: [email protected] (J. Yang).
0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.02.022
In recent years, several new types of SFET proposed by some research groups [12,13] attracted much interest. Recently, some theoretical work and experimentations focused exclusively on the influence of potential barrier strength in MTJs [4,14]. In this paper, taking into account the presence of external magnetic field, the Rashba spin–orbit coupling (SOC) and spin polarization in the ferromagnetic electrodes, we studied how the spin-dependent conductance in SFET is modulated by the potential barrier strength, Rashba SOC and external magnetic field. It is shown that the tunneling conductance in SFET exhibits excellent switching effect with the varying potential barrier strength, and the double switching effect appears by increasing the strength of the Rashba SOC. The switching effect becomes more and more pronounced while external magnetic field between the contacts and channel is applied.The results show that we can control the SFET on or off by varying the strength of the barrier or Rashba SOC.
2. Model and theory In this paper, we consider SFET as a Q1DEG spin-modulator devices. As shown in Fig. 1, we consider a Q1D Datta–Das model with interface barriers and an in-plane magnetic field in the Q1DEG channel. In this model, we assumed that the width of the transverse confining potential well for the Q1DEG is small enough so that the intersubband mixing can be neglected [11]. Taking into account that the magnetic field lies in the Q1DEG plane, orbital magnetic effects are absent in this geometry since they arise only from perpendicular fields [15]. In the one-band effective-mass approximation, the model Hamiltonians in the F regions of x o 0 and x 4 L and in the S region of 0 ox o L can be
J. Yang et al. / Solid State Communications 161 (2013) 46–49
Gate Source
ðÞ
ðÞ
ikx1 x
Drain
þ A e Z
1
There
Y
eiyðkx1 Þ
are
two
47
!
ðÞ
ðÞ
ikx2 x
þB e
eiyðkx2 Þ
! :
ð6Þ
1
dispersion relations in the Q1DEG : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðg mB BÞ2 =4g mB BaR kx sin g þ ðaR kx Þ2
Eð 7 Þ ðkx Þ ¼ _2 kx =2 mns þ dEc 7 2
B
and their corresponding spin eigenstates are ðeiyðkx Þ , 7 1Þy , where tan yðkx Þ ¼ tan g þ 2aR kx =ðg mB B cos gÞ. Each of the equations
X
ð7Þ
E 7 ðkx Þ ¼ EF has two roots kx1ð2Þ . All the reflection and transmission coefficients are determined by the boundary conditions. The matching conditions at the interfaces between the F and Q1DEG ^ C ¼ EC from x ¼ e to x ¼ þ e can be obtained by integrating H
Fig. 1. Schematic of the Q1DEG channel.
and from x ¼ Le to x ¼ L þ e in the limit of e!0 þ . According to the two spin channels, the transmission probability is given by [17]
given by [16] 2 ^ L ¼ p^ x þh0 s^ z , H F 2mnf
ð1Þ
2
2
T Pm ¼ 9C m 9 þ ðkk =km Þ9C k 9 : Similarly, we can obtain
^R H F
2 p^ ¼ x n 7h0 s^ z , 2mf
ð2Þ
and 2 ^ S ¼ p^ x aR s^ y p^ þ dEc þ 1 g m Bs^ n , H x 2 B 2mns _
ð3Þ
respectively. The magnetization in the left contact is assumed to be along the þz direction, whereas that in the right contact is taken to be along the þz direction (parallel configuration) or along the z direction (antiparallel configuration). The mns (mnf ) in the above equations is the effective mass of carriers in the Q1DEG n (F’s), spin matrix s^ s^ x cos g þ s^ y sin g and g is the angle between the Q1D channel and the magnetic field, and s^ i (i ¼ x,y,z) are the Pauli spin operators. The second term in Eq. (1) (Eq. (2)) describes the exchange interaction in the left (right) F with h0 being the exchange splitting energy. dEc in Eq. (3) is the band mismatch between the source and drain contacts of the channel. The second term stands for the Rashba SOC and the fourth stands for the Zeeman energy in the Q1DEG, where aR and g are the Rashba SOC parameter and the Lande´ g-factor of the S channel, respectively. We consider two interface barriers as dtype at x ¼0 and x ¼L: VðxÞ ¼ U dðxÞ þ U dðxLÞ with U the potential barrier height. In what follows we study conductance in the channel. First of all we should find out the electron eigenfunctions in the various regions. In the parallel configuration, since the spin quantization axis is along the z-axis, the spin-up and spin-down eigenstates in the two F’s have the forms ð1,0Þy and ð0,1Þy , respectively. Taking into account the spin components, a spin-up electron incident from the left F on the d-type barrier at x ¼0, the eigenfunction of ^ can be described as H 0 1 CL ðxÞ ¼ ðeikm x þ Rm eikm x Þ , ð4Þ þ Rk eikk x 1 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kmðkÞ ¼ ð2mnf =_2 ÞðEF 8h0 Þ is the Fermi wavevector of the spin-up (spin-down) electron for x o 0, and Rm (Rk ) is the corresponding reflection coefficient. In the right side of channel, the transmitted wave is given by 1 0 CR ðxÞ ¼ C m eikm x þ C k eikk x , ð5Þ 0 1 with C m and C k the transmission coefficients for x 4 L. In the Q1DEG region, the wave function (0 o x oL) can be written as ! ! ðþÞ ðþÞ ðþÞ ðþÞ eiyðkx1 Þ eiyðkx2 Þ CS ðxÞ ¼ A þ eikx1 x þ B þ eikx2 x 1 1
T Pk
ð7Þ for an incident spin-down electron in
AP the parallel configuration, T AP m and T k in the antiparallel config¨ uration. Using the Laudauer–Buttiker formula [13,17], the total conductance of the F/Q1DEG/F structure in the parallel (antiparallel) configuration is given by
GPðAPÞ ¼ GPðAPÞ þ GPðAPÞ ¼ m k
e2 PðAPÞ ðT þ T PðAPÞ Þ: k h m
ð8Þ
3. Results and discussions In this paper, we consider parallel configuration in SFET. From Eq. (8) together with Eqs. (4)–(7), we have evaluated numerically the conductance properties where Z 2mnf U=_2 kF , kR mns aR =_2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and kF 2mnf EF =_2 . The parameters used in the present calculations are as follows: mns ¼ 0:036me , mnf ¼ me with me the free electron mass, EF ¼ 2:47 eV, g ¼14.7 [18], g ¼ p=2 and dEc ¼2:40 eV. The exchange splitting energy h0 in the FMs is equal to 2PEF =ðP 2 þ1Þ with P the spin polarization. Fig. 2 displays the total tunneling conductance G versus Z for P¼0, 0.6, and 0.9. In this calculation, the Rashba SOC strength kR ¼ 1:0 107 , L ¼ 0:3 mm and external magnetic field strength B¼ 0. Fig. 2 shows a rapid monotonous drop of the conductance G with Z increasing for P¼ 0, and it can be understood by the
Fig. 2. (Color online) Total conductance G (parallel configuration conductivity) against Z for P¼ 0, 0.6 and 0.9. The Rashba spin–orbit coupling strength kR ¼ 1:0 107 and L ¼ 0:3 mm. Other parameters are described in the text.
48
J. Yang et al. / Solid State Communications 161 (2013) 46–49
Fig. 3. (Color online) The total conductance G against Z by taking into account the presence of an external in-plane magnetic field for B ¼ 0:1 T. The Rashba spin– orbit coupling strength kR ¼ 1:0 107 and L ¼ 0:5 mm. Other parameters are described in the text.
standard quantum tunneling theory. But the value of G appears with oscillations when the P is not equal to zero (P ¼0.6, P¼0.9), and this can be understood from the two spin-channel current models: taking into account the spin–orbit coupling effect, the spin up and spin down electrons have different ballistic tunneling behaviors and quantum interferences in the Q1EDG, which lead to the total conductance oscillations. As shown in Fig. 2, the peaks of G become more and more pronounced with the value of P increasing. Recently spintronic devices with applied external magnetic are attracting more and more interest [19]. In what follows we consider the total tunneling conductance G against the Z by taking into account the presence of an external in-plane magnetic field (B ¼ 0:1 T and L ¼ 0:5 mm). As shown in Fig. 3, the magnitude of conductance appears as sharp peaks with the strength of Z varying while the external magnetic field is applied, and the conductance varying from 0 to 1 while the value of P is large enough (for example, P ¼0.9), and this corresponds to a barrierdependent switching effect. According to this effect, the conductance of SFET can be realized on or off by tuning the strength of the barrier potential. Fig. 3 also shows that the barrier-dependent switching effect appears more and more pronounced with the increasing P. We can now turn our attention to how the conductance of SFET is modulated by the strength of the applied external magnetic. As shown in Fig. 4, the position of conductance peak moves from right to left with the magnitude of external magnetic field increasing and the conductance switching effect decay as the magnitude of magnetic field increases (B ¼ 0:2 T,0:5 T). That is to say, the conductance switching effect becomes pronounced only with certain strength of the B. For example, as shown in Fig. 4, an excellent switching effect appears for B ¼ 0:1 T. We further investigate the influence of Rashba SOC on the conductance of SFET. Fig. 5 shows the total conductance G versus Z for the kR ¼ 1, 3, and 4. In this calculation, we set P¼0.6 and other parameters are the same as the parameters in Fig. 2. It is found that the position of conductance peaks is also modulated by the spin–orbit coupling strength. Form Fig. 5, we can see that the larger strength of SOC is enhanced for the oscillation of conductance in SFET. In Fig. 6, we plot the total conductance G against B for Z ¼0.1, 0.5, and 1.0, where kR ¼ 1:0 107 , P ¼ 0:9 and L ¼ 0:5 mm. It is found that when the strength of the barrier is appropriate, there is also conductance switching effect in SFET and it is called
Fig. 4. (Color online) The total conductance G against Z for B ¼ 0:1 T, 0:2 T and 0:5 T. The Rashba spin–orbit coupling strength kR ¼ 1:0 107 , P¼ 0.9 and L ¼ 0:5 mm. Other parameters are described in the text.
Fig. 5. (Color online) The total conductance G against Z for kR ¼ 1, 3, and 4. Other parameters are described in the text.
Fig. 6. (Color online) The total conductance G against B for Z¼0.1, 0.5, and 1.0. kR ¼ 1:0 107 , P¼ 0.9 and L ¼ 0:5 mm. Other parameters are described in the text.
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and the channel. It is shown that the conductance of the SFET has high peaks while the strength of the potential barriers varies. These peaks become more and more pronounced with external magnetic field being applied. It is also found that the conductance shows quantum oscillating behavior by varying the spin–orbit coupling strength and double switching effect exhibits the increasing potential barriers strength. According to the theoretical analysis and the numerical calculation, switching on or off (switching effect) can be realized in SFET by varying the potential barriers or Rashba SOC strength. The results may be of relevance to the implementation of spintronic devices.
Acknowledgments
Fig. 7. (Color online) The total conductance G against kR for Z ¼0.1, 0.2, and 1.0. P ¼0.9, B ¼ 0:1 T and L ¼ 0:5 mm. Other parameters are described in the text.
magnetic switching effect. Fig. 6 also shows that the lower value of the Z is applied, the lower magnetic field strength is required to realize the same switching effect, and the low magnetic field is easier to realize in the real devices. Finally, we study the properties of Rashba SOC strength versus conductance. In this calculation, setting P¼0.9, B ¼ 0:1 T and L ¼ 0:5 mm. As shown in Fig. 7, the magnitude of G shows a decay oscillations with increasing kR while Z¼ 0.1, 0.2, and the magnitude of G decreases with increasing Z. It is easily understood that the G oscillations arise from the ballistic transport of electrons and quantum interference in the Q1DEG, while the decay comes from the diffusive transport in the real devices [4]. Fig. 7 also shows that the two sharp peaks appear while Z ¼1 corresponds to double switching effect. The sharp peaks can be understood as follows: the increasing potential barriers strength enhances the quantum interference in the Q1DEG that leads to resonance transmission in SFET.
This work is supported by the Innovation Program of Shanghai Municipal Education Commission under Contract no. 11YZ138, the Pre-Research Foundation of PLA University of Science and Technology under Contract no. 20110502, and the Special Fund for Academic Development of Young Teachers in Shanghai Colleges and Universities under Contract no. AAYQ0920.
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4. Summary In conclusion, we have studied the barrier-dependent conductance properties in SFET by taking into account the presence of external magnetic field, the Rashba spin–orbit coupling and the potential barriers strength between the ferromagnetic contacts
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