- Email: [email protected]
Spin transport through quantum dots
Microelectronics Journal 34 (2003) 337–339 www.elsevier.com/locate/mejo
Spin transport through quantum dots A.T. da Cunha Lima, E.V. Anda* Departamento de Fı´sica, Pontifı´cia Universidade Cato´lica do Rio de Janeiro, C.P. 38071, 22453-900 Ga´vea, Rio de Janeiro, RJ, Brazil
Abstract We investigate the spin polarized transport properties of a nanoscopic device constituted by a quantum dot connected to two leads. We show that through the manipulation of the gate potential applied to the dot it is possible to control, in a very efficient way, the intensity and polarization of the current that goes along the system. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: Quantum dot; Magnetic field; Polarization
1. Introduction We investigate the spin polarized transport properties of a nanoscopic device constituted by a quantum dot connected to two leads. The electrical current circulates with a spin polarization that is modulated via a gate potential that controls the intensity of the spin – orbit coupling (the Rashba effect) [1,2]. We study a polarized field-effect transistor [2,3] when one of its parts is constituted by a small quantum dot, which energies are controlled by another gate potential operating inside the confined region. The high confinement and correlation suffered by the charges inside the dot gives rise to novel phenomena. We show that through the manipulation of the gate potential applied to the dot it is possible to control, in a very efficient way, the intensity and polarization of the current that goes along the system. Other crucial parameters to be varied in order to understand the behavior of this system are the intensity of the external applied electric and magnetic field.
2. The model The system is represented by the impurity Anderson Hamiltonian summed to a spin –orbit interaction, which describes the Rashba effect. We write the Hamiltonian using * Corresponding author. Tel.: þ55-21-3114-1263; fax: þ 55-21-5123222. E-mail address: [email protected] (E.V. Anda).
a tight-binding representation [3] X þ X s tij cis cjs þ e i nis þ t0 ðcþ HA ¼ 0s c1s þ c0s c1s Þ þ hc s;i;j
a;i
þ ðe s0 þ Vg Þn0s þ Un0" n0# þ Hso Hso ¼ 2
X
so þ ti;j ðci;s ðisy Þss0 cj;s0 2 cþ i;s ðisx Þss0 cj;s0 þ hc
ss0 ;i;j
where the first term represents the contacts and the dot, located at site 0 characterized by a local state of energy e 0 þ Vg ; where Vg represents a variable gate potential and U is the local electronic correlation. The dot is connected to the contacts through the non-diagonal matrix element t0 : The external electric field responsible for the current and the magnetic field is simulated adopting a site and spin dependent diagonal matrix element e si : The second term is a tight-binding representation of the spin – orbit interaction taking place along the system, which permits to change the spin polarization of the current modulating the position so dependent matrix element ti;j by the application of an external potential. The sx and sy are the x and y components of the Pauli matrices. To obtain the current of this out-ofequilibrium system, we use the Keldysh formalism [4]. The problem is solved using the Green function formalism considering the localized electronic correlation within the context of the Hubbard I approximation [5]. This approach permits to obtain a Dyson equation for the Green functions given by s s00
Gij 0
0026-2692/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0026-2692(03)00021-1
¼ gsij 0 dij ds0 s00 þ
X s;l;m
ss0
s0 s gsil 0 Tlm Gmj 0
338
A.T. da Cunha Lima, E.V. Anda / Microelectronics Journal 34 (2003) 337–339
where 0
Ti;jss ¼ tij dss0 þ tijso dss0 þ t0 ðdi1 þ di1 Þdss0 gs00 ¼
1 2 ns ns þ v 2 e s0 2 Vg v 2 e s0 2 Vg 2 U
and the current per spin s is given by ðe fr s0 s0 s0 s0 Is ¼ ðGþ2 ðwÞ 2 Gþ2 ðwÞÞdw i;iþ1 iþ1;i e fl
þ2s s0
whereGi;iþ1 0 0 ðwÞ corresponds to the-out-of equilibrium Keldysh Green function that gives us the probability of an electron of energy w to hop from site i to i þ 1 when the system is connected to two reservoirs of Fermi energies e fl and e fr ; such that the external applied potential is obtained from DV ¼ e fr 2 e fl : The injection of a polarized current into the system is simulated assuming that the undressed Keldysh matrix þ2 Green functions satisfy gþ2 i# – gi" : 3. Results In order to realize that confinement permits to control the polarization of the current we solve the problem of a simple quantum dot, neglecting the Coulomb interaction, when the system is under the effect of an external magnetic field. The Zeeman effect, although modifies the states at the contact, its most important effect is at the dot itself because it separates in energy the level to be occupied by electrons with up and down spin. If the dot is small enough as to be possible to consider its resonances as isolated levels, with widths less than the Zeeman splitting, then it is possible, manipulating the gate potential, to obtain a configuration in which only one of the spins is at resonance while the other is outside. In this case, the device operates as a complete spin filter, as the current circulating after the dot is completely polarized. This is shown in Fig. 1 where we depict the dependence of the charge and the current with the gate potential. In the case of a real small dot the Coulomb interaction cannot be disregarded. In Fig. 2 we show the charge inside the dot for different spins as a function of the gate voltage. We see that, when the state corresponding to the spin up
Fig. 1. (a) Charge as a function of gate voltage: U ¼ 0:0; t0 ¼ 0:1; delta ¼ 0:2: (b) Current along the system as a function of gate voltage.
Fig. 2. (a) The dependence of the dot charge as a function of gate voltage: U ¼ 1:0; t0 ¼ 0:1; delta ¼ 0:1: (b) Spin polarized current transmitted through the system as a function of gate voltage: U ¼ 0:5; t0 ¼ 0:1; tso ¼ 0:0; delta ¼ 0:2:
enters into resonance, the charge inside the dot for this spin increases abruptly. Increasing the gate potential, the spin down electron enters as well into the dot. However, due to Coulomb blockade, which hinders the entrance of an extra electron when there is one already inside it, the spin down charge increases reducing the amount of charge with spin up and conserving the total charge of the dot. Due to the electron – hole symmetry a similar situation occurs increasing the gate potential still further by a value of the order of U: The charge of the spin up electron increases to unity eliminating the content of charge of the opposite spin up to the moment in which the second electron enters into the dot, as shown in Fig. 2. Along this process the circulating current is completely polarized when there is only one type of electron inside the dot while it losses its polarization as soon as the other gets in. This behavior is as well depicted in Fig. 2, where we show the current as a function of the gate potential and spin. Differently to the situation without correlation, described above, in this case, due to the Coulomb blockade, the spin up and down current are splitted by the Coulomb interaction. It is very interesting to understand the behavior of this device when it is injected into a polarized current. This can be done connecting a ferromagnetic injector [6] or coupling to the system a dot under the effect of a magnetic field operating as we described above.
A.T. da Cunha Lima, E.V. Anda / Microelectronics Journal 34 (2003) 337–339
339
the spin precession of the current, which can be manipulated by the gate potential, as the polarization of the current depends upon it. Finally we have studied this effect when the electrons suffer the Coulomb repulsion inside the dot, having a similar behavior.
4. Conclusions Fig. 3. (a) The charge inside the dot as a function of the gate voltage: U ¼ 0:0; delta ¼ 0:0; tso ¼ 0:2; t0 0 ¼ 0:1: (b) Current vs gate voltage when exclusively spin-up polarized electrons are injected.
In this case only the spin up electrons suffer the Coulomb repulsion inside the dot because in principle they are the only charges circulating inside the system. For a small dot this imply that the splitting among the Coulomb blockade peaks follow the distribution in energy of the dot states. The polarization of the current, as proposed by Datta et al. [2] can be modulated by a precession mechanism introduced by the spin – orbit splitting, which intensity can be changed by an external gate potential (the Rashba effect). We study first the uncorrelated case. The spin – orbit splitting breaks the degeneracy between spin up and down introducing an interaction among the spin channels. This produces inside the quantum dot, an splitting of the resonances, which is reflected as two peaks in the current as a function of the gate voltage, equally mixing spin up and down. However, when a completely polarized current is injected into the system, the spin – orbit mechanism introduces a spin precession and a corresponding change in the polarization of the current. We have limited the effect of the spin – orbit interaction to the dot region and calculated the current at the site immediately after the dot. The results are seen in Fig. 3. The possibility of observing these two peaks depends upon the nature of the confinement and the intensity of the spin –orbit interaction. It is obvious that the dot introduces an extra dephasing of
In this paper we were able to show that the presence of a quantum dot introduces, via the manipulation of the gate potential inside the dot, a very interesting mechanism to control the polarization of the current circulating along a spin polarized transistor. A complete study of the physics contained in this paper including a double quantum dot, a more generalized spin – orbit interaction and the effect of the spin polarization in the Kondo effect is under way.
Acknowledgements This work was partially supported by the Brazilian Agencies, CNPq and FAPERJ.
References [1] E.I. Rashba, Solid State Ion. 2 (1960) 1109. [2] S. Datta, B. Das, Electronic analog of the electro-optic modulator, Appl. Phys. Lett. 56 (1990) 665. [3] F. Mireles, G. Kirczenow, Ballistic spin-polarized transport and Rashba spin precession in semiconductor nanowires, Phys. Rev. B 64 (2001) 024426. [4] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, UK, 1995, p. 9. [5] M. Rasetti, The Hubard Model, World Scientific, Singapore, 1991. [6] C.M. Hu, J. Nitta, A. Jensen, J.B. Hansen, H. Takayanagi, Spinpolarized transport in a two dimensional electron gas with interdigitalferromagnetic contacts, Phys. Rev. B 63 (2001) 125333.
Spin transport through quantum dots A.T. da Cunha Lima, E.V. Anda* Departamento de Fı´sica, Pontifı´cia Universidade Cato´lica do Rio de Janeiro, C.P. 38071, 22453-900 Ga´vea, Rio de Janeiro, RJ, Brazil
Abstract We investigate the spin polarized transport properties of a nanoscopic device constituted by a quantum dot connected to two leads. We show that through the manipulation of the gate potential applied to the dot it is possible to control, in a very efficient way, the intensity and polarization of the current that goes along the system. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: Quantum dot; Magnetic field; Polarization
1. Introduction We investigate the spin polarized transport properties of a nanoscopic device constituted by a quantum dot connected to two leads. The electrical current circulates with a spin polarization that is modulated via a gate potential that controls the intensity of the spin – orbit coupling (the Rashba effect) [1,2]. We study a polarized field-effect transistor [2,3] when one of its parts is constituted by a small quantum dot, which energies are controlled by another gate potential operating inside the confined region. The high confinement and correlation suffered by the charges inside the dot gives rise to novel phenomena. We show that through the manipulation of the gate potential applied to the dot it is possible to control, in a very efficient way, the intensity and polarization of the current that goes along the system. Other crucial parameters to be varied in order to understand the behavior of this system are the intensity of the external applied electric and magnetic field.
2. The model The system is represented by the impurity Anderson Hamiltonian summed to a spin –orbit interaction, which describes the Rashba effect. We write the Hamiltonian using * Corresponding author. Tel.: þ55-21-3114-1263; fax: þ 55-21-5123222. E-mail address: [email protected] (E.V. Anda).
a tight-binding representation [3] X þ X s tij cis cjs þ e i nis þ t0 ðcþ HA ¼ 0s c1s þ c0s c1s Þ þ hc s;i;j
a;i
þ ðe s0 þ Vg Þn0s þ Un0" n0# þ Hso Hso ¼ 2
X
so þ ti;j ðci;s ðisy Þss0 cj;s0 2 cþ i;s ðisx Þss0 cj;s0 þ hc
ss0 ;i;j
where the first term represents the contacts and the dot, located at site 0 characterized by a local state of energy e 0 þ Vg ; where Vg represents a variable gate potential and U is the local electronic correlation. The dot is connected to the contacts through the non-diagonal matrix element t0 : The external electric field responsible for the current and the magnetic field is simulated adopting a site and spin dependent diagonal matrix element e si : The second term is a tight-binding representation of the spin – orbit interaction taking place along the system, which permits to change the spin polarization of the current modulating the position so dependent matrix element ti;j by the application of an external potential. The sx and sy are the x and y components of the Pauli matrices. To obtain the current of this out-ofequilibrium system, we use the Keldysh formalism [4]. The problem is solved using the Green function formalism considering the localized electronic correlation within the context of the Hubbard I approximation [5]. This approach permits to obtain a Dyson equation for the Green functions given by s s00
Gij 0
0026-2692/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0026-2692(03)00021-1
¼ gsij 0 dij ds0 s00 þ
X s;l;m
ss0
s0 s gsil 0 Tlm Gmj 0
338
A.T. da Cunha Lima, E.V. Anda / Microelectronics Journal 34 (2003) 337–339
where 0
Ti;jss ¼ tij dss0 þ tijso dss0 þ t0 ðdi1 þ di1 Þdss0 gs00 ¼
1 2 ns ns þ v 2 e s0 2 Vg v 2 e s0 2 Vg 2 U
and the current per spin s is given by ðe fr s0 s0 s0 s0 Is ¼ ðGþ2 ðwÞ 2 Gþ2 ðwÞÞdw i;iþ1 iþ1;i e fl
þ2s s0
whereGi;iþ1 0 0 ðwÞ corresponds to the-out-of equilibrium Keldysh Green function that gives us the probability of an electron of energy w to hop from site i to i þ 1 when the system is connected to two reservoirs of Fermi energies e fl and e fr ; such that the external applied potential is obtained from DV ¼ e fr 2 e fl : The injection of a polarized current into the system is simulated assuming that the undressed Keldysh matrix þ2 Green functions satisfy gþ2 i# – gi" : 3. Results In order to realize that confinement permits to control the polarization of the current we solve the problem of a simple quantum dot, neglecting the Coulomb interaction, when the system is under the effect of an external magnetic field. The Zeeman effect, although modifies the states at the contact, its most important effect is at the dot itself because it separates in energy the level to be occupied by electrons with up and down spin. If the dot is small enough as to be possible to consider its resonances as isolated levels, with widths less than the Zeeman splitting, then it is possible, manipulating the gate potential, to obtain a configuration in which only one of the spins is at resonance while the other is outside. In this case, the device operates as a complete spin filter, as the current circulating after the dot is completely polarized. This is shown in Fig. 1 where we depict the dependence of the charge and the current with the gate potential. In the case of a real small dot the Coulomb interaction cannot be disregarded. In Fig. 2 we show the charge inside the dot for different spins as a function of the gate voltage. We see that, when the state corresponding to the spin up
Fig. 1. (a) Charge as a function of gate voltage: U ¼ 0:0; t0 ¼ 0:1; delta ¼ 0:2: (b) Current along the system as a function of gate voltage.
Fig. 2. (a) The dependence of the dot charge as a function of gate voltage: U ¼ 1:0; t0 ¼ 0:1; delta ¼ 0:1: (b) Spin polarized current transmitted through the system as a function of gate voltage: U ¼ 0:5; t0 ¼ 0:1; tso ¼ 0:0; delta ¼ 0:2:
enters into resonance, the charge inside the dot for this spin increases abruptly. Increasing the gate potential, the spin down electron enters as well into the dot. However, due to Coulomb blockade, which hinders the entrance of an extra electron when there is one already inside it, the spin down charge increases reducing the amount of charge with spin up and conserving the total charge of the dot. Due to the electron – hole symmetry a similar situation occurs increasing the gate potential still further by a value of the order of U: The charge of the spin up electron increases to unity eliminating the content of charge of the opposite spin up to the moment in which the second electron enters into the dot, as shown in Fig. 2. Along this process the circulating current is completely polarized when there is only one type of electron inside the dot while it losses its polarization as soon as the other gets in. This behavior is as well depicted in Fig. 2, where we show the current as a function of the gate potential and spin. Differently to the situation without correlation, described above, in this case, due to the Coulomb blockade, the spin up and down current are splitted by the Coulomb interaction. It is very interesting to understand the behavior of this device when it is injected into a polarized current. This can be done connecting a ferromagnetic injector [6] or coupling to the system a dot under the effect of a magnetic field operating as we described above.
A.T. da Cunha Lima, E.V. Anda / Microelectronics Journal 34 (2003) 337–339
339
the spin precession of the current, which can be manipulated by the gate potential, as the polarization of the current depends upon it. Finally we have studied this effect when the electrons suffer the Coulomb repulsion inside the dot, having a similar behavior.
4. Conclusions Fig. 3. (a) The charge inside the dot as a function of the gate voltage: U ¼ 0:0; delta ¼ 0:0; tso ¼ 0:2; t0 0 ¼ 0:1: (b) Current vs gate voltage when exclusively spin-up polarized electrons are injected.
In this case only the spin up electrons suffer the Coulomb repulsion inside the dot because in principle they are the only charges circulating inside the system. For a small dot this imply that the splitting among the Coulomb blockade peaks follow the distribution in energy of the dot states. The polarization of the current, as proposed by Datta et al. [2] can be modulated by a precession mechanism introduced by the spin – orbit splitting, which intensity can be changed by an external gate potential (the Rashba effect). We study first the uncorrelated case. The spin – orbit splitting breaks the degeneracy between spin up and down introducing an interaction among the spin channels. This produces inside the quantum dot, an splitting of the resonances, which is reflected as two peaks in the current as a function of the gate voltage, equally mixing spin up and down. However, when a completely polarized current is injected into the system, the spin – orbit mechanism introduces a spin precession and a corresponding change in the polarization of the current. We have limited the effect of the spin – orbit interaction to the dot region and calculated the current at the site immediately after the dot. The results are seen in Fig. 3. The possibility of observing these two peaks depends upon the nature of the confinement and the intensity of the spin –orbit interaction. It is obvious that the dot introduces an extra dephasing of
In this paper we were able to show that the presence of a quantum dot introduces, via the manipulation of the gate potential inside the dot, a very interesting mechanism to control the polarization of the current circulating along a spin polarized transistor. A complete study of the physics contained in this paper including a double quantum dot, a more generalized spin – orbit interaction and the effect of the spin polarization in the Kondo effect is under way.
Acknowledgements This work was partially supported by the Brazilian Agencies, CNPq and FAPERJ.
References [1] E.I. Rashba, Solid State Ion. 2 (1960) 1109. [2] S. Datta, B. Das, Electronic analog of the electro-optic modulator, Appl. Phys. Lett. 56 (1990) 665. [3] F. Mireles, G. Kirczenow, Ballistic spin-polarized transport and Rashba spin precession in semiconductor nanowires, Phys. Rev. B 64 (2001) 024426. [4] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, UK, 1995, p. 9. [5] M. Rasetti, The Hubard Model, World Scientific, Singapore, 1991. [6] C.M. Hu, J. Nitta, A. Jensen, J.B. Hansen, H. Takayanagi, Spinpolarized transport in a two dimensional electron gas with interdigitalferromagnetic contacts, Phys. Rev. B 63 (2001) 125333.