Quantum ratchet in two-dimensional semiconductors with Rashba spin-orbit interaction

Ratchet is a device that produces direct current of particles when driven by an unbiased force. We demonstrate a simple scattering quantum ratchet based on an asymmetrical quantum tunneling effect in two-dimensional electron gas with Rashba spin-orbit interaction (R2DEG). We consider the tunneling of electrons across a square potential barrier sandwiched by interface scattering potentials of unequal strengths on its either sides. It is found that while the intra-spin tunneling probabilities remain unchanged, the inter-spin-subband tunneling probabilities of electrons crossing the barrier in one direction is unequal to that of the opposite direction. Hence, when the system is driven by an unbiased periodic force, a directional flow of electron current is generated. The scattering quantum ratchet in R2DEG is conceptually simple and is capable of converting a.c. driving force into a rectified current without the need of additional symmetry breaking mechanism or external magnetic field.

Ratchet is a device that produces direct current of particles when driven by an unbiased force. We demonstrate a simple scattering quantum ratchet based on an asymmetrical quantum tunneling effect in two-dimensional electron gas with Rashba spin-orbit interaction (R2DEG). We consider the tunneling of electrons across a square potential barrier sandwiched by interface scattering potentials of unequal strengths on its either sides. It is found that while the intra-spin tunneling probabilities remain unchanged, the inter-spin-subband tunneling probabilities of electrons crossing the barrier in one direction is unequal to that of the opposite direction. Hence, when the system is driven by an unbiased periodic force, a directional flow of electron current is generated. The scattering quantum ratchet in R2DEG is conceptually simple and is capable of converting a.c. driving force into a rectified current without the need of additional symmetry breaking mechanism or external magnetic field. R atchet is a device that produces direct current of particles when driven by an unbiased force 1,2 . In technological applications, ratchets are particularly useful in nano-electronics as they can be utilized as miniature current rectifiers, switches or refrigerators 3,4 . Ratchet plays an important role in many biological processes such as the intracellular transport of proteins and ATP hydrolysis 5,6 . To create directed motion of particles, a ratchet structure must possess some form of spatial or temporal symmetry breaking 7 . For example, the thermal diffusion of particles can be 'chopped' by a time-modulated asymmetrical potential barrier and this leads to a directed motion of particles 8,9 . Alternatively, net flow of particles across asymmetrical potential barrier can also be driven by dichotomous Markov noise 10,11 . Such devices belongs to the class of classical Brownian ratchets since the ratchet current originates from the classical Brownian diffusion of particles. When the quantum tunneling of particles across the asymmetrical confining barrier is taken into account, the ratchet current is significantly enhanced and it exhibits a directional reversal dependent on the temperature and the period of the external fields 12,13 . The quantum ratchet effect has been experimentally demonstrated in the transport of electrons through asymmetric conducting channels in GaAs/AlGaAs heterostructure 14 . Quantum ratchet motion of Rubidium atoms has also been realized via time-modulated optical lattice 15 . Alternatively, transport asymmetry can be generated in a two-dimensional electronic system with layer asymmetry in the presence of an in-plane magnetic field. Drexler et al has elegantly demonstrated this magnetic quantum ratchet effect in semihydrogenated graphene where the layer symmetry is broken by the selective attachment of hydrogen adatoms to only one surface of the graphene layer 16 . In such structure, the in-plane magnetic field is coupled to the terahertz (THz) excitation of the electrons to produce out-of-plane Lorentz forces. The direction of the Lorentz forces are dependent on the in-plane directions of the THz-driven electrons. Electrons that are pushed towards the adatoms experience enhanced scattering and this leads to a directed flow of electrons.
In this paper, we describe a scattering quantum electron ratchet in two-dimensional electron gas with Rashba spin-orbit interaction (R2DEG) [17][18][19] . It has been shown that the Rashba spin-orbit coupling can results in zero field Hall current 20 , specular Andreev reflection 21 , and chiral tunneling 22 in semiconductors. It can also give rise to the low frequency conductance resonance in graphene 23 . In the present problem, the ratchet current originates from the asymmetrical tunneling of electrons across a potential barrier sandwiched by two interface scattering potentials of unequal strengths. We found that although the tunneling probabilities of the same-spin-subband transmission is symmetrical for electrons tunneling across the junction in both directions, this symmetry is broken in the case of the inter-spin-subband tunneling process. When the tunnel junction is periodically driven, the left-going and the right-going tunneling currents are unequal. Such asymmetrical tunneling of electrons in R2DEG leads to a net transfer of electrons across the tunnel junction driven by a sinusoidal bias voltage.
Model and Formalism. In order to investigate the transport properties in a R2DEG tunneling junction, we first review the electronic properties of R2DEG shortly. In a quantum well structure, two-dimensionally confined electrons can undergo spontaneous lifting of the spin-degeneracy if the confining potential is asymmetric. Such effect is equivalent to the relativistic case of electron moving through a surface with inhomogeneous electric field. In the rest frame of the electrons, the electric field is relativistically equivalent to a magnetic field. This effectively generates finite spin-orbit interaction and energetically separates the electron gas into two populations of different spin chirality. Spin-orbit-interaction of this form is known is the Rashba spinorbit interaction (RSOI) 17 . The RSOI manifests itself as a left-and right-shifting of the 'free' electron parabolic bands in phase-space and the degree of the splitting is characterized by a Rashba coupling parameter l 18,19 .
Although the tunneling problems in R2DEG has previously been studied [24][25][26][27][28][29][30][31][32] , it is not clear whether the presence of an interface scattering potentials can play a role in the electron transport of this system. This is the main objective of this work. In order to study the effect of the interface scattering potential on the spin-polarized transport, we model a square potential barrier V (x) in the width d. The inhomogeneities for the left and right interface scatterings are described by introducing two delta interface potentials of the strengths Figure 1(a)]. In practice, the interface scattering potential can be achieved by applying thin strips of electrostatically-gated electrodes to the R2DEG confined in a GaAs/AlGaAs heterostructure, and the square barrier height V 0 can be controlled by gate voltage on the scattering region of the tunneling structure. The Hamiltonian of infinite R2DEG is given as 17 where K~K x ,K y À Á is the wavevector, m* is the electron effective mass, s x and s y are the Pauli spin matrices and l is the Rashba coupling parameter. In our model, we shall ignore the interaction between R2DEG and phonons 33 . This equation can be written in a form h k 5k 2 12(s x k y 2s y k x ) which introduces only the following dimensionless quantities: 2m. The eigenvalue of the reduced Hamiltonian h k without the potential barrier (i.e. v 0 50) is e s 5k 2 12sk, where s561 represents the chirality of the spin-subband. The wavevector of state s511 is given as k z . Corresponding to these wavevectors, the propagation of the eigenstates manifests in different transmittal characteristics. To see this, we first look at the group velocity of the electrons. When e s .0, the group velocity x,{ is negative. In this case, the wavevector is anti-parallel with the direction of motion. This infers a hole-like characteristic for the electrons residing in the s521 and c521 branch. Now we apply these discussions to our system. For an incident in the left in eigenstate y cos h c for the wavefunction in the barrier layer. The wavefunction in the drain is given by s' represent the strengths of the sRs9 transmission. In these wavefunctions, the conserved factor e ik y y has been omitted for simplicity. The wavevector q c cos w s Tunneling without the interface scattering potential. For the case without interface scattering potential at the x50 and d interfaces, the energy dependence of the transmission probabilities is shown in Figure 2. The transmission probabilities for the same-branch process1R1and the inter-branch process1R 2 are shown in Figure 2(a) and Figure 2(c), respectively. In comparison with Figure 2(c), Figure 2(a) shows that the same-branch transmission is much stronger than the inter-branch transmission. Similar, for s521 incident state, transmission via the process 2 R 2 is also much stronger than that of the process 2 R1[see Figure 2(b) and Figure 2(d)]. For the 2 R 2 process, the probability oscillations Tunneling in the presence of symmetrical interface scattering potentials. We now consider the case when symmetrical interface scattering potentials are present, i.e. Z L 5Z R . The energy dependence of the transmission probabilities for different strength of interface scattering potentials is shown in Figure 3 and Figure 4 respectively for s511 and s521 incident states. As an anticipatory result, electron tunneling is, in general, suppressed by the interface scattering potentials. However, there is an exception for the interbranch transmissions of 1R 2 and 2 R1. For the T z Tunneling in the presence of asymmetrical interface scattering potentials. We now investigate the case when the interface scattering potentials are asymmetrical for the left and right boundaries, i.e. Z L ? Z R . The transmission spectra of the s511 incident states is shown in Figure 5. In Figure 5  Scattering quantum ratchet in a R2DEG tunneling junction. In above, we have seen that the electron tunneling can be asymmetrical in the presence of asymmetrical interface scattering potentials. We can use this property of R2DEG tunneling junction to obtain a net transfer of spin-polarized electrons across the barrier via a alternating bias voltage. In this sense, the potential barrier acts as a quantum ratchet.
To see how the R2DEG tunnel junction with asymmetrical interface scattering potential can work as a quantum ratchet when it is driven sinusoidally, we apply an a.c. bias voltage to the R2DEG tunnel junction with asymmetrical interface scattering potentials (Z L .Z R ) [ Figure 7(a)]. In the first half of the a.c. period, a forward current I f is driven from the left to the right of the barrier and the right-moving I f 'sees' the left interface 'obstacle' Z L first and then the right Z R . In the second half period of the a.c. cycle, the current is reversed and I r is driven from the right to the left of the barrier. Due to the directional reversal, the relative order of the interface scattering potentials as 'seen' by I r is reserved, i.e. it 'sees' Z R first and then   Z L . The previous calculations told us that the tunneling probabilities T s ð Þ s remains the same when Z L «Z R . Accordingly, the same-spin tunneling process (sRs) is not affected by the interchanging of Z L and Z R . In this case, I f 2I r 50 and no net charge is transferred. However, for the opposite spin tunneling process (sR2s), T zs ð Þ {s no longer remains constant when the interface scattering potentials Z L «Z R is interchanged. As a result, I f 2I r ? 0 and a net transfer of electrons through the tunnel junction is produced [ Figure 7(b)]. Since the ratchet current has its root from the unequal scattering strengths of the interface scattering potentials, the tunnel junction can be regarded as a scattering quantum ratchet.
We now look at the I V ð Þ characteristic of the junction under a d.c. bias V first. The charge current is given as: where Df(e)5f(e2e F 2ev)2f(e2e F ) with e F 5E F /E SO and v~V=E SO . At zero temperature, we obtain: where I 0 5ek SO E SO L 2 /(2p 2 h _ ) and V (s) 5sin w (s) . When the LHS of the tunnel junction is raised by eV(i.e. 'forward-bias'), the right-moving current takes the same form as Eq. (3)  , has the same form as that of the right-moving current except that Z L and Z R are interchanged. Finally, the total forward-biased and reserve-biased currents are: I f~X s,s' I s?s' f and I r~X s,s' I s?s' r respectively. We plot the current-voltage characteristics in Figure 7(c). For easy comparison, the absolute value of the negative-valued I r is taken. We see that I f and I r is unequal. The magnitude of If is about 20% larger than that of I r at a bias voltage of eV<E SO .
We now consider the junction being driven by a symmetrical a.c. bias voltage in the form of V t ð Þ~V sin 2pt=T where T is the a.c. period. Assuming that the magnitude of V t ð Þ is small, only states at the Fermi level can contribute to the current. In the first half of the cycle, a current is driven rightwards across the junction, and the differential conductance, G~LI=LV, is given as where G 0 5e 2 k SO E SO L 2 /(2p 2 h _ ). In the second cycle, the conductance is in the same form as Eq.(4) except that Z L «Z R . In Figure 8 Figure 8(d), we show the DZ and the Fermi level dependence of the ratchet conductance where Z L 5Z 0 2DZ and Z R 5Z 0 1DZ with Z 0 51.5l. A similar conductance oscillation is also present since DG tot is dominated by DG z ð Þ { . Furthermore, the direction of the ratchet current reverses when DZ changes its sign. This allows the direction of the ratchet current to be manipulated by interchanging the scattering strengths {s (E F 50.5E SO , d520 nm, V 0 5E SO , Z L 51.5 and Z R 50.5). (d) DZ and Fermi level dependence of the ratchet conductance DG tot . When the asymmetry of the interface potential is swapped from Z L .Z R to Z L ,Z R , the ratchet current reverses its direction as signified by DG tot ,0. When the Fermi level is very large, the ratchet current is suppressed. of the LHS and RHS interface scattering potentials. It should be emphasized that the results of Eq. (4) provides a qualitative picture of the quantum ratchet. This quasi-static treatment is only valid when the amplitude and the frequency of the a.c. driving field are small. We used this simple treatment to illustrate that it is possible to create a ratchet effect in R2DEG junction due to the asymmetrical sR2s transmission behaviour. For a more general a.c. driving force, timedependent methods, e.g. Floquet methods 34 and Keldysh non-equilibrium Green function technique 35 , should be utilized. The main error of the quasi-static treatment is that the quantum states in the leads are assumed to be independence of the electron-ac field-coupling. This effect can be large if the amplitude of the ac-field is large.
We now briefly compare our system with a similar tunneling junction of metal/R2DEG/metal 33 . In such junction, a magnetic d-potential is formed at both of the metal/R2DEG interfaces due to the abrupt discontinuity of the Rashba coupling strength. They observed an adjustable spin polarized transmission of up to 10% spin-polarization. Interestingly, spin-dependent transmission is also present in our system albeit the fact that there is no Rashba coupling strength discontinuity in our case. Since the spin-dependent transmission is one of the key features that results in the scattering quantum ratchet effect, we expect the ratchet effect to be affected by the presence of such d interface potential in a R2DEG tunneling junction of unequal Rashba coupling strengths at different tunneling regions.
Finally, we emphasize that the scattering quantum ratchet cannot occur in a 'normal' 2DEG without the Rashba spin-orbit coupling. We solve the transmission probability T through a potential barrier of V(x)5(H(x)2H(x2d)) V 0 1Z L d(x)1Z R d(x2d). It is found that T can be written as: where k~ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mE , k x 5k cos w and q x 5 q cos h. w is the azimuthal angle of the wavevector k, and h can be determined by the wavevector conservation condition k sin w5q sin h. It is immediately obvious that, regardless E.V or E,V, the interchanging of Z L «Z R has no effect on T. Therefore, the scattering quantum ratchet described here cannot occur in normal 2DEG.

Discussion
We have studied the electron tunneling ratchet phenomenon in R2DEG through a square potential barrier with asymmetrical interface scattering potentials in R2DEG. We found that probabilities for the same-spin tunneling (T z ) becomes unequal when the left and the right interface scattering potentials are interchanged. We then discussed a strategy to construct a scattering quantum ratchet based on these asymmetrical tunneling behaviors. The scattering quantum ratchet in R2DEG is conceptually simple and is capable of converting a.c. driving force into a rectified current without the need of asymmetrical transport channels 14,36,37 , optical tweezers 8,9,15,38 , quantum dots 39 , THz excitation and strong magnetic fields 16,40 . Since the scattering quantum ratchet involves only one square potential barrier, the physical dimension of such device can be greatly reduced.

Methods
The main results of this work, i.e. the transmission probabilities T s ð Þ s' are derived using the standard wavefunction matching at the boundaries of the potential barriers. This is outlined in detail in the Model and Formalism Section.