Strained topological insulator spin field effect transistor

The notion of a spin field effect transistor, where transistor action is realized by manipulating the spin degree of freedom of charge carriers instead of the charge degree of freedom, has captivated researchers for at least three decades. These transistors are usually implemented by modulating the spin orbit interaction in a two- or one-dimensional semiconductor structure with an electrostatic potential, which then causes controlled spin precession in the transistor's channel that modulates the current flowing between two ferromagnetic (spin-polarized) source and drain contacts. Here, we introduce a new concept for a spin field effect transistor whose channel is made of a strained topological insulator (strained-topological-insulator-field-effect-transistor or STI-SPINFET), which does not exploit spin-orbit interaction. Instead, the transistor function is elicited by straining the topological insulator (TI) with a gate voltage which modifies the energy dispersion relation, or the Dirac velocity, to vary the interference between the two spin eigenstates on the surface of the TI. This modulates the current flowing between two ferromagnetic source and drain contacts. The conductance on/off ratio of this transistor is too poor to be useful as a switch, but it may have other uses, such as an extremely energy-efficient stand-alone frequency multiplier.


Introduction
The idea of a spin-field-effect-transistor (SPINFET) is at least three decades old and was inspired by the belief that it might take much less gate voltage (and hence much less energy) to manipulate the spins of electrons than to manipulate the number of electrons (charge) in a transistor's channel. Hence, such a transistor could potentially be an extremely low-power device. Early proposals [1] visualized the transistor as having the same structure as a conventional field effect transistor, except its "source" and "drain" contacts are ferromagnetic and act as a spin polarizer and a spin analyzer, respectively. The source will inject spin polarized electrons into the channel whose spin polarizations would then be controllably rotated while transiting through the channel with a potential applied to the "gate" terminal. That potential varies the spin-orbit interaction in the channel and thus varies the spin rotation. The "drain" contact would filter the arriving spins and preferably transmit those whose polarizations are aligned parallel to its own magnetization, while blocking those that are antiparallel. Therefore, by controlling the spin rotation in the channel with the gate voltage, we can modulate the channel current and realize transistor action. The spin-orbit interaction that is modulated with the gate potential in a conventional SPINFET can be of the Rashba type [1] or the Dresselhaus type [2].
It is now understood that there are three serious challenges with transistors of this variety. First, because spinorbit interaction is not sufficiently strong in semiconductors, it takes a considerable amount of electric field or potential to induce adequate spin orbit interaction to change the channel current appreciably, meaning that these transistors will have low transconductance and high switching voltages (hence poor energy-efficiency, contrary to expectations) [3]. Second, the ferromagnetic source and drain contacts are typically inefficient spin polarizers and analyzers, so that the modulation of the current (or the conductance on/off ratio) is relatively poor [4], while again contributing to poor transconductance. Third, in two-dimensional SPINFETs, ensemble averaging over the transverse wavevector further degrades the conductance on/off ratio. These shortcomings are very difficult to overcome and decades of research have not been able to alleviate them. Nonetheless, the physics of these transistors is intriguing and they may have unusual characteristics, such as an "oscillatory" transfer characteristic, which has other uses.
In this work, we propose and analyze a novel SPINFET. It does not overcome the above shortcomings of SPINFETs, but it employs a very different operating principle, which shows the rich variety of mechanisms that can be called into play to implement SPINFETs. In our device, the gate potential does not modulate spin-orbit interaction. In fact, spin-orbit interaction is not needed for transistor action at all. The channel is made of (the surface of) a three dimensional topological insulator (3D-TI) thin film with two (wavevectordependent) spin eignestates. The ferromagnetic source injects spins with a polarization that is a superposition of the two eigenspin states. The gate voltage mechanically strains the TI film, which modulates the Dirac velocity of the surface states, thereby changing the phase relationship in the superposition. That effectively rotates the injected spin, just as the gate voltage rotates the injected spin in the channel of a conventional SPINFET.
The ferromagnetic drain contact acts as a spin analyzer, just as in a conventional SPINFET. When the spin in the channel has been rotated by the gate voltage such that it is parallel to the drain's magnetization when it arrives at the drain contact, it transmits with the highest probability (current is "on"), and if it arrives with spin antiparallel to the drain's magnetization, it transmits with the lowest probability (current is "off"). Thus, transistor action is realized in the same way as the original SPINFET [1,2], except that here the gate control of the spin rotation is achieved via strain-induced modulation of the Dirac velocity in the TI surface and not by any modulation of spin-orbit interaction.
The transistor structure is shown in Fig. 1. The 3D-TI, acting as the transistor's channel, is deposited on a vertically poled thin piezoelectric film with two mutually shorted electrodes flanking it, as shown in Fig. 1(b). This shorted pair acts as the gate, and this gate configuration is known to generate strain in the intervening region of the piezoelectric, i.e., underneath the 3D-TI film [5]. If we apply a gate voltage whose polarity is such that the resulting (vertical) electric field is directed opposite to the direction of poling, then compressive stress will be generated along the line joining the two electrodes (z-axis in Fig. 1(b)) and tensile stress in the perpendicular direction (x-axis). Reversing the polarity will reverse the signs of the stresses. The use of a piezoelectric thin film deposited on a conducting substrate, as opposed to a piezoelectric substrate, is dictated by the fact that piezoelectrics are insulators and hence a much larger voltage would have been needed to generate a given strain had we substituted the piezoelectric film with a piezoelectric substrate. As long as the piezoelectric film thickness is much larger than the thickness of the TI film, we can assume that 100% of the strain generated in the piezoelectric is transferred to the TI film. The transferred stress/strain changes the energy dispersion relation of the surface states in the 3D-TI, specifically the slope, and hence the Dirac velocity [6]. This results in spin rotation in the TI surface (transistor channel) and that, in turn, modulates the current flowing between the ferromagnetic source and the drain. The role of the thin insulating layer between the ferromagnetic source/drain contacts and the TI surface (see Fig. 1) is to act as a tunnel barrier, which is known to improve the spin injection and detection efficiencies of the source/drain contacts [7]. In other words, its presence makes the source and drain contacts better spin polarizer and analyzer. The ferromagnetic contact materials can be those with a high degree of spin polarization, e.g. half metals, to further increase the spin injection/detection efficiency.
Because the device is two-dimensional, ensemble averaging over the transverse wave vector kz inevitably dilutes the current modulation, very much like the original twodimensional SPINFET [8], resulting in very poor on/off ratio for the channel conductance. That precludes any use as a switch, but there can be other uses, such as in frequency multiplication, as we discuss later.
We describe the theory of this device in the next section.

Theory
Fig . 2 shows the conducting surface of the 3D-TI (the channel) pinched between the ferromagnetic source and drain contacts. We can assume that the TI is a common material like Bi2Te3 or Bi2Se3. The Hamiltonian describing the surface states near a Dirac point (including higher order terms in the wave vector, up to third order) is [9,10] where m* is the effective mass,  is the band warping factor, We point out that the Hamiltonian in Equation (1) neglects the effect of finite thickness and width, as well as any external magnetic field or spin-orbit interaction.
Diagonalizing the Hamiltonian yields the energy dispersion relation of the spin resolved states as In an "ideal" topological insulator (TI) surface, only the second term in the Hamiltonian in Equation (2) will be present. That will make the energy dispersion relation 0 E vk   , which are the familiar Dirac cones. Real TIs, like Bi2Se3, however do not fit this bill and the first term in the Hamiltonian will also be present, albeit it will be much smaller than the second term. difference is that in the Rashba system, the first term in the Hamiltonian is dominant over the second term, whereas in the TI, the opposite is true. The Rashba system and the real TI system exhibit similar physics; for example, there is spinmomentum locking in both. The similarities and differences between a real TI and the Rashba system have been discussed in [10].
In Fig. 3, we plot the dispersion relations in Equation (2) assuming * The eigenspinors of the Hamiltonian in Equation (1) are sin cos ; cos sin We will assume that the TI film is semi-infinite in the zdirection, in which case, the wave vector component kz is a good quantum number. From Equation (2), it is clear that for any given energy E and magnitude of the wave vector component kz, the magnitudes of the x-components of the wave vectors are different in the two spin resolved states. They are related according to Hence the angle  in Equation (3)  Let us assume that the ferromagnetic source and the drain contacts are both magnetized in the +x-direction as shown in Fig. 2. We will assume that the source injects only +x-polarized spins into the TI at the complete exclusion of -x polarized spins (perfect spin polarizer). This assumption can be relaxed [8], but it is not necessary at this point for elucidating the principle behind the transistor operation. An injected +x-polarized spin will couple into the two eigenspin states   and   in the channel (TI surface) with (wavevector dependent) coupling coefficients C + and C -. We can view this occurrence as the incident +x-polarized beam splitting into two beams, each corresponding to an eigenspinor in the TI channel. These two beams propagate in different directions  for any given energy and kz. Hence the TI channel behaves like a birefringent medium [11]. The beam splitting is expressed by the equation The coupling coefficients are found from Equation (5) as In the drain contact, the two beams interfere. The phase difference between them (accrued in traversing the channel) determine the spinor (and hence the spin polarization) of the electron impinging on the drain. This, in turn, determines the transmission probability through the drain contact (spin analyzer) and therefore the source-to-drain current. We will show that the phase difference can be altered with a gate potential which strains the TI and modifies the Dirac velocity, and this elicits the transistor functionality.
The spinor at the drain end is given by sin cos where L is the channel length (distance between source and drain contacts) and W is the transverse displacement of the electron as it traverses the channel. Neglecting multiple reflection effects, the transmission amplitude, t, is the projection of the arriving spinor on the drain's polarization (which is the +x-polarization). Hence The transmission probability T is then given by    Fig. 1. Applying a voltage to the two electrodes in that figure will strain the piezoelectric region pinched between the electrodes. As long as the TI film is much thinner than the piezoelectric film and the insulating film is also thin enough to not clamp the TI, this strain will be transferred almost entirely to the TI film. In a TI material like Bi2Se3, small stress (or strain) can change the Dirac velocity 0  along specific crystallographic directions by ~4 2 10  m/s per GPa of stress [6]. This provides a handle to vary  and hence the transmission probability T with an external gate voltage which generates strain in the TI film. That can then modulate the current flowing between the source and the drain contact, thereby realizing transistor action. Since such a transistor consists of three elements -a 3D-TI, strain, and spin interference -we have called it strained topological insulator spin field effect transistor (STI-SPINFET).
In the Appendix, we derive an expression for the linear response channel conductance (or source-to-drain conductance) of the STI-SPINFET as a function of the gate voltage to demonstrate the transistor functionality. That expression is ( 1 2 ) This expression immediately shows that if we can change the Dirac velocity 0  with a gate voltage, then we can change the channel conductance (and hence the source-todrain current for a fixed drain bias) with the gate voltage, thereby realizing transistor action.

Material considerations
In Bi2Se3, the Dirac velocity in the K   crystallographic direction is ~5 6.2 10  m/s under no stress and increases linearly with compressive stress by ~4 2 10  m/s per GPa [6]. Therefore, this material is a good choice for the TI.
For the piezoelectric layer, one would have preferred to use relaxor materials like PMN-PT because of their high d31 (piezoelectric) coefficients, but they are not compatible with TI films. The growth temperature for a TI is typically in the range of 400 0 -500 0 Celsius and the piezoelectric may not survive such high temperatures. There are recent reports of perovskites like (1-x)BiScO3-xPbTiO3 which can survive temperatures up to 460 0 C [12] and hence would be compatible with TI growth. It has a d31 value of -670 pC/N [12] and therefore is a good choice. With this d31 value, one can generate a strain  of 1000 ppm in the piezoelectric with an electric field E of 1.5 MV/m which is a very reasonable electric field (= d31E).
As mentioned earlier, we will assume that the strain generated in the piezoelectric is completely transferred to the TI. It is a good approximation when the TI film is much thinner than the piezoelectric film. The Young's modulus of Bi2Se3 nanoribbons was reported as ~40 GPa [13] and we expect it to be about the same in thin films. Hence the stress generated by a strain of 1000 ppm in Bi2Se3 is 40 MPa. This stress will increase the Dirac velocity 0  in Bi2Se3 from 5

10
 m/s to 5 6.2 10  + 800 m/s, which is enough to modulate the channel conductance of the transistor between the maximum and minimum values. Thus our material choices are (1-x)BiScO3-xPbTiO3 for the piezoelectric and Bi2Se3 for the TI.

Transfer characteristics of the transistor
The gate voltage needed to generate a stress  in the TI film is obtained as follows. If we assume that the gate voltage is dropped entirely across the piezoelectric layer (since it is much thicker than the TI layer) and that the conducting substrate has a negligible voltage drop, then we obtain

Numerical results
In Fig. 4, we plot the quantity SD Z G W as a function of stress from 0 to 40 MPa in steps of 0.5 MPa. The maximum pressure that we consider (40 MPa) is low enough that we can ignore all other pressure-related effects that can show up at extremely high pressures (several GPa). In the upper horizontal axis in Fig. 4, we plot the gate voltage Vgate needed to generate the corresponding stress. In this plot, we have assumed that the Fermi wave vector F k = The disappointing feature in the transfer characteristic is the low conductance on/off ratio which is only about 1.07:1. Imperfect spin injection and filtering at the contacts will reduce this ratio further. Clearly this precludes any application of the STI-SPINFET as a "switch". However, note that the transfer characteristic is oscillatory. We can filter out the oscillatory component in the source to drain current with a series capacitor which blocks dc signals. In the gate voltage range 0 to 1.5 V shown in Fig. 4, there are two nearly complete periods of the oscillation in the channel conductance. Hence, if we apply an ac gate voltage with a peak-to-zero amplitude of 1.5 V, the source to drain current (for a fixed drain bias VSD) will oscillate with a frequency four times that of the gate voltage. This can implement a frequency multiplier with a single transistor. In general, the frequency multiplication factor will be  

Conclusions
We have proposed and discussed a new transistor device whose channel is made of a topological insulator (TI) thin film deposited on a piezoelectric film and the source and the drain contacts are ferromagnetic. The piezoelectric is utilized to strain the topological insulator with a gate voltage, which varies the Dirac velocity to rotate spin in the transistor's channel. That allows control of the channel conductance with the gate voltage (because of the spin filtering action of the drain) to implement a transistor. There have been other transistor proposals in the past involving TIs [15][16][17], but they involve 2D-TIs that are very different from 3D-TIs. A diode based on a 3D-TI has also been proposed [18]. None of them, however, involves "spin" and hence is not a SPINFET.
Unfortunately, the present device cannot be used as a switch because of the extremely poor channel conductance on/off ratio, but the unusual oscillatory transfer characteristic (a hallmark of SPINFETs based on spin interference [1,2]) can be exploited to implement sub-systems such as a frequency multiplier with a single transistor. Since the subsystem consists of a single device, it has remarkably small footprint and also very low energy dissipation.

Appendix I: Channel conductance as a function of gate voltage
In calculating the channel conductance as a function of gate voltage, we will ignore self-consistent effects, i.e. we will not invoke the Poisson equation because the surface of a TI is highly conductive. In a highly conductive channel (metallic), any effect of the Poisson equation (such as band bending) will be negligible and hence self-consistency effects can be safely ignored. We will also assume ballistic transport.
The current density in the channel between the source and the drain is given by the Tsu-Esaki formula [19]     0 1 , where q is the electron's charge, E is the electron (spin carrier) energy, Wy is the thickness of the channel in the y-direction (the vertical extent of the TI surface), VSD is the applied source to drain voltage and   f  is the Fermi-Dirac factor (electron occupation probability) at energy in the source contact. This relation reduces to in the linear response regime when 0 SD V  .
The channel conductance is therefore Substituting the expression for T in Equation (9) into Equation (A3), we obtain