Spin-polarized current in field-effect transistors made of magnetic polymers Open Access

Organic electronics (OE)^1–3 has attracted increasing attention due to several outstanding advantages compared to inorganic electronics, such as flexibility, lower cost, and reduced environmental impact.^4–6 However, OE faces challenges in terms of conductivity, durability, and thermal stability,^7–10 particularly in high-power or high-frequency applications. Similarly, organic spintronics (OS)^11–18 is advancing at a pace comparable to inorganic spintronics (IS),^19–21 offering benefits similar to OE. One notable advantage of OS is the significantly longer spin relaxation time (SRT),^22,23 attributed to the weak spin–orbit interactions (SOIs) in organic materials because the strength of spin–orbital interaction scales with the atomic number Z as $Z 4$⁠. The OS materials, thus, commonly provide the SRT at least 1 μs.¹⁵ Nevertheless, organic materials suffer from lower spin injection efficiency, which limits their potential in specific spintronic applications.^24,25

Most methods for producing magnetic organic materials differ significantly from those used for magnetic inorganics. Due to the weak SOI, especially 1D organic materials are required to generate spin signals for OS applications, while inorganic spintronics can rely in this regard on magnetism or the strong inherent SOI within inorganic materials or phenomena like, e.g., inverse spin Hall or Rashba–Edelstein effects.^19–21 However, due to the long SRT, also PEDOT:PSS composite polymer reveals measurable inverse spin Hall voltage.²⁶ Besides, Hu et al. theoretically show that in quasi-1D organic ladder polymers, a polaron moving under a static electric field can produce a significant, rapidly oscillating spin Hall signal.²⁷ 

In the case of organics, magnetism can be achieved either in chiral molecules,^14,17,18 when the spin formation is believed to be coupled with the helical potential, or by linking organic molecules with magnetic functional groups (MFG)^28–30 through exchange coupling between the magnetic ions of the MFG and the electrons of the organic molecules.^31–34 This magnetic exchange coupling can be either ferromagnetic or antiferromagnetic (ferrimagnetic). However, in both cases, the resulting spin moments induced in the organic material may exhibit either ferromagnetic or antiferromagnetic (ferrimagnetic) coupling, depending on the specific interactions within the system. In addition to the above methods, magnetic signals induced by photoexcitation of an organic-based semiconductor were also observed.³⁵ 

The viability of OS is assessed by whether a device can generate a significant magnetic signal through electric induction. The spin-polarized current ratio (SPR), which measures the ratio of different spin-charge currents, effectively indicates magnetic materials with a magnetic order. It has been evaluated to 100% when the organic ferromagnet is connected to metallic electrodes as a good spin filter.³⁶ Even by changing the electrodes to ferromagnetic metallic ones, significant magnetoresistance can be produced.³⁷ In this study, we propose a magnetic polymer multilayer field-effect transistor (FET),^38–41 as depicted in Fig. 1, left. In this design, we note that polymer chains are horizontally oriented and coupled to metallic leads so that the current is fully controlled only by the on-chain charge transfer process. To stabilize the spin states, the length of chains is below 10 nm, i.e., below the threshold of the spin diffusion length, which is commonly several tens of nanometers.¹⁵ However, when the source-to-drain voltage $V SD$ is applied, the length L of the chain can be even longer due to the following reasons. The travel time along the chain equals to $τ= L 2 μ V SD$ (⁠ $μ$ is the charge mobility). Taking just the lowest common experimental value for the source-to-drain voltage $V SD=0.1 V$ in FETs in the Ohmic regime and possibly the lowest values of the bulk mobility in organic systems $μ= 10 − 4 c m 2 V − 1 s − 1$⁠, we find for the travel time $τ$ at the length of 10 nm the value $τ= 10 − 7 s$⁠, which is by an order of magnitude lower than the SRT being at least $1μ s$⁠.¹⁵ More, for high on-chain mobilities of ca. $μ=1 c m 2 V − 1 s − 1$⁠, the length of the chain L could be almost by two order of magnitudes longer and no spin relaxation would occur.

For respective polymer chains, we assume that MFG are attached to the polymer backbone (Fig. 1, right). However, since the polymer host is inherently non-magnetic, the magnetic coupling between the polymer and the MFG is expected to be relatively weak, resulting in a weak Zeeman effect and small spin moments within the polymer. Depending on the structure of the polymer backbone, the polymer can exhibit either ferromagnetic or antiferromagnetic (ferrimagnetic) behavior. Due to the long SRT in both ferromagnetic and antiferromagnetic (ferrimagnetic) polymers, a substantial SPR is expected when a source-to-drain voltage $( V SD)$ is applied. Contrary to OS materials exhibiting the non-zero SPR due to their chirality, we show that the SPR is formed intrinsically as a consequence of the antiferromagnetic (ferrimagnetic) order. A pronounced SPR is observed within a narrow range of gate-source voltages $( V GS)$ and can reverse signs with changing of the $V GS$ polarities, highlighting its strong potential for applications in spintronics.

In addition to the introduction, the theoretical section outlines the self-consistent approach for solving the quantum theory-based distribution of induced charges, polymer unit displacements, and the electric potentials along each polymer layer. Simultaneously, the coupling among layers and influence of the gate voltage is described by means of the classical physics in accordance with their macroscopic character and lower energy of mutual interaction. Furthermore, the Keldysh formalism was employed to compute the spin-dependent currents for the evaluation of the SPR.^42,43 The results and discussion section provide a thorough interpretation of the calculation outcomes. Finally, a clear conclusion is presented in the closing section.

The nanoscaled multilayered field-effect transistor (FET) we will investigate is schematically drawn in Fig. 1. When such FET system is in an on-regime, the following phenomena take place:

(i) The applied source-to-drain voltage $V SD>0$ switches-on the current of charge carries along the chain (the so-called x-direction). The local intra-unit Coulomb (Hubbard) interaction and the dimerization effect on the backbone induce a spin resolution of charges. This effect is amplified by the spin–spin interaction between itinerant spin-charges on repeat units and induced magnetic moments on functional side-groups. Due to the partial non-homogeneity of the spin-charge distribution near both ends, also the regular dimerization of bonds partially changes by the charge-lattice coupling. Further, as the length of chains is taken below their conjugation length, the on-chain kinetic should be treated fully by means of the quantum mechanics.

(ii) The applied gate-to-source voltage $V GS<0$ brings two effects. First, it induces the charge carries (holes) inside the whole semiconducting sample. For the second, it forms the thin conducting channel near the polymer semiconductor—insulator boundary with the strongest gradient of both charge concentration and electric field in the vertical (so-called z−) direction, perpendicular to the insulated gate electrode. However, in the steady state, both diffusion current stemming from the charge concentration gradient and oppositely oriented drift current created by the gate voltage fully compensate each other in the vertical (z−) direction. Hence, we can restrict the charge motion just to the x−, i.e., source-to-drain, direction. Next, as both the diffusion coefficient $D z$ and the mobility $μ z$ in the vertical direction are given by the inter-chain charge transfer processes, they are controlled by the thermally activated hopping kinetics.

Since the on-regime of the FET assumes both drain and gate voltages applied simultaneously, the spin-resolved charge and current distribution should be solved self-consistently, together with the total charge density and the vertical charge distribution given by the gate voltage as well as with the on-chain spin-charge delocalization controlled by the drain voltage and coupled with the lattice displacement. Such theoretical description of the operating device is given in the following paragraphs.

We can, thus, directly see that the discrete sets of induced charge densities $δn( k , i)$ and potentials $V( k , i)$ inside the sample are unambiguously externally controlled by potentials at the source, drain, and gate contacts.

The equation means that the electrons hop between nearest-neighbor on-chain repeat units (sites), i.e., $i↔i+1$⁠, through the transfer integral $t k i ; k i + 1$ and simultaneously interact with phonons, represented in the adiabatic representation. Summation goes over all N sites and both spin orientations. The second term in the square brackets represents the phonon coupling with electrons, i.e., the electron–phonon interaction, where $α$ and $u k i$ are the phonon coupling constant and the displacement of the ith repeat unit, respectively. The third term in the square brackets, $( − 1 ) i t e$⁠, is employed to break the energy’s degeneracy, which is also used to simulate the regular alteration of single and double bonds in the polymer chain. $c ^ k i σ +$ and $c ^ k i σ$ are the electron creation and annihilation operators acting on the ith site with the spin $σ$⁠. It is worth mentioning that the electron–phonon Hamiltonian includes any possible phonon modes without explicitly including their eigenfrequency so that they simulate the effective reaction coordinate of phonons.

We also note that the last “additive” term in Eq. (26) guarantees the conservation law of the spring length. Eigenvalues, eigenvectors, and ensemble of displacements ${ u k i}$ in Eqs. (22)–(26) are coupled together, and we must solve them self-consistently.

The numerical calculation proceeds as follows. First, from Eqs. (7)–(11), we obtain profiles of local potentials $V( k , i)$⁠, which will be fixed throughout the calculation. Then, the local charge densities $δn( k , i)$ will be estimated from Eq. (6). These values will be taken as an initial input for the mean densities $δn( k , i)= n k i ↑+ n k i ↓$ at repeat units. We then assume that the local densities can partly delocalize along the chain due to the quantum processes. For these processes, we take an eigensolution to Eqs. (22) and (23). We also note that during the charge delocalization, the total sum of the on-chain charges $δn( k , i)= ∑ i⁡( n k i ↑ + n k i ↓)$ is conserved so that we can adjust, accordingly, the Fermi energy $ε f(k)$ for each chain separately. Having determined the expansion coefficients $ϕ λ ; k i σ$⁠, we also determine displacements ${ u k i}$⁠, as well as new mean values for $σ( k , i)$⁠, $S( k , i),$ and $δn( k , i)$⁠. We procced this iterative procedure until we obtain self-consistency.

We remark that due to the symmetry of the Hamiltonian with respect to spin orientations $( σ =↑ or ↓)$⁠, the solution is ambiguous against the spin orientation $↑↔↓$ inversion. Thus, the simulation below always corresponds to one selected branch of the bi-stable solution. More, in the model Hamiltonian, we neglected potential coupling of spins between respective chains. This coupling contributes to the formation of domains with an identical “choice” of the polarization in the bulk. Presented results below then show dependences on applied voltages $V G S$ and $V S D$ within such domains. During calculations, values of applied voltages are continuously changed and the selected spin orientation of previous calculation is taken as the initial input for the next set of voltages. In this way, the initially chosen sign of the bulk polarization is preserved for all calculations performed for different values of voltages. Thus, the obtained results correspond to common experiments, where a time window between respective measurements is faster than the relaxation of the spin-charge distribution.

For the numerical simulation, we set the following parameter values. The dimensions $a u, u z$⁠, $t o x,$ and $S a$ in the field-effect transistor are 0.15 nm, 1.6 nm, 50 nm, and 0.24 nm², respectively. The number of repeat units is 50, corresponding, thus, to the length of respective chains of ca. 7.5 nm. The number $n L$ of stacked chains is 30. The permittivities of the insulating oxide $ϵ I$ and the polymers $ϵ s$ are 3.9 and 4.2 times the vacuum permittivity $ϵ 0$⁠, respectively. Unless otherwise stated, the parameters $α$ and $k s$ in the electron–phonon interaction are 2 eV/Å and 20 eV/Ų, respectively. The temperature in all calculations is set to 300 K. We also assume that the conducting chain contains a small monomer intrinsic density of holes δ_h. Both values of the spectral function $Γ R$ and $Γ L$ are set to $1× 10 − 2 eV$⁠. The values of the spin–spin J_s and the intra-site Coulomb interaction U parameters used below are introduced in $eV n m 6$ units.

Spin-charge distribution in each polymer layer in the multilayer FET is formed through the following effects. We observed that the spin moments along the polymer layer alternate ferrimagnetically, resulting in a net spin moment. This alternating alignment is favored because ferrimagnetic states can evolve through the second-order perturbation in $t/ U$ coupling, thereby reducing the total energy. The edge effect is pronounced due to the finite length of the polymer, with spin moments being most prominent near the polymer edges. This pronounced edge effect is caused by the electron localization, which enhances the spin moments. Away from the edges, the spin-charges become more delocalized, the spin moments decrease due to finite intra-site Coulomb (Hubbard) interaction $U$ within the polymer units, reducing the overall Coulombic energy. As expected, the spin moments are influenced by the on-chain interaction couplings within each polymer layer, such as electron–phonon and intra-site Coulomb couplings, as shown in Figs. 2(a) and 2(b). Additively to that, we also note a significant influence of the magnetic coupling of repeat units with magnetic functional groups [Fig. 2(c)] on the spin moments. Figure 2(a) demonstrates that the electron–phonon interactions diminish the amplitude of ferrimagnetic order of spin moments along the chain. Figure 2(b) shows that the intra-site Coulomb interaction $U$ enhances both the magnitude of the spin moments and the net spin moment because the coupling $U$ increases the energy split between the up- and down-spin electrons. Figure 2(c) shows that the magnetic coupling $J s$ between repeat units and side-groups increases the spin moments both near and far from the edges, resulting in a significant enhancement in the magnitude of the alternating spin moments. This is expected as that the mean-field Hamiltonian of the magnetic coupling is invariant against a change of the sign of the coupling constant $J s$ (see above). Thus, this coupling will amplify the local magnitude of the on-chain spins, but this cannot change the antiferromagnetic order between nearest-neighbor spins, where the antiferromagnetism arises from dimerization of the chain. The pronounced antiferromagnetic spin moments observed for a larger $J s=0.2$ result, thus, from the competition between $J s$ and $U$ on one side and the electron–phonon coupling $α$ on the other side [note its opposite effect in Fig. 2(a)]. Notably, the spin moments farther from the edges are primarily dominated by magnetic coupling $J s$ rather than by the intra-site Coulomb interaction $U$⁠. The electron density, closely linked to the spin moments, can be controlled either intrinsically or by the gate voltage $V G S$⁠, with positive and negative $V G S$ corresponding to electron and hole induction, respectively. Obviously, at finite $U$ and $J s$⁠, the amplitudes of spin moments increase with the gate-controlled electron density (lesser hole density), as shown in Fig. 2(d) for intrinsically generated hole density $δ h=0.02$⁠. The role of the on-chain couplings studied above can be also estimated in the current–voltage characteristics.

For the output curves $( I vs V SD)$ in Fig. 3(a), we see that the slope (conductivity) significantly decreases with the electron–phonon coupling $α$⁠. We can attribute this to the formation of the dynamical energy disorder by the lattice interaction. For the intra-site Coulomb (Hubbard) coupling, we note that the conductivity only slightly decreases with it [Fig. 3(b)]. Here, two effects compete with each other. While increased ferrimagnetism promotes the alteration of the spin distribution, which promotes the on-chain transfer of spin-charges due to the Pauli-exclusion principle, the locally increased values of the Hubbard terms contribute to the local energy disorder, in turn. Increasing conductivity with the values of the magnetic coupling $J s$ [Fig. 3(c)] may look as counter-intuitive, as it is the coupling of itinerant spin-resolved charges with the side-groups. However, we remind that this coupling promotes the formation of the on-chain ferrimagnetic state, which favors, due to the Pauliexclusion principle, the on-chain transfer of spin-resolved charges between occupied and unoccupied orbitals.

In Fig. 4, we calculated the output curves for different intrinsic densities of holes or values of the gate voltage. While in the first case, the hole density changes uniformly in the volume, and in the second case, the sample could be macroscopically charged both positively and negatively; however, due to high intrinsic on-chain hole densities, the sample would be always hole conducting even for $V G S>0$⁠. In fact, for $V G S>0$⁠, the highest hole concentration is at the top, while for $V G S<0$⁠, it is at the bottom of the sample. In Fig. 4(a), we could easily see that the conductivity (slope) is notably non-linear with respect to the intrinsic hole densities. Namely, it superlinearly increases with the hole densities. This non-linearity can also be anticipated from Fig. 4(b). We see that the highest conductivity is for the highest absolute value of $V G S$⁠, i.e., the highest cumulation of holes either at the top or bottom of the sample. Profiles are almost symmetric against $V G S↔− V G S$ inversion, i.e., induction of small charge densities by the gate voltage is less important (4) than the hole redistribution across the sample. Thus, this indicates that the conductivity is a convex function of the charge density.

The above-mentioned non-linearity (superlinearity) of the conductivity on the hole concentration means that the on-chain charge carrier mobility becomes dependent on (increasing with) the charge carrier concentration. Namely, within a macroscopic approach, the total current $I$ in thin film FETs becomes $I= w L V SD ∫ 0 Z L μ ( ρ ( z ) ) ρ ( z ) d z$⁠. When the mobility $μ$ becomes independent of the charge density $ρ$⁠, the current $I$ turns out to be directly proportional to the total surface charge density $∫ 0 Z L ρ ( z ) d z$⁠, i.e., the conductivity scales linearly with the “surface charge density.” Instead, we observed a notable superlinear dependence of the conductivity $( I V SD )$ on the charge density. This indicates that $∂ μ ∂ ρ>0$⁠. As the charge density in FETs is also controlled by the gate voltage $U SG$⁠, we should have also $∂ μ ∂ U SG>0$⁠. Such mobility dependences on the gate voltage in semiconducting polymers were also found experimentally in FETs,⁴⁴ where the mobility of charges was explicitly controlled by the gate voltage. Increasing mobility dependences with the charge concentrations were explained theoretically⁴⁵ by the following model. The energy band of highly delocalized on-chain orbitals is broadened due to the partial energy disorder in local potentials and transfer integrals. This broadening creates the so-called deep energy trap tail states. When polymer chains are coupled to metallic leads, the grand canonical ensemble is formed and holes, first, occupy these deep energy trap states with very low mobility. Upon increasing hole concentrations, higher energy states (less disordered) with higher mobility are then populated. Thus, with an increasing concentration of mean density of holes, mean mobility of holes will increase. In our case, the energy disorder states are controlled by the disorder in transfer integrals (vibration-controlled dimerization of bonds and the ferrimagnetic order) and in local potential by the Hubbard term with the ferrimagnetic order. Similarly, the correlation between conductivity and the spin order was recently shown by the dependence of the resistivity $R ISHE$ (inverse spin-Hall effect) on the gate voltage.²¹ 

In Fig. 5, we studied the SPR dependence on the applied source-to-drain voltage $V S D$ in the Ohmic limit. We see that SPR generally slightly decreases with the applied voltage $V S D$⁠, except some regions of the value of the coupling constant $J s$⁠. The formation of the SPR for the antiferromagnetic systems was shown to be controlled by the local translation symmetry breaking.⁴⁶ In our case, the latter is amplified by the finite length of polymer chain with the influence of both ends on the spin-charge distributions, the bonds dimerization and the electron–phonon coupling $α$ within the transfer integral term, see Eq. (14). In Fig. 5(a), we see that the increasing value of the electron–phonon coupling increases the SPR, further we see that this dependence is fully anticorrelated with changes in conductivity in Fig. 3(a). As expected, we see the Hubbard coupling $U$ amplifies the spin resolution of SPR in Fig. 5(b); however, it only slightly decreases the conductivity in Fig. 3(b), because increased local energy disorder due to the Hubbard coupling is partly compensated by increased amplitudes of the ferrimagnetic order in Fig. 2(b), which promotes the spin-charge transfer between occupied and unoccupied orbitals. Concerning the magnetic coupling $J s$ between repeat units and the side-groups, the situation is more complicated. The coupling $J s$ increases the SPR only up to some value. For higher $J s$⁠, SPR decreases. Anyway, it increases magnitude of local ferrimagnetic order in Fig. 2(c) so that it promotes the spin-charge transfer between occupied and unoccupied orbitals, with a high superlinear increase of the conductivity in Fig. 3(c). Nontrivial dependences of SPR on $V S D$ for different $J s$ can be explained by competing effects within density-of-states (DOS) for almost half-filled orbitals. The formation of the magnetic order in low-dimensional carbon-based nanostructures was found to be controlled with the formation of so-called zero-energy in-bandgap states.⁴⁷ The local static and dynamic energy disorder contribute to the formation of discrete in-bandgap states within the DOS. Such trap states have opposite impact on the formation of $S P R$ and the charge carrier mobility.

For almost half-filled system due to the relatively low hole concentration, the Fermi energy becomes close to these spin-polarized (zero-energy in-bandgap) states. By tuning the gate voltage, we will not only change the total mean on-chain hole density (intrinsic + gate induced) in the sample, but we can considerably change their vertical distribution between respective layers.

The so-called transfer curves (I vs V_GS) are shown in Fig. 6. We see that when the absolute value of the gate voltage satisfies $| V G S|≤4 V$⁠, the smooth profile of the total current changes into the region with local resonant minima, which are sensitive to interaction parameters $α$⁠, $U,$ and $J s$⁠. However, inside this region, the current rather increases with |V_GS| so that sets of local resonant minima are close to $V G S≈0$⁠. We also note that the current $I$ seems to be partly symmetric against the polarity inversion of the gate voltage. In this region, the current decreases with the electron–phonon coupling $α$⁠, while it increases with the magnetic coupling $J s$ for negative gate voltage $V G S$⁠, but it decreases with the magnetic coupling $J s$ for positive gate voltage $V G S$⁠. On the other hand, for regions with $| V G S|> 4 V$⁠, the current $I$ significantly decreases with |V_GS| for $| V G S|> 4 V$⁠. In Fig. 7, we see that these changes in the current (Fig. 6) are strongly correlated with the formation of the $S P R≠0$⁠, which appears when $| V G S|≤ 4 V$⁠. Here, we note a remarkable phenomenon, when for low negative gate voltage, the $S P R<0$⁠, while for low positive gate voltage $S P R>0$⁠.

To avoid misinterpretation for the change of the sign of the $S P R$⁠, we calculated in Fig. 8 the mean value of the total spin $σ t≡ ∑ k i σ ^ k i$ inside the sample. We see that during change in polarity of the gate voltage $V G S$⁠, the total spin does not change its sign, but it increases continuously. More, near values of $V G S≈0$⁠, the change in $σ t$ is lower. To understand, how it is possible, we remind the definition of $S P R$ in Eq. (28).