Introduction

Implementation of molecules in integrated circuits (IC) offers great advantages due to extreme miniaturization and perfect reproducibility1,2,3. But despite long-term and intensive efforts since its origin in the early 70s4, molecular electronics (ME) has not yet presented any experimentally realized candidate to replace the silicon transistor as a “wheel-horse” of the modern IC industry. High expectations were held and are still in place with graphene5 and post graphene organic Dirac materials6. During past period ME mainly concentrated on the attempts to reproduce typical elements of silicon electronics7,8,9,10,11,12. In the case of graphene and related materials this approach has been based on the efforts to develop band opening methods13, which, however, haven’t resulted yet in a new IC technology either. On the other hand, due to complex geometry and topology of molecular structures one could expect that the devices with working principles different from the ordinary field-effect and bipolar transistors, could be designed.

The energy spectrum of a molecule manifests itself in transport phenomena by means of resonances. If the molecule possesses different carrier paths, destructive interference can result in formation of asymmetric Fano-Feshbach resonance14, which combines a resonance (transmission peak) and an antiresonance (transmission dip) nearby. Existence of the interference effect in transport through molecules, which is intensively discussed in the literature15,16,17,18,19, is now well established experimentally20,21,22,23. In ref.24 quantum interference transistor (QIT) was described with the “off” state corresponding to perfect interference destruction of both transmission and current. One of the main challenges in CMOS electronics is reduction of the operating voltage that doesn’t follow Moore’s law (ITRS 2.0). In ref.25 it was argued that the interference control of the carrier transport over different paths can substantially reduce the operating gate voltage, because the suppression of the transmission function can be achieved at lower gate voltage compared with the one required to move the transmission function peak away from the distribution function window. However, antiresonances, which arise from the destructive quantum interference (DQI), are determined by the topology of the structure that includes different interfering carrier paths. Hence, variation of the on-site potential and/or intersite hopping can only shift the antiresonance in energy rather than destroy it, because interfering paths are retained under such variations. The voltage required to shift an antiresonance away from the operating energy region is determined by the carrier distribution in the leads on a scale no less than kT and, hence, is not small. Therefore, the proposed control of the transmission resonance by low voltages should rely on a mechanism more complex than multipath interference solely. For a logical gate to operate, its constituting elements (transistors) should undergo transitions between the “off” and the “on” states, with the latter state being even more important than the former one as it provides switching of the successive gate. The “on”/“off” ratio for the transistor conductance should be as high as possible to provide a reliable gate operation. However, this requirement is scarcely achievable in quantum interference transistors operating near the antiresonance because of the low transmission away from the antiresonance26. Hence, a quantum transistor is required, which possesses a combination of antiresonance and nearby resonance that is responsible for high conductance in the “on” state.

In this paper we show that, indeed, the transmission probability of a special class of molecules can be controlled in a wide range by applying small gate voltages. This phenomenon can be easily understood with a deep connection between \({\mathscr{P}}{\mathscr{T}}\)-symmetry and scattering problems27,28. Coupling of spatially symmetric molecule to electrodes results in \({\mathscr{P}}{\mathscr{T}}\)-symmetry breaking, which is accompanied by coalescence of resonances29 at the exceptional point of an open quantum system comprised of the molecule and the electrodes28,30 and transmission decrease. This effect is enhanced by the shift of Fano-Feshbach antiresonance to the EP point. The mentioned special class of molecules consists of systems with degenerate energy levels, e.g. diradicals31,32,33 (but not restricted to), which possess mirror symmetry.

Results

Phenomenological model

Consider an open quantum system comprised of a molecule and contacts that possesses EP in a sense of ref.28. At this EP two unity resonances coalesce and cancel each other making the transition to the “off” state very sharp. An open quantum system should be spatially symmetric in order to possess EP. To take advantage of both DQI and coalescence of resonances at the EP one should consider a system with two resonances and one antiresonance. The transmission coefficient of an arbitrary two-terminal quantum system can be written in the compact form28,30:

$$T(\omega )=\frac{|P(\omega {)|}^{2}}{|P(\omega {)|}^{2}+|Q(\omega {)|}^{2}}.$$
(1)

Here P(ω) and Q(ω) are some functions of an energy ω. Real zeroes of function P(ω) correspond to transmission nodes (antiresonances), while real zeroes of function Q(ω) determine exact positions of perfect (unity) resonances on the energy axis28,30. In the vicinity of the resonances and antiresonance P(ω) and Q(ω) can be expressed as28:

$$\begin{array}{rcl}P(\omega ) & = & 2{\rm{\Gamma }}B(\omega -{\varepsilon }_{0}){D}_{P},\\ Q(\omega ) & = & (\omega -{\varepsilon }_{1}^{+})(\omega -{\varepsilon }_{1}^{-}){D}_{Q},\end{array}$$
(2)

where ε0 and \({\varepsilon }_{1}^{\pm }\) determine exact position of the transmission antiresonance and resonances, correspondingly, Γ is the imaginary part of the contact self-energy describing interaction of a molecule with the leads34,35 and B is some positive dimensionless coefficient. Factors DP and DQ take into account the contributions from the remote energy levels and can be estimated as DP ~ DQ ~ ΔN−2, where Δ is an average distance between the remote energy levels and N is the dimension of the molecular orbital Hilbert space. Phenomenologically, functions P(ω) and Q(ω) are defined up to an arbitrary common factor, hence, we can redefine the parameter \(B\mapsto B{D}_{P}/{D}_{Q}\) and replace three phenomenological parameters B, DP and DQ by just B. Further we will use B as such generalized parameter.

Consider a model that possesses degenerate antiresonance and resonance levels in the symmetric phase, which can be distorted by an external perturbation described by parameter δ. Energies of the antiresonance ε0 and resonances \({\varepsilon }_{1}^{\pm }\) can be expressed as:

$$\begin{array}{rcl}{\varepsilon }_{0} & = & {x}_{0}\delta ,\\ {\varepsilon }_{1}^{\pm } & = & {x}_{1}\delta \pm \sqrt{{y}^{2}{\delta }^{2}-{z}^{2}{{\rm{\Gamma }}}^{2}}.\end{array}$$
(3)

Here x0,1, y and z are some dimensionless parameters depending on the structure of a particular system. Terms in Eq. (3), which are linear in δ, describe the shift of the (anti)resonance positions due to the external perturbation and non-analytical term (square root) in the expression for \({\varepsilon }_{1}^{\pm }\) describes the coalescence of resonances phenomenon. Energy of the degenerate state (at δ = 0) is set to the energy origin. One should bare in mind that degenerate levels occur only in multiply-connected structures (in simple-connected, e.g. linear, all energy levels are non-degenerate). In such systems antiresonances naturally appear as well (P(ω) = 0). If the external perturbation δ is high enough (δ > δEP = zy−1Γ), then the transmission has two unity peaks at \(\omega ={\varepsilon }_{1}^{\pm }\), which coalesce at δ = δEP. This is the EP, which is associated with \({\mathscr{P}}{\mathscr{T}}\)-symmetry breaking. For small δ or, equivalently, for strong enough coupling with the leads (δ < δEP) the roots of Q(ω) in (2) are complex. Therefore |Q(ω)| is non-zero for any real energy ω and the transparency is always less than unity. The poorest transmission profile (i.e. the “off” state) corresponds to δ = 0. From Eqs (1–3) one can see, that in this case there are two peaks at ω = ±zΓ with

$${T}_{peak}(\omega =\pm \,z{\rm{\Gamma }};\delta =0)=\frac{{B}^{2}}{{B}^{2}+{z}^{2}},$$
(4)

separated by a zero dip at ω = 0.

Microscopic model

The above described phenomenological properties of the transmission coefficient can be realized within the following microscopic model. There are two degenerate states |1〉 and |2〉 with the same energy ε. This system is attached symmetrically to two leads (left and right) in such a way that the mirror symmetry operation σLR, which maps the left lead into the right one and vice versa, is also an element of the symmetry group G of the bare Hamiltonian of the system, i.e. σLR ∈ G. Due to the degeneracy, there must be an irreducible representation of the symmetry group G acting on the subspace \({{\mathscr{H}}}_{12}=Span(|1\rangle ,\,|2\rangle )\) of the total Hilbert space of states of the isolated system. Let us choose the basis in \({ {\mathcal H} }_{12}\) as the basis of a symmetric |s〉 and an anti-symmetric |a〉 states, which are the eigenstates of the reflection operator σLR: σLR|s〉 = |s〉 and σLR|a〉 = −|a〉. These states conserve their symmetry with introduction of the perturbation, which is invariant under σLR. The tunnelling matrix elements between the leads and the symmetric state are of the same sign, whereas, the tunnelling matrix elements between the leads and the anti-symmetric state are of opposite signs (see Fig. 1a). Therefore in this basis couplings to the leads can be written as

$$\begin{array}{ccc}{{\bf{u}}}_{L} & = & \sqrt{{\rm{\Gamma }}}(\begin{array}{c}{\gamma }_{s}\\ {\gamma }_{a}\end{array}),\\ {{\bf{u}}}_{R} & = & \sqrt{{\rm{\Gamma }}}(\begin{array}{c}{\gamma }_{s}\\ -{\gamma }_{a}\end{array}).\end{array}$$
(5)

Here Γ governs the coupling strength and positive dimensionless parameters 0 ≤ γs,a ≤ 1 describe relative couplings of symmetric and anti-symmetric states to the leads. Parameters γs,a can be calculated, for example, as projections of the vector \({{\rm{\Gamma }}}^{-\mathrm{1/2}}{{\bf{u}}}_{L,R}^{site}\) onto |s〉 or |a〉 respectively, where \({{\bf{u}}}_{L,R}^{site}\) describes the coupling to the leads in the site (atomic orbitals) basis. If each lead is attached to only one site, then γs,a is just a contribution of the state localized in the connection site to the symmetric or anti-symmetric state correspondingly (see Supplementary Materials).

Figure 1
figure 1

Microscopic model and its transmission coefficient. (a) Schematic view of the microscopic model of the molecular system depicting symmetric |s〉 and anti-symmetric |a〉 states connected to the leads by couplings (5). Γ is set as energy unit and ka = −ks = 1 for convenience. Evolution of the transmission coefficient profile with variation of δ (γs is set to 1 and ka = −ks = 1) for (b) some discrete values of δ for γa = 0.9, and (c–e) in the form of density plots for (c) γa = 1, (d) γa = 0.5 and (e) γa = 0.1. Red solid lines indicate the position of perfect resonances and dashed cyan–zeros of the transmission.

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Application of the gate voltage introduces external perturbation that lowers the symmetry of the system, resulting in removal of the degeneracy. Suppose that the external perturbation lowers the symmetry of the system from the group G to its some non-trivial subgroup H ⊂ G, such that σLR ∈ H. This perturbation introduces detuning of the energy of symmetric and anti-symmetric states: εs,a(δ) = ε + ks,aδ with δ > 0 and dimensionless parameters −1 ≤ ks,a ≤ 1 accounting for the different influence of the perturbation on the energies of symmetric and anti-symmetric states (see Fig. 1a). Parameters ks,a can be estimated, for instance, from the perturbation theory (see Supplementary Materials for details); note that ka ≠ ks as the considered perturbation removes the degeneracy. Assume that couplings (5) are not affected by this perturbation. In fact, Γ and γs,a are some smooth functions of the perturbation strength, i.e. δ. However, taking this into account does not change the qualitative picture described below. The transport through the states |s〉 and |a〉 (neglecting the contribution from remote states to the transport process) within the wide-band limit35 can be described by the transmission coefficient in the form (1) with the following P and Q functions (see Methods section):

$$\begin{array}{rcl}P(\omega ) & = & 2{\rm{\det }}(\omega \hat{I}-{\hat{H}}_{0})\times {{\bf{u}}}_{L}^{\dagger }{(\omega \hat{I}-{\hat{H}}_{0})}^{-1}{{\bf{u}}}_{R},\\ Q(\omega ) & = & {\rm{\det }}(\omega \hat{I}-{\hat{H}}_{aux}).\end{array}$$
(6)

Here

$${\hat{H}}_{0}=(\begin{array}{cc}\varepsilon +{k}_{s}\delta & 0\\ 0 & \varepsilon +{k}_{a}\delta \end{array})$$
(7)

is the bare Hamiltonian of the system (without taking the leads into account),

$${\hat{H}}_{aux}={\hat{H}}_{0}+i{{\bf{u}}}_{L}{{\bf{u}}}_{L}^{\dagger }-i{{\bf{u}}}_{R}{{\bf{u}}}_{R}^{\dagger }=(\begin{array}{cc}\varepsilon +{k}_{s}\delta & 2i{\rm{\Gamma }}{\gamma }_{s}{\gamma }_{a}\\ 2i{\rm{\Gamma }}{\gamma }_{s}{\gamma }_{a} & \varepsilon +{k}_{a}\delta \end{array})$$
(8)

is the non-Hermitian auxiliary Hamiltonian with its real eigenvalues corresponding to energies of perfect transmission28, and \(\hat{I}\) is the 2 × 2 identity matrix. Hamiltonian \({\hat{H}}_{aux}\) is \({\mathscr{P}}{\mathscr{T}}\)-symmetric, where \({\mathscr{P}}={\sigma }_{LR}\) denotes to the mirror reflection and \({\mathscr{T}}\) is the time reversal operator (complex conjugation). Indeed, one can easily check that operator \({\mathscr{P}}{\mathscr{T}}{\hat{H}}_{aux}{\mathscr{P}}{\mathscr{T}}\) acts on any vector \({\bf{v}}\in {{\mathbb{C}}}^{2}\) in the same way as operator \({\hat{H}}_{aux}\). Thus, \({\hat{H}}_{aux}\) is \({\mathscr{P}}{\mathscr{T}}\)-symmetric36. Therefore, it can possess real eigenvalues, which correspond to perfect transmission peaks, and for certain parameters they can coalesce and the \({\mathscr{P}}{\mathscr{T}}\)-symmetry of the Hamiltonian \({\hat{H}}_{aux}\) will be broken, leading to coalescence of perfect resonances into one peak with amplitude lower than 1. Moreover, such resonance coalescence is accompanied by symmetry breaking of electron occupation at the energy corresponding to the transmission peak (see Methods for details).

Using Eqs (5–8), the transmission coefficient can be written as:

$$T(\omega )=\frac{4{{\rm{\Gamma }}}^{2}{[({k}_{a}{\gamma }_{s}^{2}-{k}_{s}{\gamma }_{a}^{2})\delta -({\gamma }_{s}^{2}-{\gamma }_{a}^{2})(\omega -\varepsilon )]}^{2}}{4{{\rm{\Gamma }}}^{2}{[({k}_{a}{\gamma }_{s}^{2}-{k}_{s}{\gamma }_{a}^{2})\delta -({\gamma }_{s}^{2}-{\gamma }_{a}^{2})(\omega -\varepsilon )]}^{2}+{\{{[\omega -\varepsilon -\frac{1}{2}\delta ({k}_{a}+{k}_{s})]}^{2}+4{\gamma }_{s}^{2}{\gamma }_{a}^{2}{{\rm{\Gamma }}}^{2}-{\delta }^{2}\frac{{({k}_{a}-{k}_{s})}^{2}}{4}\}}^{2}}.$$
(9)

From this formula one can see that for a sufficiently large detuning δ there are two unity peaks of transmission (zeros of Q, i.e. eigenvalues of \({\hat{H}}_{aux}\)) at \(\omega =\varepsilon +\frac{1}{2}\delta ({k}_{a}+{k}_{s})\pm \sqrt{\frac{1}{4}{\delta }^{2}{({k}_{a}-{k}_{s})}^{2}-4{\gamma }_{s}^{2}{\gamma }_{a}^{2}{{\rm{\Gamma }}}^{2}}\). Decreasing the detuning one can achieve the coalescence of resonances at δ = 4γsγaΓ|ka − ks|−1, which corresponds to the EP of \({\hat{H}}_{aux}\). Further decreasing δ results in further lowering of the transmission coefficient peak. There is also a zero-valued antiresonance (zero of P) at \(\omega =\varepsilon +\delta ({k}_{a}{\gamma }_{s}^{2}-{k}_{s}{\gamma }_{a}^{2})/({\gamma }_{s}^{2}-{\gamma }_{a}^{2})\), which additionally lowers the transmission with decreasing δ. One can see that, moving the energy origin to ε, Eq. (9) takes the phenomenological form described by Eqs (1–3) with the following phenomenological parameters:

$$B=|{\gamma }_{s}^{2}-{\gamma }_{a}^{2}|\,{x}_{0}=\frac{{k}_{a}{\gamma }_{s}^{2}-{k}_{s}{\gamma }_{a}^{2}}{{\gamma }_{s}^{2}-{\gamma }_{a}^{2}},\,{x}_{1}=\frac{{k}_{a}+{k}_{s}}{2},\,y=\frac{|{k}_{a}-{k}_{s}|}{2},\,z=2{\gamma }_{s}{\gamma }_{a}.$$
(10)

Thus, according to Eq. (4) one can see that the poorest transmission peaks (at δ = 0) are

$${T}_{peak}(\omega =\varepsilon \pm 2{\gamma }_{s}{\gamma }_{a}{\rm{\Gamma }};\delta =0)=\frac{{({\gamma }_{s}^{2}-{\gamma }_{a}^{2})}^{2}}{{({\gamma }_{s}^{2}+{\gamma }_{a}^{2})}^{2}}.$$
(11)

From (10) we see that there is a limiting case γs/γa → 1 that results in B → 0 and x0 → ∞, while the product \(B{x}_{0}\to {\gamma }_{s,a}^{2}({k}_{a}-{k}_{s})\) remains finite. In this case complete opaqueness, i.e. T ≡ 0, can be obtained for δ = 0. In practice, however, the transmission never vanishes because of the transport through remote energy levels, which are not taken into account in this model. Evolution of the transmission coefficient profile (9) with variation of δ for different ratios of the parameters γs and γa is illustrated in Fig. 1c–e. It is important to note that for a single-level system (i.e. γa = 0 or γs = 0), where the \({\mathscr{P}}{\mathscr{T}}\)-symmetry breaking is absent, Tpeak equals unity identically.

Quantum interference inverters based on \({\mathscr{P}}{\mathscr{T}}\)-symmetric interference transistors

Consider a quantum analogue of CMOS inverter consisting of two quantum switches, connected between one common output lead and two reference voltage sources with voltages Vref1 and Vref2, respectively. Input signal Vin is applied to the common gate of these switches, which is galvanically isolated from the system. Figure 2 depicts two examples of such quantum interference inverters. For a high-resistance load we can implicitly evaluate the voltage transfer characteristic Vout(Vin) of this inverter and estimate its maximum negative gain, which is achieved at \({V}_{in}=\frac{1}{2}({V}_{ref1}+{V}_{ref2})\) due to the symmetry (see Methods):

$$\begin{array}{rcl}{G}_{max} & = & G({V}_{in}=\tfrac{{V}_{ref1}+{V}_{ref2}}{2})\\ & = & \sinh \,\tfrac{e{\rm{\Delta }}V}{2kT}\times \tfrac{256{\alpha }^{2}{{\rm{\Gamma }}}^{2}{\gamma }_{a}^{2}{\gamma }_{s}^{2}{({k}_{a}-{k}_{s})}^{2}kTe{\rm{\Delta }}V}{[16{{\rm{\Gamma }}}^{2}{({\gamma }_{s}^{2}-{\gamma }_{a}^{2})}^{2}+{\alpha }^{2}{({k}_{a}-{k}_{s})}^{2}{e}^{2}{\rm{\Delta }}{V}^{2}][16{{\rm{\Gamma }}}^{2}{({\gamma }_{s}^{2}+{\gamma }_{a}^{2})}^{2}+{\alpha }^{2}{({k}_{a}-{k}_{s})}^{2}{e}^{2}{\rm{\Delta }}{V}^{2}]}.\end{array}$$
(12)

Here ΔV = Vref2 − Vref1 is fixed by the external supply voltage. In the saturation regime (\(e{\rm{\Delta }}V\gg kT\)) the maximum value of G grows exponentially with ΔV due to the factor \(\sinh \,\frac{e{\rm{\Delta }}V}{2kT}\). For \(e{\rm{\Delta }}V\lesssim kT\) (in the ohmic regime) it becomes independent of the temperature and we can estimate the minimum difference of the reference voltages (supply voltage) ΔVcrit needed to make the inverter operate, i.e. which provides Gmax = 1:

$${\rm{\Delta }}{V}_{crit}\approx \frac{4{\rm{\Gamma }}}{e\alpha |{k}_{a}-{k}_{s}|}\sqrt{4{\gamma }_{a}^{2}{\gamma }_{s}^{2}-{\gamma }_{a}^{4}-{\gamma }_{s}^{4}-2{\gamma }_{a}{\gamma }_{s}\sqrt{5{\gamma }_{a}^{2}{\gamma }_{s}^{2}-2{\gamma }_{a}^{4}-2{\gamma }_{s}^{4}}} \sim |{\gamma }_{s}-{\gamma }_{a}|\,{\rm{as}}\,\frac{{\gamma }_{a}}{{\gamma }_{s}}\to 1.$$
(13)

From Eq. (13) one can see that ΔVcrit can become infinitesimal as γa/γs → 1. On the other hand, however, Gmax remains bounded in the ohmic regime even if γa/γs → 1. From the analysis of Eq. (12) one can conclude that for \({\rm{\Delta }}V=4\frac{{\rm{\Gamma }}}{e\alpha |{k}_{a}-{k}_{s}|}\sqrt{|{\gamma }_{s}^{4}-{\gamma }_{a}^{4}|}\) the gain Gmax reaches its maximum: 2(γs/γa)2 for γs < γa or 2(γa/γs)2 for γs > γa. Hence, the steepest negative gain of the voltage transfer characteristic is limited to −2. Nevertheless, it is suitable for operation of the inverter.

Figure 2
figure 2

Diradical based quantum inverters. Structural model of diradical configuration of (a) trimethylenemethane molecule (non-disjoint) and (b) divinylcyclobutadiene molecule (disjoint). Schemes of quantum interference inverters, composed of two quantum switches based on (c) non-disjoint diradicals and (d) disjoint diradicals. Molecules are shown in the form of their Hückel theory tight-binding graphs corresponding to their carbon skeleton. Shaded regions indicate the atoms, which are electrostatically affected by the input gate.

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Model examples of real molecular structures

Possible candidates for a physical realization of the proposed quantum switch are molecules with degenerate states, e.g. diradicals31, which are already known for providing transmission antiresonances32. Moreover, linkers can stabilize diradical character of such molecules37. Hence, we can expect that connection of certain contacts to them will not destroy the degeneracy of the states, but rather stabilize it. Diradicals can be classified into two types: disjoint and non-disjoint depending on how their non-bonding orbitals intersect (i.e. whether they have common atomic orbitals or not). It was shown that simple starring procedure can distinguish between these two types38,39. Disjoint diradicals seem to be the most appropriate candidate for our quantum switch. Indeed, applying contacts to atoms comprising different degenerate orbitals means that symmetric and antisymmetric combinations of these orbitals will be connected to the leads by equivalent coupling strength, i.e. parameters γs and γa [introduced in Eq. (5)] in this case can be made equal (at least within the nearest neighbour tight-binding approximation). As was highlighted above, according to Eq. (11) this leads to zero conductance in the “off” state.

Operation principle of quantum interference inverter requires that one switch must be in the “on” state and another in the “off” state. There are two possible ways of dealing with this task. First of all, one can choose two different quantum systems (molecules) to make two quantum switches that is similar to the conventional CMOS, where there are two different types of transistors: n-channel MOS and p-channel MOS. This approach requires a technology of synthesis of two different molecules with strictly given parameters. On the other hand, we can use the same quantum system (molecule) to create both switches, but influence their spectrum in different ways by additional gates. This method needs only one type of molecule to be synthesised, but the introduction of additional gates results in some complication of the conventional technological process. In the following subsections we consider some schematic examples of quantum inverters with the same molecules in both switches. Different energies of the on-site atomic states are assumed to be achieved by a certain configuration of additional gates (e.g. backgates).

Model of non-disjoint diradical

The first example structure we consider is a model of the trimethylenemethane molecule, which is a non-disjoint diradical32. Schematically the quantum inverter structure composed of two such four-atomic (carbon skeleton) molecules is shown in Fig. 2a. Presented schematic model corresponds to a tight-binding Hückel structure of one of the resonance configurations of the trimethylenemethane, which is stabilized as it coincides in symmetry with the leads couplings. Hence, hopping integral τ is assumed to be greater than τ1 as it corresponds to a higher bond order. The transmission coefficient, phenomenological, and microscopical parameters of such switches are presented in the Supplementary Material.

We apply the reference voltages as follows: Vref1 = 0 and Vref2 = V0 is the supply voltage. The range of the input voltage, thus, is 0 ≤ Vin ≤ V0. Applied input potential changes only some on-site energies of the system (in the shaded region in Fig. 2a). We take this into account in the following form:

$${\varepsilon }_{\mathrm{1,2}}={\varepsilon }_{\mathrm{1,2}}^{0}+\alpha (e{V}_{in}-{\varepsilon }_{\mathrm{1,2}}^{0}).$$
(14)

The electrostatic influence of the reference and output leads can also be taken into account in a way similar to (14). It can be shown that this influence only distorts the voltage transfer characteristic and taking it into account is not obligatory to illustrate the operation principles of the quantum interference inverters.

Consider the following example: \({\varepsilon }_{1}^{0}=e{V}_{ref1}=0\), \({\varepsilon }_{2}^{0}=e{V}_{ref2}=e{V}_{0}\), α = 0.5, τ1 = 2.75 eV, τ = 3.65 eV and Γ = 1 meV. Energies are measured from the Fermi level of the first reference lead. Here we take typical values of the hopping integrals for single and double bonds (τ1 and τ correspondingly)40. Figure 3a shows voltage transfer characteristics of the inverter for V0 = 5 mV and Fig. 3b for V0 = 10 mV (by dot-dashed lines in both cases). In the latter case the voltage transfer characteristic is obviously better because of higher negative gain achieved.

Figure 3
figure 3

Numerically calculated voltage transfer characteristics for the quantum inverter based on \({\mathscr{P}}{\mathscr{T}}\)-symmetric interference transistors for room and zero temperature. Supply voltage is V0 = 5 mV (a) and V0 = 10 mV (b). The inverter operates at zero temperature better than at room temperature. Dashed black line shows the −1 slope for comparison.

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Model of disjoint diradical

Another example we consider is a model of the divinylcyclobutadiene molecule, which is a disjoint diradical32. Schematically the quantum inverter structure composed of two such molecules is shown in Fig. 2b. Presented model corresponds to a simple tight-binding Hückel structure of the divinylcyclobutadiene molecule with all bonds treated as equal, providing equal tunnelling matrix elements τ between p-orbitals of carbon atoms. The transmission coefficient, phenomenological, and microscopical parameters of such switches are presented in the Supplementary Material.

We assume that the applied input voltage changes only on-site energies in the shaded region in Fig. 2b, which is taken into account similarly to Eq. (14). Figure 3a shows voltage transfer characteristics of this inverter for V0 = 5 mV and Fig. 3b for V0 = 10 mV (by solid lines in both cases) for the following parameters: \({\varepsilon }_{1}^{0}=e{V}_{ref1}=0\), \({\varepsilon }_{2}^{0}=e{V}_{ref2}=e{V}_{0}\), α = 0.5, Γ = 1 meV, and typical value of the hopping integral τ = 3 eV for conjugated hydrocarbons40.

For higher supply voltage transfer characteristic of the inverter based on the disjoint diradical switches (solid lines in Fig. 3b) show higher maximum absolute value of the gain rather than for the inverter based on the non-disjoint diradical switches (dot-dashed lines in Fig. 3b). This is expectable as disjoint diradicals provide γs/γa = 1 and, thus, the “off”-state current of such switch becomes smaller (it differs from zero only due to the presence of the “background” transmission arising from remote resonance peaks). However, for smaller supply voltage (Fig. 3a), this “background” component may become high enough to cancel out the key benefit of the disjoint diradical (γs/γa = 1). Moreover, for disjoint diradicals the degeneracy is removed only in the second order of perturbation (i.e. ka, s become functions of δ). Hence, for lower supply voltages, the sensitivity to the gate voltage decreases compared to non-disjoint diradicals. In this case the transfer characteristic of a non-disjoint diradical based quantum inverter turns out to have slightly more gain (Fig. 3a). From Fig. 3 we see that for our particular parameters chosen only divinylcyclobutadiene can provide a working inverter at room temperature with 10 mV supply voltage.

Discussion

We have shown that there is a fundamental difference between resonant tunnelling through a non-degenerate and a doubly degenerate state. It arises from the effect of an antiresonance formation (because of destructive interference of electron flows through both degenerate states) and the coalescence of resonances, which can be well described by the concept of the \({\mathscr{P}}{\mathscr{T}}\)-symmetry breaking28. Thus, one can utilize this phenomenon and use the degenerate quantum system spectrum to construct the quantum interference switch. In comparison with numerous other proposals of molecular interference transistors we should emphasize that our solution has important advantages: it is an all-electrical device (i.e. electric current is controlled by an applied voltage), it can possess extremely low operating voltage (even at room temperature) and our \({\mathscr{P}}{\mathscr{T}}\)-symmetry based model provides a straightforward design rules for constructing such a transistor, which we demonstrated by the examples of specific disjoint and non-disjoint diradicals. Thus, this might be a way to dramatically lower the supply voltage, which now cannot be made lower than 0.5–1 V41 for the conventional silicon electronic devices, even for promising tunnel field-effect transistors (FET)42. Advanced technology of FETs with carbon nanotube (CNT) channel43 also provides a variety of advantages over the bulk Si electronics10,44, but sufficient reduction of the supply voltage is not among them. A separate approach is based on a development of non-electronic logical gates, based on, e.g., exciton45,46 or even heat flow47 control. Typically such devices has input and output signal of different nature and are designed for optoelectronic45 or optomechanical46 applications, rather than for large-scale integration.

Performance of real devices is limited by noise. It is especially significant for low supply voltages. Noise in quantum systems is not distinguished into thermal and shot, it is always a superposition of both and it can be described by a closed expression48. Nevertheless, it is illustrative to discuss these contributions independently. Shot noise spectral power is proportional to the current through the system and, thus, it becomes negligible as the voltages and, correspondingly, the currents are scaled down. On the other hand, at finite (room) temperature thermal noise can influence the transport dramatically. Thus, thermal noise is one of the limiting factors of lowering the supply voltage41.

The mean-square voltage uncertainty is \({\rm{\Delta }}{V}_{therm}=\sqrt{kT/C}\), where C is the capacitance of the load, which is typically the gate capacitance of the next switch. Therefore, using several molecules in parallel in the single switch and, consequently, a bigger gate contact, will increase the capacitance C and lower the noise. But, on the other hand, the greater C is, the worse switching rate ν ~ (τ)−1 = (RC)−1 can be achieved. Here the resistance R can be estimated from the current in the “on” state (see Supplementary Material): \(R\approx \frac{h}{{e}^{2}}\times \frac{2kT}{\pi {\rm{\Gamma }}}\). Assuming the gate is about 10 × 10 nm lying on 2 nm thick dielectric with k ≈ 5, we can estimate its capacitance to be about 2aF. This means that without noise taken into account we can expect switching frequencies to be of order of 500 GHz even for low conducting “on” state with R ≈ 1 MΩ (Γ ≈ 0.04 kT for room temperature). However, noise dramatically lowers the possible operating frequency. Indeed, restricting the minimum operating frequency νmin, one can estimate the minimum allowed supply voltage, which we take to be 8 times the noise voltage uncertainty to provide an error probability at about 10−1541. Finally we arrive at the following restriction:

$${V}_{0} > \sqrt{2\frac{h}{{e}^{2}}\times \frac{\mathrm{2(}kT{)}^{2}}{\pi {\rm{\Gamma }}}\times {\nu }_{min}},$$
(15)

which for the room temperature and Γ ≈ 1 meV gives \({V}_{0}\gtrsim 10\,{\rm{mV}}\times \sqrt{{\nu }_{min}}\), where νmin is in GHz. Thus, sub-kT/e supply voltages seems to be possible up to ν ≈ 7 GHz. More detailed analysis of noise impact on operation of quantum interference gates will be presented elsewhere, as well as consideration of technological parameter variation resulting in asymmetry of the inverter structure.

Our model is based on Hückel theory and does not account for electron repulsion. In ref.17 it has been shown that DQI predicted by Hückel model does survive in alternant hydrocarbons even with Coulomb effects being taken into account. Diradicals, which we consider, are alternant, hence, we expect that DQI will persist in a more realistic model either. On the other hand, a role of electron repulsion (Coulomb blockade), which is detrimental in the case of weak coupling of a molecule with contacts, is less significant in the case of strong coupling. We have performed tight-binding simulations with increased value of electrode-molecule coupling Γ = 0.1 eV and Γ = 0.2 eV (see Supplementary). We have shown that such a value of Γ still retains switching properties of interference transistor but makes operating voltage larger. Thus, the supply voltage cannot be made arbitrary low, as can be provided by an idealized model, but its lower bound should be determined from the optimal value for Γ, which, on the one hand, prevents the Coulomb blockade and, on the other hand, provides sharp enough transmission peaks resulting in sufficient contrast of the switching current. By no means the study of the interplay of electrode-molecule coupling strength and electron repulsion in DQI sensitive molecular transport based on ab initio simulations will be of great interest. In our model DQI acts in concert with \({\mathscr{P}}{\mathscr{T}}\)-symmetry breaking at the exceptional point, which is accompanied by coalescence of resonances and sharp decrease of electron transmission. \({\mathscr{P}}{\mathscr{T}}\)-symmetry breaking in quantum conductor possesses a close resemblance with equilibrium phase transitions in condensed matter28,30. Hence, we expect that the model proposed in our manuscript should retain efficiency with Coulomb interactions included due to the fact that it is related to the symmetry properties of the system. But the general impact of Coulomb interactions on the behaviour of a quantum conductor near the \({\mathscr{P}}{\mathscr{T}}\)-symmetry breaking transition at the exceptional point is quite a new interesting problem, which deserves special study.

It is also important to admit that we consider an idealized model of the quantum transport process within this paper. Nevertheless, a few steps toward a realistic description can be made, e.g. (see Supplementary): going beyond the wide-band approximation and taking into account electrostatic influence of the reference voltage sources. It turns out that qualitatively our results are valid in more realistic situations as well. However, in order to verify them, one should provide a full numerical calculation for the electronic structure of the considered system, which we plan to perform in our further publications.

At the moment practical realization of high scale integration of quantum molecular gates is beyond the reach of modern technology. However continuous progress in self-assembling methods and, especially, development of atomic precision lithography could make almost inevitable the implementation of molecular gates as building blocks of ICs. We hope that the idea of using degenerate energy levels (e.g. diradicals, but not obligatory) to create molecular switches could open a new field of research.

Methods

Microscopic model: auxiliary Hamiltonian and its exceptional point

Conductance of a quantum conductor is defined by its tunnelling transmission coefficient49, which can be calculated by the standard formula50 derived from the nonequilibrium Green’s function formalism34,51,52:

$$T=4Tr({\hat{{\rm{\Gamma }}}}_{R}{\hat{G}}^{r}{\hat{{\rm{\Gamma }}}}_{L}{\hat{G}}^{a}),$$
(16)

where \({\hat{G}}^{r,a}\) is the retarded/advanced Green function and \({\hat{{\rm{\Gamma }}}}_{L,R}={{\bf{u}}}_{L,R}{{\bf{u}}}_{L,R}^{\dagger }\)53 is the coupling matrix (imaginary part of corresponding contact self-energy) to the left or to the right lead. Here uL,R are vectors, describing couplings of the states of the isolated system to the left/right lead. The traditional approach within the wide-band limit (neglecting real parts of the contact self-energy) leads to the following expression for the transmission28:

$$T=\frac{4{|{\rm{\det }}(\omega \hat{I}-{\hat{H}}_{0})|}^{2}{|{{\bf{u}}}_{R}^{\dagger }{(\omega \hat{I}-{\hat{H}}_{0})}^{-1}{{\bf{u}}}_{L}|}^{2}}{{|det(\omega \hat{I}-{\hat{H}}_{eff})|}^{2}},$$
(17)

where \({\hat{H}}_{eff}={\hat{H}}_{0}-i{{\bf{u}}}_{L}{{\bf{u}}}_{L}^{\dagger }-{{\bf{u}}}_{R}{{\bf{u}}}_{R}^{\dagger }\) is the Feshbach effective Hamiltonian. Following general formalism form ref.28, one can show that transmission (17) can be written in the form (1). For our microscopic model this can be easily checked using Eqs (5) and (7). Indeed, the following identity holds true:

$$\begin{array}{c}{|Q|}^{2}={|det(\omega \hat{I}-{\hat{H}}_{0}-i{{\bf{u}}}_{L}{{\bf{u}}}_{L}^{\dagger }+i{{\bf{u}}}_{R}{{\bf{u}}}_{R}^{\dagger })|}^{2}={|det(\omega \hat{I}-{\hat{H}}_{0}+i{{\bf{u}}}_{L}{{\bf{u}}}_{L}^{\dagger }+i{{\bf{u}}}_{R}{{\bf{u}}}_{R}^{\dagger })|}^{2}\\ \,-4{|det(\omega \hat{I}-{\hat{H}}_{0})|}^{2}{|{{\bf{u}}}_{R}^{\dagger }{(\omega \hat{I}-{\hat{H}}_{0})}^{-1}{{\bf{u}}}_{L}|}^{2}={[4{{\rm{\Gamma }}}^{2}{\gamma }_{a}^{2}{\gamma }_{s}^{2}+(\omega -\varepsilon -{k}_{s}\delta )(\omega -\varepsilon -{k}_{a}\delta )]}^{2}.\end{array}$$
(18)

Hence, from Eq. (18) we see that within the wide-band limit, the transmission of our system can be written in the form (1) with \(P=2{\rm{\det }}(\omega \hat{I}-{\hat{H}}_{0}){{\bf{u}}}_{R}^{\dagger }{(\omega \hat{I}-{\hat{H}}_{0})}^{-1}{{\bf{u}}}_{L}\) and \(Q={\rm{\det }}(\omega \hat{I}-{\hat{H}}_{aux})\), where \({\hat{H}}_{aux}\) is from Eq. (8).

Real eigenvalues of the auxiliary Hamiltonian define the exact location of perfect transmission resonance and, being \({\mathscr{P}}{\mathscr{T}}\)-symmetric it can experience \({\mathscr{P}}{\mathscr{T}}\)-symmetry breaking, which results in resonance coalescence. This is accompanied by the symmetry breaking of electron occupation in the transmission maximum. The matrix of occupations per unit energy \(\hat{n}\) can be calculated within NEGF formalism34:

$$\hat{n}=\frac{1}{2\pi }[{f}_{L}(\omega ){\hat{G}}^{r}{\hat{{\rm{\Gamma }}}}_{L}{\hat{G}}^{a}+{f}_{R}(\omega ){\hat{G}}^{r}{\hat{{\rm{\Gamma }}}}_{R}{\hat{G}}^{a}],$$
(19)

where fL,R is the Fermi-Dirac distribution function in the left/right lead. Now suppose that symmetric and anti-symmetric states in the site basis are |s〉 = (s1, s2, ..., sN)Τ and |a〉 = (a1, a2, ..., aN)Τ. Thus, neglecting the contribution from distant energy levels, the occupations per unit energy of the i-th site (i.e. (i, i) diagonal element of the occupation matrix in the site basis) is following:

$${n}_{i}={s}_{i}^{2}{n}_{ss}+{a}_{i}^{2}{n}_{aa}+{s}_{i}{a}_{i}({n}_{sa}+{n}_{as}).$$
(20)

Here nss, nsa, nas, and naa are elements of the occupation matrix in the basis of symmetric and anti-symmetric states. If the sites i and j are mapped into each other by the mirror reflection σLR (i.e. j = σLR(i)), then corresponding components of the symmetric and anti-symmetric states must be: si = sj and ai = −aj. Therefore, the difference between occupations of this sites is

$${n}_{i}-{n}_{{\sigma }_{LR}(i)}=2{s}_{i}{a}_{i}({n}_{sa}+{n}_{as})\propto {n}_{sa}+{n}_{as}.$$
(21)

This difference appears to be proportional to the sum of non-diagonal elements of the occupation matrix in the symmetric/anti-symmetric states basis. Utilizing Eq. (19) we can calculate this sum for our system:

$${n}_{i}-{n}_{{\sigma }_{LR}(i)}\propto {n}_{sa}+{n}_{as}=\frac{{\rm{\Gamma }}{\gamma }_{s}{\gamma }_{s}({f}_{L}-{f}_{R})}{\pi [4{{\rm{\Gamma }}}^{2}{\gamma }_{a}^{4}+{(\omega -\varepsilon -{k}_{a}\delta )}^{2}][4{{\rm{\Gamma }}}^{2}{\gamma }_{s}^{4}+{(\omega -\varepsilon -{k}_{s}\delta )}^{2}]}\times |Q|,$$
(22)

where |Q| is given by Eq. (18). Thus, it is obvious, that at perfect transmission resonances (real zeroes of Q) electron occupation is distributed symmetrically (with respect to σLR operation). Whereas, for energies, which correspond to complex roots of Q and transmission lower than 1, there is always asymmetric distribution of electrons. Therefore, \({\mathscr{P}}{\mathscr{T}}\)-symmetry breaking at the EP (coalescence of two perfect resonances into one non-perfect) manifests itself by a symmetry breaking of electron distribution, that was shown for linear systems in ref.30.

Quantum interference inverter transfer characteristic

For a given general structure of the inverter one can calculate all terminal currents if certain voltages are applied. To do so, the transmission coefficients between the leads T1out, T2out and T12 should be determined first. Reference voltages Vref1 and Vref2 (assume Vref1 < Vref2) are given by some external ideal voltage sources, i.e. we treat them as constants. As the input lead is isolated from the system, the voltage Vin influences only transmission coefficients. For high-resistance loads the output voltage Vout is derived from the condition Iout = 0, where Iout is the total current through the output lead, which is composed of the currents from the first and the second reference voltage leads (with appropriate sign).

Let us consider an inverter composed of two identical quantum switches (\({\mathscr{P}}{\mathscr{T}}\)-symmetric interference transistors). Assuming that resonance width is sufficiently small, we can approximate condition Iout = 0 as follows:

$$\begin{array}{c}[f({\varepsilon }_{1}-e{V}_{out})-f({\varepsilon }_{1}-e{V}_{ref1})]\times \frac{({\gamma }_{s}^{2}+{\gamma }_{a}^{2})[{\delta }_{1}^{2}{({k}_{a}-{k}_{s})}^{2}+4{{\rm{\Gamma }}}^{2}{({\gamma }_{a}^{2}-{\gamma }_{s}^{2})}^{2}]}{{\delta }_{1}^{2}{({k}_{a}-{k}_{s})}^{2}+4{{\rm{\Gamma }}}^{2}{({\gamma }_{s}^{2}+{\gamma }_{a}^{2})}^{2}}\\ =\,[\,f({\varepsilon }_{2}-e{V}_{ref2})-f({\varepsilon }_{2}-e{V}_{out})]\times \frac{({\gamma }_{s}^{2}+{\gamma }_{a}^{2})[{\delta }_{2}^{2}{({k}_{a}-{k}_{s})}^{2}+4{{\rm{\Gamma }}}^{2}{({\gamma }_{a}^{2}-{\gamma }_{s}^{2})}^{2}]}{{\delta }_{2}^{2}{({k}_{a}-{k}_{s})}^{2}+4{{\rm{\Gamma }}}^{2}{({\gamma }_{s}^{2}+{\gamma }_{a}^{2})}^{2}}.\end{array}$$
(23)

Here subscripts 1 and 2 correspond to the first and to the second quantum switches. Energies ε1,2 are the energies of degenerate states in the first and in the second system respectively. Assume that they are adjusted to ε1,2 = eVref1,2, i.e. to the biased Fermi level of each reference lead. The applied input voltage influences parameters δ1,2 of the switches. The following model dependence of δ1,2 on the input voltage Vin provides a symmetrical transition from the “on”-mode to the “off”-mode of each quantum switch as Vin varies in the interval [Vref1, Vref2]:

$${\delta }_{\mathrm{1,2}}=\alpha (e{V}_{in}-{\varepsilon }_{\mathrm{1,2}}),$$
(24)

where 0 < α < 1 is an electrostatic lever arm of the input lead (common gate). One can substitute Eq. (24) into Eq. (23) and derive the implicit dependence Vout = Vout(Vin), which is then used to get an expression for the gain (12).