Robust Electric-field Input Circuits for Clocked Molecular Quantum-dot Cellular Automata

Figure 4 provides a schematic depiction of a three-dot cell. The three molecular dots (black circles) are labeled 0, 1, and $N$. The state $\ket{x}$ is the localized state of one mobile electron (red disc) on dot $x\in{0,1,N}$. These are assigned device values “0,” “1,” and “Null,” respectively. States “0” and “1” represent a bit and are designated as “active states” of the molecular cell. Not depicted here is a fixed neutralizing charge, assumed to be located at the null dot. The neutralizing charge $+q_{e}$ gives the molecule net charge neutrality, modeling a zwitterionic QCA molecule. For simplicity, the cell is treated as a system of two point charges: the mobile charge, $-q_{e}$, and the fixed charge, $+q_{e}$, where $q_{e}$ is the elementary charge. Black lines connecting dots represent tunneling paths for the mobile charge. As discussed above, direct tunneling between active states is suppressed, enabling the latching of a bit onto a cell. The active dots are separated by distance $a$. Here, we will consider molecules adsorbed onto a surface (the $z=0$ plane) so that dot $N$ is on the surface and active dots are elevated to the plane $z=+h$.

Consider an isolated cell at position $\vec{r}$ immersed in an electrostatic field $\vec{E}(\vec{r})$, which is approximately constant over the volume of the cell. The Hamiltonian of the cell may be written as

where $\hat{P}_{\alpha,\beta}\equiv\ket{\alpha}\bra{\beta}$ is a transition operator from $\ket{\beta}$ to $\ket{\alpha}$, and $\hat{P}_{\alpha}\equiv\hat{P}_{\alpha,\alpha}$ is a projection operator onto $\ket{\alpha}$. The operator $\hat{Z}\equiv\hat{P}_{1}-\hat{P}_{0}$ is analogous to the Pauli operator $\hat{\sigma}_{z}$ and employs the same sign convention for $\hat{\sigma}_{z}$ as in Ref. [32]. The first term in Equation (1) describes electronic tunneling between the active states and the “Null” state, and is characterized by $\gamma$, the hopping energy between either of the active states and the “Null” state. It is assumed that the molecule is symmetric so that $\gamma$ describes both the $\ket{0}\rightarrow\ket{N}$ transition and the $\ket{1}\rightarrow\ket{N}$ transition. The second term in Equation (1) is characterized by $\Delta$, the detuning between states “0” and “1”:

The detuning may be separated into two components:

where $\Delta_{\text{Neigh}}$ is the bias due to the charge distribution of neighboring QCA cells, and $\Delta_{E}$ is a bias driven by the applied field, $\vec{E}$. The field-driven bias is given by

where $\vec{a}$ is the vector of length $a$ pointing from dot 1 to dot 0 (see Figure 4). The third term of Equation (1) is proportional to $\hat{P}_{N}$ and describes the occupation energy of the “Null” state $\ket{N}$ relative to the unperturbed active states. Here, the clock biases the “Null” state relative to the active states by the potential energy

and the energy $E_{a}$ describes the affinity the mobile charge has for the fixed neutralizing charge on dot $N$. An $E_{a}>0$ biases the system toward the “Null” state. $E_{a}$ is a property of a particular QCA molecule, which may be estimated using quantum chemistry techniques or measured experimentally.

A cell’s charge state may be characterized by two real numbers: its polarization, $P$, and its activation, $A$:

The sign of $P$ encodes a classical bit, and $A$ may be understood as the probability that the molecule will be measured in an active state.

We introduce the kink energy, $E_{k}$, which is interpreted as the cost of a bit flip or the strength of a bit [33]. Consider a driver cell of characteristic length $a$ prepared in state $\ket{0}$. The kink energy for a target cell a distance $a$ away from the driver cell is the difference in energy between the target cell’s frustrated state and its relaxed state. Figure 5(a) shows the relaxed state, which is anti-aligned with the driver. In its frustrated state, the target cell aligns with the driver cell, as in Figure 5(b). The kink energy may be calculated directly using electrostatics:

where $\epsilon_{0}$ denotes the permittivity of free space.

In this paper, all calculations assume parameters $a=1~{}\text{nm}$, $h=a/2$, $\gamma=50~{}\text{meV}$, and $E_{a}=1~{}\text{eV}$. These parameters are chosen to represent a molecule like that of Figure 2(b). It is also assumed that the mobile charge is one electron, and that the fixed neutralizing charge is positive. For $a=1~{}\text{nm}$, $E_{k}=422~{}\mbox{meV}$.

The isolated cell’s ground state response $P$ to various applied fields $\vec{E}$ is shown in Figure 6. In this plot, both the clocking component ($E_{z}$) and the biasing (input) component ($E_{y}$) of $\vec{E}$ are scaled to $E_{o}$, the field strength required to produce a kink in two cells, as in Figure 5(b):

In the absence of a clock ($E_{z}=0$), or under weak clocking ($E_{z}\gg-5E_{o}$), the cell remains in the null state (green region of the plot) because of the positive electron-hole affinity, $E_{a}$. Therefore, some $E_{z}<0$ is required to clock the cell to an active state. If $|E_{z}|$ is large enough, even a weak input field ($|E_{y}|\ll E_{o}$) is adequate to select a bit on the cell (yellow and blue regions of the plot).

A QCA cell also may be driven by a neighboring molecule, as in Figure 7. Here, the response is calculated for a target cell in the presence of a fully-activated driver cell ($A_{\text{drv}}=1$). The target molecule’s response, $P_{\text{tgt}}$, is plotted as a function of the driver polarization, $P_{\text{drv}}$, and the clocking electric field, $E_{z}$. The $y$-component of the field is zero here, since no external input field is applied. The repulsion by the driver cell’s mobile charge and the attraction to the driver cell’s neutralizing charge further bias the target cell toward the null state so that a stronger clock than before is required to activate the target cell. This effect has been termed “population congestion,” and has been studied in detail [34].

A convenient basis, $\{\ket{\mathbf{x}}\}$, for an $M$-cell circuit may be formed by taking the direct products of localized single-cell states:

Here, $\mathbf{x}$ represents the $M$-trit state $x_{M}x_{M-1}\cdots x_{2}x_{1}$, and $x_{k}\in\{0_{k},N_{k},1_{k}\}$ labels a state of the $k$-th cell.

The circuit Hamiltonian $\hat{H}$ may be written as

where $\hat{H}_{k}$ is the free Hamiltonian of the $k$-th cell, and $\hat{H}_{\text{int}}$ describes the interactions between neighboring cells. Here, $\Delta_{\text{Neigh}}$ is not included in $\hat{H}_{k}$, but rather is accounted for in $\hat{H}_{\text{int}}$. In the basis $\{\ket{\mathbf{x}}\}$, the interaction $\hat{H}_{\text{int}}$ is diagonal and may be written as:

Here, $U_{\mathbf{x}}$ is the electrostatic energy of interaction between all cells given circuit state $\ket{\mathbf{x}}$:

Integers $j$ and $k$ label a cell in the circuit, and they are chosen to avoid double-counting pairwise intercellular interactions or including unphysical, infinite self-interaction energies. The $m$-th cell is treated as a system of point charges $\{q_{\ell}^{(m)}\}$, with charges at positions $\{\mathbf{r}_{\ell}^{(m)}(x_{m})\}$. In the case of the mobile charges, $\mathbf{r}_{\ell}^{(m)}(x_{m})$ depends on the state $x_{m}\in\{0_{m},N_{m},1_{m}\}$ of cell $m$. Positive integers $\ell$ and $\ell^{\prime}$ index the point charges consituting cells $j$ and $k$, repsectively.

A clocked molecular QCA input circuit is shown in Figure 8(a). The molecular circuit consists of an input segment (cells 1-3) and a binary wire (cells 4,5,…).

The input segment also is referred to as a longitudinal array, since the cells are aligned along their longitudinal axes, the axis which points from one active dot to another. The minimal field-sensitive segment is two cells long. A longitudinal section of two or more cells has high sensitivity to the field component along its longitudinal axis, since each member cell interacts with the field as a dipole, aligning with both the field and one another. The input segment could be longer than three cells, but we limit its length to three cells in order to limit the dimension of the circuit Hamiltonian, as well as to provide a circuit with symmetry about the $\hat{x}$ axis. This keeps the calculation manageable, and it allows the circuit to respond in a symmetric way to the input field, $E_{y}\hat{y}$.

The binary wire is used to shift the bit selected on the input segment downstream to QCA logic for processing. The binary wire exhibits some insensitivity to the $\hat{y}$-component of the field, since adjacent cells tend to antialign, and pairs of cells interact with the field more weakly as quadrupoles. We refer to the binary wire as a shift register henceforth, because its constituent cells are able to latch bits under the clock.

There is no theoretical limit to the length of the shift register. An unclocked binary wire, on the other hand, is limited in length, since the entropy of a bit error (a kinked state) increases with the length of the wire, lowering the free energy of such errors [33]. For an adequately long wire, the free energy of a kink will fall below that of an error-free wire, giving rise to an error despite bit energies towering high above the thermal noise floor. A shift register, however, is actively driven by the clock: it is not in thermal equilibrium, and the clock provides power gain to restore weakened signals. Thus, a shift register can be longer than an unclocked wire [35].

The clocked input circuit is a synchronous version of the ballistic input circuit developed previously [8]. To select the input bit, the input circuit is immersed in an applied electric field parallel with the input section. The field may be established using input electrodes. This circuit is similar to an input from a proposals in which electrodes establish an electric field to write a bit onto a single molecule [7], but is designed to support input electrodes lacking single-molecule specificity. Previously, the clocked circuit of Figure 8(a) was studied [9] using an intercellular Hartree approximation (ICHA) [36], which discards intercellular correlations. This model includes all intercellular correlations and treats the circuit exactly in the limit of three-state quantum devices.

In this paper, two limiting cases of the field are studied. First is an idealized limit, which we call the nano-electrode limit. Here, the input field may be constrained to only the input segment, as illustrated Figure 8(b). We also consider the large-electrode limit, in which the entire circuit is immersed in the input field, as in Figure 8(c). This represents the worst case, in which the $y$-component of the unwanted fringing $\vec{E}$ is as strong as the input $E_{y}$ intentionally applied to the input portion of the circuit. A more realistic situation is intermediate to these two limits: input electrodes establish an input $|E_{y}^{\left(\text{in}\right)}|$ across the input segment, and field fringing applies some $|E_{y}^{\left(\text{fringe}\right)}|<|E_{y}^{\left(\text{in}\right)}|$ to the downstream QCA circuitry. If the circuits function properly in the large-electrode limit, then it is reasonable to expect that the circuits will function properly in the more realistic, intermediate regime.

In this paper, circuits are limited to $M\leq 8$ cells, since presently, calculations are unwieldy for $M>8$. Given the small size of these circuits, we apply in all calculations a uniform clock $E_{z}\hat{z}$ to all cells, which approximates a clock having a negligible gradient over the circuit modeled.

The ground state response of a clocked input circuit is shown in Figure 9, and both the idealized and worst-case of fringing fields are considered. In the ideal case, the sign of the field selects the output bit of the circuit, which is taken here to be encoded on cell 6. The same holds true in the large-electrode limit; here, however, fringing fields disrupt circuit operation for $|E_{y}|>E_{o}/2$. In this case, it is possible to apply an input field of strength $|E_{y}|<E_{o}/2$, which is adequate to select a bit on the input+shift-register circuit, but too weak to drive a wrong state on the shift register. When $|E_{y}|>E_{o}/2$, the field is strong enough to induce a kink between cells 2 and 4, so that the wrong bit propagates down the shift register (see insets in the lower left and uppper right of Figure 9). This result is consistent with the result previously published for a an unclocked input QCA circuit [8]. All calculations use a strong clocking field ($E_{z}=-10E_{o}$), since a clock of this magnitude results in a strong output bit ($|P_{6}|\rightarrow 1$).

An even stronger fringing input field ($|E_{y}|\gg E_{o}/2$), can simply inject multiple kinks into the shift register, causing all cells to align with $E_{y}$. We do not study the circuit up to this point, since it is only necessary to find where the first failure occurs in order to determine the limits of operation for this circuit.

The vulnerability of the binary wire cells to the $y$ component of fringing input fields suggests a simple modification for more robustness: the binary wire cells may be rotated by $90^{\circ}$, as shown in Figure 10. For the rotated cells (4, 5, 6, etc.), $\hat{y}\perp\vec{a}$ leads to $E_{y}\hat{y}\cdot\vec{a}=0$ so that the $y$-component of the fringing input field cannot affect the detuning between their active states. Thus, the rotated cells provide a binary wire immune to the $y$-component of the field, but the non-rotated cells (1-3) retain their sensitivity to $E_{y}$.

Results from the circuit of Figure 10 are shown in Figure 11. To obtain a result from this circuit directly comparable to the results of Figure 9, we group the active dots from the rotated molecular cells into functional cells, labeled $4^{\prime},5^{\prime},\ldots,$ etc. (see Figure 10). In particular, $P_{6^{\prime}}$, the polarization of cell $6^{\prime}$ is plotted, where

Notably, the large-electrode $P_{6^{\prime}}$ response under unwanted fringing fields in the $y$ direction is identical to the nano-electrode $P_{6^{\prime}}$ response (ideal case), even under $|E_{y}|>E_{o}/2$. The functional cell polarization $P_{6^{\prime}}$ remains unaffected by a strong fringing field with a $y$-component, and the circuit will not fail, even under arbitrarily large $|E_{y}|$ values. As before, the clocking field was set to $E_{z}=-10E_{o}$ for all calculations.

In addition to immunity to fringing fields in the $y$-direction, the input circuit and shift register of Figure 10 demonstrates robustness against fringing fields in the $x$-direction. This is shown in Figure 12, where $P_{6^{\prime}}$ is plotted as a function of $E_{y}$ for various values of $E_{x}$. Here, all calculations are in the large-electrode limit so that $E_{x}$ is applied to the entire circuit. The input field $E_{y}$ selects the input, and $P_{6^{\prime}}$ encodes the correct–albeit weakened–bit until $E_{x}\sim E_{o}$. Circuit failure requires alignment between physical cells 6 and 7. Here, physical cells 6 and 7 align with the field and with one another (i.e., $P_{6}=P_{7}$) so that functional cell $6^{\prime}$ fails to an undesirable state: $P_{6^{\prime}}\rightarrow 0$. Note that a bit weakened by an unwanted non-zero $E_{x}$ may be strengthened by increasing the strength of the clocking field, $|E_{z}|$; however, it is simplest and most illustrative to perform all calculations at a single clocking field strength, chosen here to be $E_{z}=-7E_{o}$. We use a weaker clock than before, as it provides a more gradual onset of failure with increasing $|E_{x}|$ than in the case of the stronger $E_{z}=-10E_{o}$.

A broader exploration of the effects of the $x$ component of the field on the circuit of Figure 10 is shown in Figure 13. Here, the blue and yellow regions are where the shift register functions properly: $P_{6^{\prime}}$ is strong and non-zero, despite some unwanted $E_{x}$. The green region is where functional cells of the shift register fail to $P_{k^{\prime}}\rightarrow 0$ as the physical cells simply align with $E_{x}$. In the absence of cells 6 and 7, the shift register (comprised only of cells 4 and 5) would fail when $E_{x}\sim-E_{o}$: here, the field has to do the work of injecting a kink between cells 4 and 5 (this is consistent with the result of Figure 8 of Ref. [8]). However, with the addition of cells 6 and 7, as in Figure 10, the attraction between neutralizing (mobile) charges in cells 4 and 5 and oppositely-charged mobile (neutralizing) charges of cells 6 and 7 lowers the energy of the configuration in which all shift register cells align with the field component $E_{x}$. Thus, the shift register fails for $E_{x}\lesssim-0.6E_{o}$. Similarly, in the absence of cells 6 and 7, a shift register comprised only of cells 4 and 5 would be resistant even to some fields $E_{x}>E_{o}$, since the field must not only inject a kink, but also overcome the Coulomb repulsion between the mobile charge of cells 1-3 and the mobile charge of cells 4 and 5. Interactions between the (4,5) cell pair and the (6,7) cell pair lowers the field strength to $E_{x}\gtrsim E_{o}$ for the onset of shift register failure. Nonetheless, the circuit tolerates significant non-zero, unwanted $E_{x}$ before the shift register fails: $-0.6E_{o}\lesssim E_{x}\lesssim E_{o}$. Here, all calculations were performed using a clock of strength $E_{z}=-7E_{o}$, although a stronger clock may compensate for the effects of $E_{x}$ to some extent.

It is reasonable to design the circuit for immunity to the $y$ component of fringing input electrode fields, but only tolerance to an unwanted non-zero $x$ component. The $y$-component of fringing fields will be the dominant component applied over the innterconnecting shift register or even the down-stream logic, since the applied $\vec{E}$ will be designed to be dominant in its $y$ component.