Exponentially Adiabatic Switching in Quantum-Dot Cellular Automata

Abstract

We calculate the excess energy transferred into two-dot and three-dot quantum dot cellular automata systems during switching events. This is the energy that must eventually be dissipated as heat. The adiabaticity of a switching event is quantified using the adiabaticity parameter of Landau and Zener. For the logically reversible operations of WRITE or ERASE WITH COPY, the excess energy transferred to the system decreases exponentially with increasing adiabaticity. For the logically irreversible operation of ERASE WITHOUT COPY, considerable energy is transferred and so must be dissipated, in accordance with the Landauer Principle. The exponential decrease in energy dissipation with adiabaticity (e.g., switching time) distinguishes adiabatic quantum switching from the usual linear improvement in classical systems.

1. Introduction

Understanding the limits of energy dissipation imposed by physics has become increasingly important as alternate approaches to Si-based transistors are being explored. This is because the continual need for enhanced performance in these transistors has led to shrinking of critical dimensions to a few nanometers with corresponding increasing areal power dissipation [1]. It has become clear that this is a fundamental issue independent of how logic devices are implemented. Making devices small is of no benefit if the heat generated threatens to melt the circuit.

Some have argued that encoding information with electronic charge, and therefore moving charge in switching events, must necessarily lead to power dissipation of at least $k_{B}Tlog\left(2\right)$ [2]. Other state variables, it is argued, would be free of this limitation. This was shown to be incorrect by experiments on charge bits which measured energy dissipation of less than 0.01 $k_{B}T$, with a bit energy of $100k_{B}T$ [3]. Movement of charge is not the problem.

The Landauer Principle (LP) states that only logically irreversible operations (e.g., erasure of an unknown bit) must be accompanied by a minimum heat generation of $k_{B}Tlog\left(2\right)$ [4]. LP sets no lower limit for reversible operations and is independent of the choice of state variable which represents information.

Quantum-dot cellular automata (QCA) is a promising candidate for storing and processing binary information. QCA uses the configuration of localized charge to represent bit information and quantum mechanical tunneling to enable switching between states [5]. Here we calculate the excess energy dissipated as a function of the adiabaticity of the switching operation for two-state and three-state quantum-dot systems. A two-state system captures the most basic elements of QCA operation. The three-state system is needed to implement clocked QCA, which enables both data flow control and power gain [6]. Implementations of three-state systems have been demonstrated in metal-dot QCA [7] and molecular systems [8].

In molecular QCA the role of the dot is played by a redox center within the molecule—a site that can be reversibly reduced or oxidized by adding or removing an electron. These are typically single metal atoms or boron cages. An electric field moves the electron from one redox site to another. The field can be that of an externally applied field acting as a clock [9] or that due to a neighboring molecule [10]. The molecule-molecule field coupling represents the flow of information within the QCA circuit [11]. Here we are focusing on the fundamental aspects of the electron switching from one dot (redox center) to another, as driven by the energy difference produced by the applied field.

The key to our approach here is that we are calculating the energy delivered as an excitation to the QCA cell. This is the energy that will subsequently be dissipated as heat to the environment. By focusing on the excitation of the system, we can treat switching as a unitary quantum process and avoid the complication of including explicit dissipation mechanisms which may introduce model-dependent features. The excitation energy due to the switching event is the main focus. We calculate the way this excitation energy depends on the adiabaticity, as quantified in what follows, of the switching event.

We are not here addressing the dissipation of energy that occurs in gradually charging up and discharging the electrodes that create the clocking signals for QCA. Because of the non-zero resistance of the conductors, this can be non-trivial, though in all but the highest operating frequency dissipation in the active devices is dominant [12].

2. Two-Dot System

2.1. Model

A two-state quantum-dot system is the simplest QCA element, consisting of two dots and a mobile charge, as illustrated schematically Figure 1. A dot is simply a region of space that localizes charge. In the molecular implementation, it is usually a metal center. The two states quantum states are $\left|L\right.〉$ or $\left|R\right.〉$ representing states with charge localized on the left dot (L) or right dot (R). If the charge is on the left dot, it represents a binary 0, and if it is on the right dot it represents a binary 1. Charge transport between dots is through quantum-mechanical tunneling.

The Hamiltonian for such a system can be written as

$$\widehat{H}\left(t\right)=E_c\left(t\right)\left[\left|L\right.〉\left.〈L\right|\right]-\gamma \left[\left|L\right.〉\left.〈R\right|+\left|R\right.〉\left.〈L\right|\right]+E_0\left[\left|R\right.〉\left.〈R\right|\right].$$

(1)

Here $E_0$ is the on-site energy of the right dot, independent of time, and $E_c$ is the on-site energy of the left dot. The tunneling energy between dots is $\gamma$. The characteristic time scale for dynamics $\tau$ is set by the Rabi oscillation period.

$$\tau =\frac{\pi \hslash }{\gamma }$$

(2)

The switching dynamics are modeled by solving the quantum Liouville equation for the unitary evolution of the the density operator $\widehat{\rho }=\left|\psi \left(t\right)\right.〉\left.〈\psi \left(t\right)\right|$ [13].

$$i\hslash \frac{\partial \widehat{\rho }}{\partial t}=[\widehat{H}\left(t\right),\widehat{\rho }]$$

(3)

The instantaneous eigenstates of $\widehat{H}$ are denoted $|E_1\left(t\right)\rangle$ and $|E_2\left(t\right)\rangle$ with eigen-energies $E_1$ and $E_2$ ($E_1\le E_2)$ respectively. From the density operator we can calculate the time-dependent probabilities of the charge being found on the left dot, the right dot, the ground state, or the excited state, and the expectation value of the energy.

$$\begin{array}{ccc}\hfill P_L\left(t\right)& =& \mathrm{Tr}\left(\widehat{\rho }\left(t\right)|L\rangle \langle L|\right)\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill P_R\left(t\right)& =& \mathrm{Tr}\left(\widehat{\rho }\left(t\right)|R\rangle \langle R|\right)\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill P_{E_1}\left(t\right)& =& \mathrm{Tr}\left(\widehat{\rho }\left(t\right)|E_1\left(t\right)\rangle \langle E_1\left(t\right)|\right)\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill P_{E_2}\left(t\right)& =& \mathrm{Tr}\left(\widehat{\rho }\left(t\right)|E_2\left(t\right)\rangle \langle E_2\left(t\right)|\right)\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill 〈E〉\left(t\right)& =& \mathrm{Tr}(\widehat{\rho }\left(t\right)\widehat{H}\left(t\right))\hfill \end{array}$$

(8)

For the two-dot system, the probability of being found in an excited state $P_{E_x}$ is just equal to $P_{E_2}$ (for comparison later with the three-dot system). We define the excess energy of the system as

$$E_{excess}\left(t\right)\equiv 〈E〉\left(t\right)-E_1\left(t\right)$$

(9)

2.2. Switching the Two-Dot System

We consider switching from the binary 0 to the binary 1 state, transferring the electron between left and right dots, by linearly ramping $E_c\left(t\right)$ from $E_{c,initial}$ to $E_{c,final}$ over the switching time T.

$$E_c\left(t\right)=E_{c,initial}+\frac{t}{T}\left(E_{c,final}-E_{c,initial}\right)$$

(10)

This is shown schematically in Figure 2. We define the change $\Delta E_c$ during the switch event

$$\Delta E_c\equiv E_{c,final}-E_{c,initial}.$$

(11)

The initial density operator is taken to be $\widehat{\rho }\left(0\right)=|E_1\rangle \langle E_1|$. For convenience, we use energies appropriate to a molecular system for which relevant energy scales are in the electron volt range. For larger systems the energies would scale to smaller values appropriately. Equation (3) is solved with a Runge-Kutte method [14].

Figure 3 shows the switching dynamics of a two-dot system where we take $E_0=0$, $\gamma =0.1$ eV, and measure time in terms of $\tau$ from Equation (2). The energy of the left dot $E_c\left(t\right)$ is switched from −2.5 to 2.5 eV over a switching time of $T=5\tau$ as shown in Figure 3a. The instantaneous eigenstate energies $E_1\left(t\right)$ and $E_2\left(t\right)$ are shown in Figure 3b and demonstrate the expected anti-crossing behavior. Post-crossover, the expectation value of the energy $〈E〉$ is higher than the ground state $E_1$ and the difference between them is $E_{excess}$ as defined in Equation (9). Because the switching in this case is very rapid compared to the system’s natural response time, there is considerable non-adiabiticity and excess energy is transferred into the system. Figure 3c shows the probability of dot occupancy shifting from left to right, with ringing evident after the crossover at $t=2.5\tau$. In Figure 3d we plot the probabilities of ground and excited state occupancy.

We quantify the adiabaticity of the switching using two quantities: the excess energy deposited in the system at the end of the switching event, $E_{excess}$, and the probability of being found in the excited state (the “wrong” state) after the switching, $P_{E_x}$. In this example, switching is significantly non-adiabatic. Most of the activity happens between $t=2\tau$ and $t=3\tau$. Because the natural resonance time of the system is just $\tau$, it can barely keep up with the switching and becomes excited. Please note that by a little time after $t=3\tau$, the probability $P_{E_x}$ is basically constant. The additional rise in $〈E〉$ can be understood as the trapped probability density being steadily raised in energy by the signal $E_c\left(t\right)$.

In a real system, this deposited energy $E_{excess}$ would be dissipated to the environment through inelastic mechanisms, during or after the switching event itself. The unitary evolution we are considering here allows us to quantify the amount of excess energy that would have to be dissipated, without the need to adopt a specific model for coupling the system to the environment. If the dissipation was happening on a timescale more rapid than the switching, the net energy dissipation might be further reduced (the case of strong heat-sinking).

Now consider the more adiabatic switching operation shown in Figure 4, where the $E_c\left(t\right)$ is switched between the same values (−2.5 to 2.5 eV), but over a time of 50$\tau$. The switching operation is smoother as seen from the absence of ringing in $P_L\left(t\right)$ and $P_R\left(t\right)$ in Figure 4c. The expectation value of the energy $〈E〉$ tracks the ground state much more closely (Figure 4b) so $E_{excess}$ is nearly zero (not visible on this scale). By switching more gradually, less energy builds up in the system and therefore less needs to be dissipated as heat.

Switching will be more adiabatic if the switching time T is greater, or if $\Delta E_c$ is smaller, or if the tunneling energy $\gamma$ is larger. The effect of each of these can be captured by defining the adiabaticity parameter for the switching $\beta$:

$$\beta \equiv \frac{2\pi {\gamma }^{2}T}{\hslash |\Delta E_c|}$$

(12)

Except for the constant, this result can be obtained by dimensional analysis. The correct value of the constant and a derivation was obtained by Landau and Zener [15,16] in the context of atomic scattering. For a recent derivation and discussion see reference [17].

Numerical results for $P_{E_x}$ obtained by directly solving Equation (3) for switching for with various values of $\gamma$ and $\Delta E_c$ are shown in Figure 5a. The parameter $\beta$ here can be viewed as an appropriately scaled version of the switching time T. Changing $\gamma$, T, and $\Delta E_c$ such that $\beta$ remains constant yields the same result. The exponential decay in excitation probability with increasing $\beta$, i.e., longer switching times, agrees with the Landau-Zener result for nine orders of magnitude (down to the limits of the numerical tolerance).

$$P_{E_x}=e^{-\beta }.$$

(13)

The excess energy deposited in the system during switching $E_{excess}$ also decreases exponentially with increasing $\beta$ as evident in the numerical results shown in Figure 5b. For $\gamma \ll E_c$, we have $E_{excess}\approx P_{E_x}{E}_{c,final}$, so to a very good approximation,

$$E_{excess}\approx E_{c,final}{e}^{-\beta }.$$

(14)

For the switching event shown in Figure 3, $\beta =2$ and for that shown in Figure 4, $\beta =20$. The result is a reduction in the excess energy by a factor of $e^{-10}\approx 2\times {10}^{-9}$. There is no fundamental lower limit to $E_{excess}$, it can be lowered to any arbitrarily value by increasing $\beta$ by changing either T, $\Delta E_c$ or $\gamma$.

3. Three-Dot Clocked QCA System

3.1. Model

A three-dot QCA system, illustrated schematically in Figure 6, includes a middle dot which allows clocked control and memory. The three quantum basis states are $|L\rangle$, $|M\rangle$ and $|R\rangle$ denoting charge occupancy in left , middle, or right dot. As in the previous section, left and right dot occupancy represents a binary 0 or 1. If the middle dot is occupied, the system represents a null state, which contains no information (like a blank on a Turing machine tape). Clocking in QCA is achieved by raising and lowering the energy of the middle dot. This can be realized without requiring individual molecule-scale contacts, but rather by using local electric fields [10]. In Figure 6, the clocking field would point along the vertical axis of the figure, pulling the charge down into the middle null dot or pushing it up into the active dots.

The Hamiltonian of a three-dot system can be written as:

$$\begin{array}{ccc}\hfill \widehat{H}\left(t\right)& =& (E_0+q{V}_b)\left[\left|L\right.〉\left.〈L\right|\right]+E_c\left(t\right)\left[\left|M\right.〉\left.〈M\right|\right]+E_0\left[\left|R\right.〉\left.〈R\right|\right]\hfill \\ & & -\gamma \left[\left|L\right.〉\left.〈M\right|+\left|M\right.〉\left.〈L\right|+\left|M\right.〉\left.〈R\right|+\left|R\right.〉\left.〈M\right|\right].\hfill \end{array}$$

(15)

Here $q{V}_b$ is an applied bias which shifts the relative energies of the left and right dot. The bias may be from a neighboring molecular cell, or from an external field. The onsite energy of the middle dot $E_c\left(t\right)$ represents the clocking signal. When $E_c$ is low, the ground state of the system is the null state, with the charge occupying the middle dot. If $E_c$ is high, the ground state of the system can be a 1 or a 0 state (or a superposition), depending on $q{V}_b$. As in Section 2, $\gamma$ is the tunneling energy between dots and the characteristic time-scale for dynamics $\tau$ is given by Equation (2).

3.2. Switching in the Three-Dot QCA System

We model the the switching dynamics by solving the quantum Liouville Equation (3) for a given $E_c\left(t\right)$ and $V_b$. The instantaneous eigenstates of the Hamiltonian $\widehat{H}$ are denoted $|E_1\left(t\right)\rangle$, $|E_2\left(t\right)\rangle$ and $|E_3\left(t\right)\rangle$ with eigen-energies $E_1$, $E_2$ and $E_3$$(E_1\le E_2\le E_3)$ respectively. Time-dependent probabilities are obtained from Equations (4)–(8), with the obvious extension for three-state system: $P_M$ and $P_{E_3}$. The probability of finding the system in an excited state is now:

$$P_{E_x}=1-P_{E_1}=P_{E_2}+P_{E_3}$$

(16)

For a three-dot-system, there are three important operations to be considered, as shown schematically in Figure 7: writing a bit, erasing a bit when a copy of the existing bit is available, and erasing a bit when there is no copy.

3.3. Writing a Bit

Prior to writing the bit, $E_c$ is low and the system is in the null state $\left|M\right.〉$, which is also the ground state. Figure 7a shows the on-site energies during the WRITE operation. A bias $V_b$ is applied, as shown in the figure, which we will take here to be positive so that a binary 1 will be written. The applied bias shifts the energy of the left dot so that it is higher than the energy of the right dot, though both are still higher than the middle dot. This bias can result from the Coulomb influence of neighboring QCA cells in a circuit, or can be due to an externally applied in-plane field. The clocking energy $E_c\left(t\right)$ then ramps up, passing the energy $E_R$, and then $E_L$, and continuing to increase until it is substantially higher than either. As a result, the new ground state for the system at the end of the writing event is $\left|R\right.〉$, representing the 1 bit.

Figure 8 shows the results of a numerical solution of Equation (3) in the case when a 1 bit is written very rapidly into the cell. Here we have $\gamma =0.05$ eV, and $E_c\left(t\right)$ is switched linearly from −2.5 to +2.5 eV over a time of 17.5$\tau$, as shown in Figure 8a. The applied bias $V_b$ is 1 V. Figure 8b shows the instantaneous energy eigenvalues as the bit is written. The anti-crossing behavior is visible both when $E_M$ crosses $E_R$ and again when it crosses $E_L$. Because the switching happens within a time of a few $\tau$, excess energy is transferred into the system; $〈E〉$ (dotted line) is elevated above the ground state $E_1$. As with the two-state case, this energy excitation is evident both in the oscillations of dot probabilities seen in Figure 8c, and the finite excitation probability $P_{E_x}$ (dashed line) shown in Figure 8d. This excess energy must eventually be dissipated as heat.

How does this excitation scale when we make the operation more adiabatic? We solve Equation (3) for the WRITE operation with various values of $\gamma$, $\Delta E_c$, and switching time T. The results are shown in Figure 9 in terms of terms of the adiabatic parameter $\beta$ defined in Equation (12). We characterize the WRITE operation using $P_{E_x}$, the probability of the system being in an excited state after the operation is complete, and $E_{excess}$, the expected value of the excitation energy. As shown in Figure 9a, $P_{E_x}$ decreases exponentially with $\beta$. The probability depends on $\beta$, irrespective of the individual value of $\gamma$, T and $\Delta E_c$. This is the same relationship observed in a two-state system.

$$P_{E_x}=e^{-\beta }$$

(17)

Furthermore, the excess energy deposited in the system $E_{excess}$ also decreases exponentially with increasing $\beta$ as shown in Figure 9b. To a very good approximation one can write

$$E_{excess}\approx P_{E_x}\left(q{V}_b\right)\approx \left(q{V}_b\right)e^{-\beta }.$$

(18)

This can be understood by examining Figure 8b. After the switching, excitations from the ground state are dominated by excitations to the first excited state, which is above the ground state by an amount $q{V}_b$.

This exponential decay in the excess energy means there is a corresponding exponential decay in the amount of energy that needs to be dissipated due to a WRITE operation. This is in accord with the Landauer principle—there is no fundamental lower limit to the amount of energy that must be dissipated as heat to write a bit of information.

3.4. Erasure with Copy

The ERASE operation begins with the system in either the $\left|L\right.〉$ or $\left|R\right.〉$ (binary 0 or 1) state and $E_c$ at a high value, and ends with the system in the $\left|M\right.〉$ (null) state and $E_c$ at a low value. The initial state is a one-bit memory and as such cannot be an energy eigenstate of the system. A memory bit must have a state which depends on its past (which bit was written last) and not simply on the Hamiltonian of the system at the present moment. The stored bit must always be in a long-lived metastable state; this is a general feature of any physical memory.

One of Landauer’s key insights was that if there is a copy of the bit available, then the erasure process can use that information to construct an erasure protocol that is different for a 1 bit than for a 0 bit. This ERASE WITH COPY operation, as we will see, can be done with arbitrarily little energy dissipation. The essential step is to use the copy of the bit to initially bias the system into the state it is already in, as shown in Figure 10a. This has the effect of transforming the metastable memory-holding state into the system ground state. Figure 7b shows this schematically for a system holding a 1 bit with the biased already applied. The bias $q{V}_b$ raises the energy of the left dot relative to the right dot so the initial state is in fact the ground state of the system. The middle dot is initially high, creating a large kinetic barrier for tunneling between left and right. As the on-site energy of the middle dot is lowered (clock goes to low), the charge will transfer from the right to the middle dot as the energy of the middle dot passes the energy of the right dot. Once the appropriate bias is applied, the ERASE WITH COPY operation (for a stored 1 bit) shown in Figure 7b is then just the time-reversed version of the WRITE operation for a 1 bit, shown in Figure 7a. Please note that if the initial bit were 0, then the sign of the applied bias would need to be reversed, again biasing the system into the state it is in. The ERASE WITH COPY operation is sometimes referred to as “erase a known bit”, or “conditional bit erasure” (because the erasure protocol is conditioned on the stored bit state).

Figure 10 shows the calculated energetics and probabilities for a rapid ERASE WITH COPY operation for a stored 1 bit. The parameters are chosen to display the residual non-adiabiticity. To see the scaling of the erasure with adiabticity, we calculate the unitary evolution of the ERASE WITH COPY operation using Equation (3) and find the excitation probability $P_{E_x}$ and the excess energy $E_{excess}$ for various values of the parameters. The clocking potential on the middle dot changes from $E_{c,initial}=+\Delta E_c/2$ to $E_{c,final}=-\Delta E_c/2$ in time T. The system is initially in a binary 1 state so the bias applied is $q{V}_b=1$ eV (this biases the system into the state it is in). Figure 11a shows the exponential reduction in $P_{Ex}$ as $\beta$ is increased, just as in Equation (17). Figure 11b shows the exponential reduction in $E_{excess}$ with increasing $\beta$.

$$E_{excess}\approx -P_{E_x}{E}_{c,final}\approx -e^{-\beta }{E}_{c,final}$$

(19)

3.5. Three-State System: Erasure without Copy

Unlike ERASE WITH COPY, where the system is biased in the state it is in, during an ERASE WITHOUT COPY operation the system cannot be biased into an the initial ground state of the system. The on-state energies of the left dot and right dot are equal throughout. During the switching, when the energy of the middle dot crosses the energies of left and right dot, there is an equal probability that the system will end up in the ground state or the first excited state. Figure 12 shows the calculated energetics and probabilities for such an operation. The probability oscillates rapidly and the system has a residual energy equal to 1.25 eV. This lack of adiabaticity, and the corresponding energy dissipation is unavoidable, without knowing the initial value of the bit and biasing appropriately. Figure 13 shows the excited state probability and the excess energy as a function of the adiabaticity parameter $\beta$. When information is destroyed without a copy being left behind, significant energy dissipation as heat is inevitable, just as the Landauer Principle asserts.

4. Discussion

The two-dot QCA system is the simplest switching device possible. We see that, in contrast to the classical result, there is an exponential reduction in excess energy delivered to the system when the switching time T is reduced. For adiabatic CMOS , for example, one achieves a linear decrease [18]. This is a dramatic difference and points to the possibility of realizing the ultra-low power dissipation required for molecular scale devices. We have connected this exponential decrease in power dissipation to the classic Landau-Zener result from atomic scattering theory.

The three-dot QCA system allows clocking and an exploration of erasure to a null state (this is the only unambiguous way to define erasure). We see that the WRITE operation and the ERASE WITH COPY operations require no fundamental minimum energy dissipation. In fact, as with the two-dot case, the reduction is exponential in the adiabatic parameter (typically the switching time). The ERASE WITHOUT COPY operation, necessarily dissipates the full bit energy (many times $k_{B}T$). This is exactly what the Landauer Principle would require, though it does not predict the exponential quantum behavior.

Our analysis makes it clear that the origin of the exponential improvement is that by applying the correct bias, either for writing or erasing, quantum adiabatic time-evolution is possible. The adiabatic theorem [19] assures that if the time-dependent Hamiltonian is varied smoothly enough eigenstate probabilities will remain unchanged. The issue with erasure of an unknown bit (that is, without a copy) is that the initial state cannot be in an eigenstate, because the device itself is a memory bit. If a copy of the bit exists then the system can be biased into the state it is in, making the initial state an eigenstate. This state can then be smoothly moved past other eigenstates (with appropriate anti-crossing behavior) as seen in Figure 10b. This is the quantum mechanical basis for the Landauer Principle at its most fundamental level.

Using the copy of a bit to erase does not require increased circuit complexity. In typical QCA circuits such as shift registers and majority gates, this happens automatically [20]. The bit is represented by a bit packet that includes several cells, each of which provides a bias to the neighbors. Biasing each cell to the state it is already in therefore happens through Coulomb interaction with neighboring cells.

The exception is the “loser” input to a majority gate, which necessarily loses information and therefore dissipates the bit energy. If all three input bits to the majority gate are the same, there is no loss of information, no erasure, and no necessary energy dissipation. If exactly two of the input bits are the same, say 1, then the output will be a 1 no matter whether the third input is a 0 or a 1. That information is therefore lost (erased) and an amount of heat greater than $k_{B}Tlog\left(2\right)$ will be dissipated.

The exponential adiabatic decrease in the excitation energy associated with switching QCA devices, and therefore the associated power dissipation, represents a potentially very significant advantage for this emerging paradigm.