Michael H Goerz
email: goerz@physik.uni-kassel.de

Tommaso Calarco

Christiane P Koch
email: christiane.koch@uni-kassel.de

The Quantum Speed Limit of Optimal Controlled Phasegates for Trapped Neutral Atoms

We study controlled phasegates for ultracold atoms in an optical potential. A shaped laser pulse drives transitions between the ground and electronically excited states where the atoms are subject to a long-range 1/R 3 interaction. We fully account for this interaction and use optimal control theory to calculate the pulse shapes. This allows us to determine the minimum pulse duration, respectively, gate time T that is required to obtain high fidelity. We accurately analyze the speed limiting factors, and we find the gate time to be limited either by the interaction strength in the excited state or by the ground state vibrational motion in the trap. The latter needs to be resolved by the pulses in order to fully restore the motional state of the atoms at the end of the gate.

I. INTRODUCTION

The physical realization of a quantum computer requires the implementation of a set of universal gates [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]. The most difficult part is generally the two-qubit gate since it involves interaction between two otherwise isolated quantum systems. In proposals for quantum computing with ultracold neutral-atom collisions [START_REF] Jaksch | [END_REF]3], the two-qubit gate operation involves atomic motional degrees of freedom [4,5], most often following adiabatic processes. This implies frequencies much lower than those characteristic of the trap, typically around a few tens of kHz. When long-range interactions, like dipole-dipole forces between Rydberg atoms [6][7][8], are employed, the relevant energy scales are larger and gate speed can in principle reach a few GHz.

Here, we study the limits to the two-qubit gate operation time for resonant excitation of two ultracold atoms into an electronically excited molecular state. This may be a low-lying state, like those used for photoassociating two atoms to form a molecule [9,10], or high-lying Rydberg states [6][7][8]. In this scenario, the system dynamics becomes more complex, involving motion under the influence of the excited state potential. A high-fidelity gate can then no longer be designed "by hand". Fortunately, since any gate operation corresponds to a unitary transformation on the qubit basis, its implementation can be formulated as a coherent control problem [11,12]. Solutions to the control problem can be found within the framework of optimal control [START_REF] Tannor | Time-dependent quantum molecular dynamics[END_REF][START_REF] Somlói | [END_REF][15]. Such an approach has been explored theoretically for molecular quantum computing with qubits encoded in vibrational states [11,16]. Experimentally the control problem is solved using femtosecond laser pulse shaping combined with feedback loops [17]. Implementation of single-qubit gates with shaped picosecond pulses has recently been demonstrated for a qubit encoded in hyperfine levels of an atomic ion [18]. In this experiment, the fundamental limit to the gate operation time is set by the inverse of the hyperfine splitting since the hyperfine dynamics are required to realize arbitrary qubit rotations. The gate duration is in particular much shorter than the period of atomic motion in the trap [18].

Our goal is to implement a controlled phasegate, expressed in the logical basis |00 , |01 , |10 , and |11 as

Ô = diag e iχ , 1, 1, 1 , (1)
between two qubits carried by neutral atoms: A phase χ is applied to the target qubit provided the control qubit is zero. Ô is implemented by shaped short laser pulses that drive transitions into an electronic state where the two atoms are interacting. Obviously, the minimum gate operation time will depend on the interaction strength.

A second timescale comes into play because the interaction couples electronic and nuclear dynamics, inducing vibrational excitations of the two atoms. The vibrational period of the trap might thus also affect the minimum gate operation time.

The limit to how fast a quantum gate can be performed is closely related to the minimum time it takes for a quantum system to evolve from an initial state to an orthogonal state, see for example [19] and references therein. This bound has been named the quantum speed limit; it is given in terms of the average energy and energy uncertainty of the quantum evolution [20]. Since evaluation of the bound requires knowledge of the full spectrum, one can typically evaluate it analytically only for simple model systems. Numerically, the bound can be determined using optimal control where the breakdown of convergence indicates that the quantum speed limit has been reached [21]. The following procedure allows us to determine the quantum speed limit for our desired two-qubit gate: We first set the gate operation time to a sufficiently large value to obtain a high fidelity implementation by optimal control. Then we reduce the gate time until the optimization algorithm cannot find a highfidelity solution anymore. Analysis of the fairly complex system dynamics reveals which part of the overall system dynamics limits the gate operation time.

We consider neutral atoms trapped by optical tweezers or in neighboring sites of an optical lattice (Fig. 1). The sites to the right and left of the atom pair are assumed to be empty. We first consider alkaline-earth atoms which possess extremely long-lived excited states, cf. Ref. [22] and references therein. The qubits can therefore be encoded directly in the electronic states; that is, the atomic ground state 1 S 0 together with the 3 P 1 clock-transition state form a qubit. Such an encoding in electronic states is in contrast to the one commonly employed for alkaline atoms, where hyperfine states carry the qubits.

For alkaline earth atoms, the laser is slightly detuned from the dipole allowed atomic transition between the 1 S 0 ground state and the 1 P 0 excited state, exciting the atom pair into the B 1 Σ + u molecular state. When both atoms are in the ground state, their interaction is of vander-Waals type and practically zero at the trap distance. Due to different exchange interaction in the electronically excited state, the B 1 Σ + u state scales as 1/R 3 at long range. This interaction may be employed to entangle the qubits provided the time that the atom pair resides in this state is much shorter than its lifetime of a few nanoseconds. The interaction strength is determined by the distance of the atoms. For realistic lattice parameters, d ≥ 200 nm, the interaction is too weak to generate entanglement in a sufficiently short time. On the other hand, we would like to probe what are the factors that limit the achievable speed on a general level. To this aim, we first explore a regime that cannot be realized in experiments, but presents a clear separation of timescales that allows for an easier interpretation of the dynamics. We start by assuming a fictitious distance of d = 5 nm and a correspondingly unrealistic atomic trap frequency of 400 MHz. This allows us to identify the limiting factors for fidelity and gate time.

Formally, our Hamiltonian is equivalent to the one yielding a Rydberg phasegate for alkali atoms with the qubits encoded in hyperfine levels of the electronic ground state [6]. We therefore study in a second step the implementation of a controlled phasegate based on very strong dipole-dipole interaction in the excited state for realistic lattice spacings. In particular, we seek to answer the question whether excitation of the vibrational motion can be avoided if the excited state interaction is strong enough to allow for very short gate pulses.

The paper is organized as follows. Section II introduces our model for the two atoms and summarizes how quantum gates can be implemented using optimal control theory. The numerical results are presented in Sec. III with Sec. III A devoted to generation of entanglement for two calcium atoms via interaction in the B 1 Σ + u state. A generic dipole-dipole interaction, -C 3 /R 3 , is considered in Sec. III B where we determine the gate operation time for varying C 3 . We draw our conclusions in Sec. IV.

II. THEORETICAL APPROACH

A. Modeling Two Atoms in an Optical Lattice

We consider the following qubit encoding in a single calcium atom: The 1 S 0 ground state corresponds to the qubit state |0 and is used to define the zero of energy. The 3 P 1 first excited state, taken to be the qubit state |1 , then occurs at E 1 = 15210 cm -1 . We consider the 1 P 1 level as an auxiliary state |a , with energy E a = 23652 cm -1 . The Hamiltonian describing such a threelevel atom reads

Ĥ1q =   E 0 0 µ 0a (t) 0 E 1 0 µ 0a (t) 0 E a   , (2)
where µ 0a is the transition dipole moment between states |0 and |a , and (t) is the amplitude of the driving laser field. For two atoms, we obtain nine electronic states |00 , |01 , |0a , |10 , |11 , |1a , |a0 , |a1 , and |aa in the product basis, with asymptotic energy E ij = E i + E j for state |ij (i, j = 0, 1, a). This is depicted in Fig. 2.

With the laser tuned close to the transition |0 ↔ |a , no further electronic states will be resonantly populated.

Considering the motion of the two atoms in the trap, we approximate the trapping potential by two displaced harmonic oscillators. This allows us to separate centerof-mass coordinates and relative coordinates. The centerof-mass coordinates are integrated analytically, reducing the motional degrees of freedom of the two atoms from six to three. Furthermore, the parameters of the optical lattice are chosen as ω ⊥ ω 0 , such that the motion of the atoms is restricted to just one spatial dimension, the internuclear distance R (see Fig. 1).

The Hamiltonian of our model, taking nine electronic states, relative motion in the harmonic trap, and interaction between the atoms into account, is given by

Ĥ = Ĥ1q ⊗ ½ 1q + ½ 1q ⊗ Ĥ1q ⊗ ½ R + ½ 1q ⊗ ½ 1q ⊗ Ĥtrap + Ĥint (3) = i,k |ik ik| ⊗ T + Vtrap (R) + Vik BO (R) + Êik + (t) i =j,k |ik jk| + |ki kj| ⊗ μij . (4)
Here, Ĥ1q denotes the Hamiltonian for a single threelevel system, Eq. ( 2), and ½ 1q and ½ R are the identities for SU(3) and the motional degree of freedom, respectively. The Êik = E ik • ½ R are the asymptotic energies of each electronic state. Ĥtrap = T + Vtrap (R) describes the relative motion in the trap with T the kinetic energy and the harmonic trap potential,

Vtrap (R) = 1 2 mω 2 0 (R -d) 2 , (5)
with m the reduced mass, ω 0 the trap frequency, and d the lattice spacing. The interaction Hamiltonian Ĥint contains the Born-Oppenheimer potentials VBO . The laser amplitude (t) couples to the transition dipole operator, μij = µ ij • ½ R . The Hamiltonian, Eq. ( 4), is represented on a Fourier grid with variable step size [23][24][25].

The Born-Oppenheimer potentials describing the interaction between the two atoms in the five lowest electronic states (up to the B 1 Σ + u state generating the interaction) are shown in Fig. 3. The X 1 Σ + g ground state potential corresponding to the two-qubit state |00 shows a 1/R 6 behavior at long range such that the atoms are effectively non-interacting at any relevant distance, while the auxiliary B 1 Σ + u state corresponding to |0a and |a0 goes as 1/R 3 . The remaining potentials corresponding to the two-qubit states |aa , |a1 , |1a , |11 , |10 , and |01 , are essentially zero at the relevant distances. The potentials and transition dipole moment functions employed in the calculations are gathered from Refs. [26,27].

V (R) (10 3 cm -1 ) interatomic distance R (Bohr radii) µ |a0 |0a |01 |10 |00 X 1 Σ + g 3 Πg 3 Σg 1 Πg B 1 Σ + u S + 1 P S + 1 D S + 3 P S + S
For a Rydberg phasegate with alkali atoms, the qubit encoding is the standard one in hyperfine levels of the electronic ground state; and the auxiliary level corresponds to the Rydberg state (we omit in this description any intermediate level that might be needed for nearresonant two-photon excitation of the Rydberg state). In this case, the ground state potential is simply set to zero, while the auxiliary state is modeled by a generic C 3 /R 3 potential.

A two-qubit gate is successfully implemented if the four basis states |00 , |01 , |10 , |11 transform according to the desired unitary transformation Ô. Initially, the wavefunction for the motional degree of freedom is given in terms of two atoms in the ground state of the displaced harmonic oscillator, ϕ 0 (R) = R |ϕ 0 . Hence we consider the four initial states given by the product states |ijϕ 0 = |ij ⊗ |ϕ 0 , i, j = 0, 1. The dynamics induced by a laser pulse will populate other states in the 3 × 3 × N Rdimensional Hilbert space (with N R the number of grid points to represent the motional degree of freedom); that is, it will induce internuclear motion leading out of the logical subspace. In order to calculate the gate operation in the logical subspace at the end of the pulse, the final states are reduced to the logical basis by projecting onto |ϕ 0 and integrating over the motional degree of freedom.

B. Optimal Control Theory for a Two-Qubit Gate

In order to implement the target operation defined in Eq. ( 1), a suitable pulse (t) must be found that drives the system evolution such that Û(T, 0; ) = Ô. Optimal control treats the fidelity F , F ∈ [0, 1], which measures how close the evolved two-qubit basis states Û(T, 0; ) |ijϕ 0 come to the desired target states Ô |ijϕ 0 , as a functional of the control field . A pos-sible starting point to define the fidelity is given by the complex-valued inner matrix product [12],

τ = i,j=0,1 ijϕ 0 Ô † Û ijϕ 0 . (6)
Out of the several choices for obtaining a real-valued functional, we employ

F = 1 N Re [τ ] , (7)
which is sensitive to a global phase [12]. N is the dimension of the subspace in which Ô acts, N = 4 in our case. Our definition of the fidelity is related to a distance measure [28] by √ 1 -F . We also include a running cost functional, g( ), aimed at limiting the integrated pulse intensity. Together, this leads to the total functional J,

J = -F + T 0 g( ) dt , (8)
which is to be minimized. Variation of the functional J with respect to the evolving two-qubit basis states and the control field yields a set of coupled optimization equations that need to be solved iteratively [12][START_REF] Tannor | Time-dependent quantum molecular dynamics[END_REF][START_REF] Somlói | [END_REF][15]. We use the linear Krotov algorithm as outlined in Ref. [12] to perform the optimization. Starting with an arbitrary guess pulse (k) (t), k = 0, the algorithm sequentially updates the pulse at every point in time to yield an optimized pulse (k+1) (t) that is guaranteed to improve the target functional J in each step of the iteration k. We take the running cost g( ) in Eq. ( 8) to be the change in integrated pulse energy,

g( ) = α S(t) (k+1) (t) -(k) (t) 2 , (9)
rather than the integrated pulse energy itself. This ensures that as we approach the optimum, g( ) goes to zero so that the minimum of J becomes equal to the maximum of F [12]. Note that a different choice of g( ), simply as the integrated pulse energy, could lead to the undesirable effect of improving J by decreasing the pulse intensity, at the expense of a reduced fidelity [12]. In Eq. ( 9), α denotes an arbitrary positive scaling parameter, and the shape function S(t),

S(t) = sin 2 (πt/T ) , (10)
ensures that the pulse is switched on and off smoothly. The Hilbert space of the optimization problem can be reduced to the 8N R -dimensional subspace excluding |11 ⊗ ½ R , since the |11 level is not coupled to any other level, see Fig. 2. The evolution of the |11ϕ 0 state cannot be controlled by the laser pulse but it is known to be

Û(T, 0; ) |11ϕ 0 = e iφT |11ϕ 0 ; φ T = E 11 T/ .
Including the information about the phase φ T explicitly as a global phase for all target states Ô |ijϕ 0 , the evolution of |11ϕ 0 can be omitted from the system dynamics. Entanglement is generated only by the dynamics of the states in the left-most column of Fig. 2 with all other levels remaining uncoupled from the vibrational dynamics.

The dynamics can therefore be further separated into those of the four molecular states in the left-most column of Fig. 2 and those of a two-level system representing the states in the two middle columns of Fig. 2. Care must then be taken to extract the true two-qubit phase from the evolution of the |00ϕ 0 state which contains twoqubit and single-qubit contributions. This is described in Appendix A. The numerical results presented below are obtained both within the 8N R -dimensional model and the 4N R -dimensional model plus two-level system, and the optimal pulses have been cross-checked.

For the fidelity defined in Eq. ( 7) and the running cost given by Eq. ( 9), the optimization equations read [12]

∆ (t) = (k+1) (t) -(k) (t) = S(t) 2α Im   i,j=0,1 Ψ bw ij (t) μij (R) Ψ fw ij (t)   , (11)
Ψ bw ij (t) = ijϕ 0 | Ô † Û † (t, T ; (k) ) , (12)
Ψ fw ij (t) = Û(t, 0; (k+1) ) |ijϕ 0 . (13)
Here, Ψ bw ij (t) denotes the backward propagated target state Ô |ijϕ 0 at time t in the full Hilbert space. The backward propagation is carried out using the old field, (k) (t). Ψ fw ij (t) represents the forward propagated initial state |ijϕ 0 at time t. The new field, (k+1) (t), is employed in the forward propagation. Ψ bw ij (t) and Ψ fw ij (t) are obtained by solving the time-dependent Schrödinger equation numerically with the Chebychev propagator [29]. The time is discretized in n t steps of width ∆t, between 0 and T . Since no rotating wave approximation is employed, ∆t has to be fairly small (0.025 fs to 0.05 fs).

For the desired gate implementation, there are two as-pects to the optimization problem: On one hand, the two-qubit phase χ has to be realized. This is possible due to the interaction between the two qubits in the electronically excited auxiliary state. On the other hand, control over the motional degree of freedom has to be exerted.

That is, at the end of the gate operation, the motional state |ϕ 0 has to be fully restored (except for the phases φ ij ). Final wavefunctions containing contributions from eigenstates other than |ϕ 0 imply leakage from the quantum register. These two aspects of the optimization result are quantified independently, allowing for a more thorough analysis of the solutions to the control problem than just the fidelity. The success of control over the motional degree of freedom for the ground state is measured by projecting the final state, Û(T, 0; opt ) |00ϕ 0 , onto the desired state,

F 00 = 00ϕ 0 Û(T, 0; opt ) 00ϕ 0 2 . ( 14
)
The phase acquired by each of the propagated basis states is given by

φ ij = arg ijϕ 0 Û(T, 0; opt ) ijϕ 0 . ( 15
)
The phases φ ij contain both single-qubit and two-qubit contributions. This is due to the weak molecular interaction which corresponds to small detunings of the laser from the atomic transition line. The pulse therefore drives single-qubit purely atomic local transitions in addition to true two-qubit molecular nonlocal transitions.

The Cartan decomposition of a two-qubit unitary into local and non-local contributions provides a tool to extract the desired non-local phase χ from the φ ij . The non-local contribution is characterized by the local invariants [30].

Calculating and comparing the local invariants according to Ref. [30] for the controlled phasegate Ô and the actual evolution Û projected onto the logical subspace allows us to identify the non-local phase

χ = φ 00 -φ 01 -φ 10 + φ 11 . (16)
The non-local character of the implemented gate Û can also be measured in terms of the entangling power or concurrence [31]. For the controlled phasegate it is obtained from the non-local phase χ,

C = sin χ 2 . ( 17
)
In the presentation of our numerical results below we will use the motional purity F 00 and the non-local phase χ in addition to the fidelity F to analyze the performance of the optimal pulses.

III. RESULTS

In order to obtain a clear physical picture of the limiting factors that influence the speed of the two-qubit gate operation, we start by exploring a regime in which the atom-atom interaction would be so strong to yield a time scale shorter than any other in the problem. This regime is experimentally unfeasible with optical potential both in terms of length scales and of confinement strengths, and in this sense the calculation represents little more than a toy model. Nevertheless, we simulate the dynamics taking into account in detail the physical features of a real atomic species as if the geometry considered were realizable in the laboratory via some trapping force. This allows us to gain a thorough understanding of the relevant energy and time scales, which we subsequently apply to a more realistic case of Rydberg-excited atoms interacting at longer distances, compatible with realistic optical potentials.

A. Optimization for Two Calcium Atoms at Short Distance

We consider two calcium atoms in an optical lattice at a distance of d = 5 nm that will be excited into a lowlying excited state. While such a distance is not feasible with the trapping techniques currently available in experiments, larger distances do not provide a sufficient interaction strength in the excited state to reach any significant fidelity in a reasonable amount of time. Nevertheless, this unrealistic assumption allows us to determine the physical mechanisms that limit the gate operation time.

At close distance, the ground state wave functions of the two atoms in a harmonic trap can have a significant overlap,

Ψ + |Ψ -≈ e -mωd 2 2 .
In order to be able to treat qubits carried by the two atoms as independent, we compensate for the small value of d with an artificially large trap frequency. A choice of ω = 400 MHz ensures that the overlap of the wave functions is smaller than 10 -4 . The grid parameters need to be chosen such that a reasonable number of trap eigenstates (about 50 in our case) is correctly represented. This is accomplished by taking R min to be 5.0 a 0 , R max = 300.0 a 0 , the number of grid points N R = 512, with mapping parameters β = 0.5, E max = 1 × 10 -8 , cf. Ref. [23].

The minimum gate operation time to achieve a highfidelity implementation can be due to (i) the strength of the exchange interaction in the excited state, or (ii) the vibrational motion in the trap. We investigate both hypotheses. The time scale associated with the interaction strength is estimated from the maximum phase that can accumulate in the interacting state during time T . For a non-local phase of one radian, we find where E 0a denotes the energy of two infinitely separated, i.e. non-interacting atoms in the |0a or |a0 state, and V 0a (d) the interaction potential at distance d. For d = 5 nm, this yields T rad int ≈ 1.23 ps for a non-local phase of one radian, and T π int ≈ 4.4 ps for a non-local phase of π. The time scale associated with the vibrational motion in the trap is estimated by considering the mean energy difference of the trap ground state energy to its neighboring levels, i.e. the last bound state and the first excited trap state. For the chosen trap frequency, we obtain T v ≈ 800 ps.

T rad int (d) = 1 E 0a -V 0a (d) , (18)
The optimization results for gate operation times varied between the two limits T rad int and T v are shown in Figure 4. We compare fidelity F , Eq. ( 7), non-local phase χ, Eq. ( 16), and vibrational fidelity F 00 , Eq. ( 14). The optimizations are converged to within ∆F < 1 × 10 -4 except for T = 30 ps and 50 ps which are converged to within ∆F < 2 × 10 -4 . For durations below 150 ps, with errors remaining larger than 10 -2 no satisfactory fidelity is obtained. As the gate operation time approaches T v , optimization is successful in the sense that fidelities arbitrarily close to one can be reached. The results shown in Fig. 4 can be understood as follows: The two-qubit phase χ increases with the pulse duration T , and at T = 5 ps, the time that was roughly estimated to reach a non-local phase of π, about half that phase is actually obtained. This is not surprising since the wavepacket is not in the excited state for the complete gate duration T due to the switch-on and switch-off phases of the pulse and its general shape. The non-local phase reaches the desired value of π at about 50 ps. We thus find that a prolonged action of the exchange interaction leads indeed to a nonlocal gate. However, for short gate durations, no control over the motional degree of freedom can be exerted. For T = 5 ps, the vibrational fidelity drops below 50%, and it increases rather slowly for larger T . This is due to the wave packet spending enough time in the excited state to be accelerated by the 1/R 3 potential. When the laser pulse returns the wavepacket to the electronic ground state, it has acquired significant vibrational energy. Since the pulse is too short to resolve the vibrational motion in the trap, optimization cannot identify the desired trap state and thus it cannot counteract the excitation. Population of excited trap states after the gate can be avoided only once the pulse is long enough to resolve different trap states. As the gate duration becomes comparable to T v , fidelities close to one are obtained.

An analysis of the dynamics induced by the optimized pulses is instructive for short gate durations despite the low fidelity. Figures 5 and6 display the optimized pulse, its spectrum, and the dynamics for T = 5 ps.

The guess pulse that is used to start the iterative optimization has a Gaussian envelope with a peak intensity of about 4.9 × 10 7 V/m. The intensity was chosen to drive one complete Rabi cycle for a single qubit in the |0 state (2π-pulse). The pulse fluence is increased by a factor of about 7 during the course of iterations. Optimization results in a pulse shape that clearly shows more features than a Gaussian, cf. top panel of Fig. 5. The first peak of the pulse, centered around ≈ 100 fs, drives significant

-1 0 1 -1 0 1 [τ 00 ] [τ 00 ] -1 0 1 -1 0 1 [τ 01 ] [τ 01 ] -1 0 1 -1 0 1 [τ 0 ] [τ 0 ] -1 0 1 -1 0 1 [τ 1 ] [τ 1 ]
FIG. 6: Phase dynamics induced by the optimized pulse (T = 5 ps) in the complex plane for the two-qubit states (top) and the single-qubit states (bottom). The initial state is represented by a black circle, the final state by a black square, and the target state by a blank circle.

population transfer to the auxiliary state. The last peak of the pulse, centered around ≈ 1 ps, restores all population to the electronic ground state. The dynamics yielding the non-local phase occur at intermediate times.

Since there is no interaction between the two atoms in the |01 and |a1 states, the population dynamics of the |01 state (red dot-dashed line in Fig. 5) are equivalent to those of a single atom. Comparison of the population dynamics of the |00 state and the |01 state (black solid and red dot-dashed lines in the top panel of Fig. 5) therefore yields the difference between single-qubit and two-qubit dynamics. For the short gate operation time shown in Fig. 5, the two curves are fairly similar. This is in agreement with the non-local phase of only 0.63 π that is achieved. The spectrum of the optimized pulse, cf. bottom panel of Fig. 5, is tightly centered around the |0 → |a transition frequency of 23652.30 cm -1 . It is sufficiently narrow to guarantee that no undesired transitions, for example into the 1 Π g (S +D) state, are induced. The dynamics of the two-qubit system and of a single qubit can be analyzed by projecting the time-evolved two-qubit basis state onto the initial two-qubit and single-qubit states,

τ ij (t) = ijϕ 0 Û(t, 0; opt ) ijϕ 0 , (19)
τ j (t) = j Û(t, 0; opt ) j . ( 20
)
The phase dynamics τ ij (t) and τ j (t) obtained with the optimized pulse of Fig. 5 are shown in Fig. 6 with the top (bottom) corresponding to the two-qubit (single-qubit) dynamics. The phase of the initial state is indicated by a filled black circle, the phase of the final state by a black square, and the phase of the optimization target by a blank circle. The phase dynamics of the |01 state (upper right panel of Fig. 6, no connecting lines are shown) consist of a mixture of the natural time evolution and the dynamics induced by the pulse. The |1 state (lower right panel of Fig. 6) displays only the natural time evolution. Since it is not coupled to any other state by the pulse, the population remains one, i.e., on the unit circle, at all times. The fact that the |00 state does not return to the unit circle at the final time T indicates leakage out of the quantum register. The overall fidelity amounts to only 0.805, cf. Figure 4, with the target phase for both the |00 state and |01 state missed by almost equal amounts, cf. black squares and empty circles in the upper panel of Fig. 6. This reflects that the optimization is balanced with respect to all targets, i.e. the terms in the sum of the target functional, Eqs. ( 6)-( 7), all enter with the same weight. A comparison of the |00 and |0 phase dynamics (upper and lower left panels of Fig. 6) illustrates how a true non-local phase is achieved, even though the optimization is only partially successful: Without interaction the phase on the |00 state would evolve according to φ 00 = 2φ 0 . The extent to which this is not the case demonstrates how the interaction leads to the non-local phase.

The optimized pulses, their spectra, and the corresponding population dynamics for intermediate and long gate durations are shown in Figs. 7 (T = 50 ps) and 8 (T = 800 ps). The guess pulses were again chosen to be Gaussian 2π-pulses. During the course of the iterations, the pulse fluence was increased by a factor of 28 for T = 50 ps and by a factor of 44 for T = 800 ps. The overall structure of the optimized pulse for T = 50 ps is similar to that obtained for T = 5 ps: Two peaks at the beginning and the end induce population transfer to and from the auxiliary state while the intermediate part of the pulse drives Rabi oscillations in the course of which the non-local phase is achieved. The first peak of the pulse triggering population transfer to the auxiliary state remains clearly visible as the gate operation time T is further increased, cf. Fig. 8. Overall, however, the optimal pulse shows less discernible features for T = 800 ps than for the shorter gate durations where a sequence of subpulses was found. This is reflected in the population dynamics: While for T = 50 ps, each subpulse drives a partial transfer, resulting in step-wise population dynamics, an almost adiabatic behavior is observed for T = 800 ps. Comparing the population dynamics for a single qubit and the two-qubit system for T = 50 ps (black solid and red dot-dashed lines in the upper panel of Fig. 7), more differences are obtained than for T = 5 ps, cf. Fig. 5, but overall the single qubit and two-qubit dynamics are still fairly similar. This changes dramatically for T = 800 ps (black solid and red dot-dashed lines in the upper panel of Fig. 8), where the populations dynamics for |00 and |01 are clearly distinct, reflecting that the desired nonlocal phase is fully achieved (χ = 0.998π for T = 800 ps as compared to χ = 0.975π for T = 50 ps). The spectrum of the optimal pulse for T = 50 ps is fairly similar to that obtained for T = 5 ps, cf. the lower panels of Figs. 5 and7: It basically consists of a single narrow peak centered around the |0 → |a transition frequency. The spectrum for T = 50 ps shows somewhat more features within the peak which is attributed to the better spectral resolution for larger T . For T = 800 ps, the spectrum of the optimal pulse consists of a narrow peak at the |0 → |a transition frequency and sidebands. These sidebands remain sufficiently close to the |0 → |a transition, that resonant excitation into other electronic states can be excluded, cf. Fig. 3. The phase dynamics induced by the optimized pulse of Fig. 8 are shown in Fig. 9. Since the target phases include the natural time evolution, their locations in Fig. 9 differ from those in Fig. 6. The overlap of the final states (black square) and the target states (open circle) confirms success of the optimization. All phases end up on the unit circle demonstrating that no leakage from the quantum register occurs at the end of the gate for T = 800 ps.

We also carried out optimizations for non-local target phases that are a fraction of π such as π 2 or π 3 . If highfidelity implementations of such fractional phasegates are found, several of these gates can be combined sequentially to yield a total non-local phase of π. However, for short gate operation times, optimization for non-local target phases smaller than π did not prove any more successful than optimization for π. In particular, population of excited trap states at the end of the gate could not be avoided for fractional phasegates either. Moreover, we investigated whether pulses driving multi-photon transitions, for example pulses with their central frequency a third of |0 → |1 transition frequency, yield better fidelities for short gate operation times. However, we did not observe any substantial difference in the results compared to the pulses reported in Figs. 5,7, and 8. These additional investigations confirm that for our example of two ultracold calcium atoms in an optical lattice, the limits on the gate operation time is set by the requirement to restore the ground vibrational state of the trap.

-1 0 1 -1 0 1 [τ 00 ] [τ 00 ] -1 0 1 -1 0 1 [τ 01 ] [τ 01 ] -1 0 1 -1 0 1 [τ 0 ] [τ 0 ] -1 0 1 -1 0 1 [τ 1 ] [τ 1 ]
FIG. 9: Phase dynamics induced by the optimized pulse (T = 800 ps) in the complex plane for the two-qubit single-qubit states, analogously to Fig. 6.

B. Optimization for Two Atoms at Long Distance under Strong Dipole-Dipole Interaction

To determine whether it is really the ground state motion in the trap and not the non-local interaction in the excited state that sets the speed limit for two atoms resonantly excited to an interacting state, we vary the interaction strength C 3 of the dipole-dipole interaction potential,

V(R) 0a = V(R) a0 = - C 3 R 3 , (21)
keeping the trap frequency constant. We consider the atoms to be separated by d = 200 nm which corresponds to a realistic optical lattice in the UV regime. In order to keep the overlap of the ground state wave functions smaller than 10 -4 at a distance of 200 nm the trap frequency has to be set to at least 250 kHz. This corresponds to T v ≈ 2 ns. For the interaction potential of two calcium atoms in the B 1 Σ + u state used in Sec. III A, the C 3 coefficient takes a value of 16.04 a.u.= 0.5217 × 10 3 nm 3 cm -1 [26,27,32]. This results in an interaction energy of about 4 cm -1 at d = 5 nm. Based on the results of Sec. III A, we know that such an interaction energy is sufficient to yield a nonlocal phase in a few tens of picoseconds. For d = 200 nm, the same interaction energy is obtained by choosing C 3 to be roughly 1 × 10 6 a.u. Just for comparison, the C 3 coefficient for highly excited Rydberg states is about 3 × 10 6 a.u., resulting in an interaction energy of about 1.3 × 10 -3 cm -1 at a typical distance of 4 µm for two atoms trapped in optical tweezers [33].

We vary the C 3 coefficient from 1 × 10 6 a.u. to 1 × 10 9 a.u. If the gate duration is solely determined by the requirement of a sufficiently strong interaction to realize the non-local phase, we expect to find highfidelity implementations with optimal control by increasing the C 3 coefficient. In particular, we pose the question whether picosecond and sub-picosecond gate durations can be achieved given that the interaction is sufficiently strong, i.e. given that the C 3 coefficient is sufficiently large. Based on the results of Sec. III A where a nonlocal phase of π was achieved within 50 ps, we estimate that C 3 needs to be increased from 1 × 10 6 by a factor of 50 (100) to obtain a high-fidelity gate for a duration of 1 ps (0.5 ps).

Figure 10 presents optimization results for a controlled phasegate with gate operation times of T = 0.5 ps and T = 1 ps. The central frequency of the guess pulse was adjusted in each case to compensate for the increased interaction energy and ensure resonant excitation. The grid parameters were chosen to be R min = 5 a 0 , R max = 13000 a 0 ≈ 688 nm, and N R = 2048. This choice of R max guarantees that at least fifty eigenstates of the trap are accurately represented. We verified that the grid is sufficiently large, i.e., the wave packet does not reach the boundaries of the grid during propagation. Moreover we checked that doubling the number of grid points did not yield substantially different results. Figure 10 clearly shows that increasing the interaction strength leads to larger non-local phases. A non-local phase of π is reached for C 3 = 4 × 10 8 a.u. However, increasing the interaction strength also results in a complete loss of control over the motional degree of freedom, with the vibrational fidelity F 00 reduced to 75% and below. A stronger interaction in the excited state accelerates the wave packet more, increasing its vibrational excitation. This results in a larger spread over many trap states upon the wave packet's return to the ground state. Since 0.5 ps or 1 ps are much, much shorter than the time scale of the motion in the trap, T v , the optimization algorithm cannot resolve the eigenstates of the trap. Thus it cannot identify the target F 00 which is consequently missed completely. Note that trap frequencies up to a few MHz are possible and imply shorter T v . Nevertheless, any realistic trap frequency results in vibrational time scales much larger than a few picoseconds. While excited state potentials providing for a strong interaction between two neutral atoms exist, resonant excitation into such an excited state will not yield an ultrafast non-local gate due to coupling with the motional degree of freedom, unless we consider a gate scheme that is completely insensitive to vibrational excitations

IV. SUMMARY AND CONCLUSIONS

We have studied high-fidelity implementations of a controlled phasegate for two trapped ultracold atoms via resonant optical transitions to an electronically excited state with long-range diatomic interaction. To the best of our knowledge, we have for the first time explicitly accounted for the detailed R-dependence of the interaction and thus for the coupling between internal and motional degrees of freedom that may cause leakage out of the quantum register within in a scalable setup. We have employed optimal control theory to calculate laser pulses that carry out the gate. This has allowed us to determine gate implementations of basically arbitrarily high fidelity provided the gate operation time is sufficiently long (and at the same time short enough to neglect dissipation). Our main goal was to achieve the fastest possible gate implementation and to identify what limits the gate operation time, i.e., to determine the quantum speed limit for a controlled phasegate for two neutral trapped atoms.

The standard reasoning considers the interaction strength to be the limiting factor, i.e., the gate operation time is estimated by the inverse of the two-qubit interaction. Our calculations show that a second time scale might come into play: For resonant excitation, the interaction between the two atoms causes a coupling between internal and motional degrees of freedom. This induces vibrational excitation which can be carried away by the laser pulse only if the target state is fully resolved during the optimization. The gate operation time is thus limited either by the two-qubit interaction strength or by the vibrational motion in the trap, which ever one of the two yields the larger time.

This finding has important implications for the design of two-qubit gates where the qubits are carried by neutral atoms. For example, excitation of atoms into Rydberg states yields an interaction that one might expect to allow for nanosecond to sub-nanosecond gate operation times. However, the motional state of the atoms needs to be restored at the end of the gate, and traps with subnanosecond vibrational motion seem difficult to realize. The question of how an ultrafast two-qubit gate can be realized in a scalable setup therefore remains open.

FIG. 1 :

 1 FIG.1:(color online) Two calcium atoms in neighboring sites of an optical lattice (a), separated by distance d. The lattice potential can be taken in a harmonic approximation, allowing to separate the motion of both atoms into the center-of-mass motion and the interatomic motion (b). A phasegate is implemented by applying a shaped short laser pulse.

FIG. 2 :

 2 FIG.2: Levels and laser-induced transitions for two qubits encoded in electronic states of two calcium atoms.

FIG. 3 :

 3 FIG. 3: (color online) Potential energy curves describing the interaction of two calcium atoms in the five lowest electronic states. The asymptotic values corresponding to R = ∞ are indicated on the right.

FIG. 4 :

 4 FIG. 4: (color online) Fidelity F , non-local phase χ (in units of π) and vibrational fidelity, i.e., projection onto the vibrational target state, F00, for different gate times T . The inset shows the infidelity 1 -F and the respective quantities 1 -χ and 1 -F00.The interatomic distance is d = 5 nm.

  FIG. 5: (color online) Optimized pulse (gray) and population dynamics (|00 state: solid black line, |01 state: dot-dashed red line, top) and pulse spectrum (bottom) for T = 5 ps (F = 0.805).

  FIG. 7: (color online) Pulse dynamics (top) and spectrum (bottom) for the optimized pulse with T = 50 ps after 255 iterations(F = 0.988), analogously to Fig. 5

  FIG. 8: (color online) Pulse dynamics (top) and spectrum (bottom) for the optimized pulse with T = 800 ps after 104 iterations (F = 0.999), analogously to Fig. 5

FIG. 10 :

 10 FIG. 10: (color online) Fidelity F , non-local phase χ and vibrational fidelity, i.e. projection onto the vibrational target state, F00 for increasing interaction strength C3 in the excited state for two different gate times T . All optimizations have converged to ∆F < 1 × 10 -4 The interatomic distance is d = 200 nm.

  i(φ+φ T ) |00 |01 → e iφ T |01 |10 → e iφ T |10 |11 → e iφ T |11 |00 → e i(φ+φ T ) |00 |0 → e iφ T /2 |0 phase χ = φ00 -φ01 -φ10 + φ11 χ = φ00 -2φ0

TABLE I :

 I Comparison between the full model and the reduced model for optimization of a controlled phasegate.

Acknowledgments

We would like to thank Peter Zoller for many stimulating discussions. Financial support from the Deutsche Forschungsgemeinschaft (Grant No. KO 2302), as well as the BMBF (project QUOREP) and the EC (IP AQUTE) is gratefully acknowledged .

APPENDIX A: REDUCED OPTIMIZATION SCHEME

Since the dynamics relevant for obtaining a non-local phase involves only the |00 state out of the four twoqubit states, cf. Fig. 2, we can reduce our full model, Eq. ( 4), to one describing only the left-most column of Fig. 2. This is a direct consequence of optimizing for a diagonal two-qubit gate. We are then operating in a 4N R -dimensional Hilbert space instead of a 3 × 3 × N Rdimensional Hilbert space. However, care must be taken to extract the correct non-local phase χ. The phase φ 00 describing the time evolution of the two-qubit |00 state alone is not sufficient to obtain the non-local phase χ since φ 00 contains contributions from both the singlequbit and two-qubit dynamics. We therefore need to augment our reduced model for the dynamics starting from the |00ϕ 0 by a two-level system (|0 , |a ). This captures the purely single-qubit dynamics in the phase φ 0 . The non-local phase is then obtained as the difference between φ 00 and (twice) φ 0 , see Table I. The optimization targets for the full model and the reduced model are also listed in Table I. They correspond to the unitary transformation for the full model and to two state-to-state transitions for the reduced model. Note that this type of state-tostate transition requires a phase-sensitive functional such as the one in Eq. (7). We have checked numerically that the full and reduced model are indeed equivalent: Propagating the Schrödinger equation with an optimal pulse obtained for the reduced model but employing the full Hamiltonian, Eq. ( 4), we obtained the same fidelity as for the reduced model.


Michal Duda

Janusz Wolny

Coherent joining of the domains in Mg 2 Al 3 β and β' phases

Keywords: intermetallic compounds, structure analysis, crystal structure Mg2Al3 compound, Samson phase complex metallic alloys, Mg-Al compound, Samson phase

HAL is

Introduction

The β-Mg 2 Al 3 Samson structure is an example of the complex intermetallic structure intensively investigated by Samson [1] and many other authors (for the review see [2]). The Samson phase is stable at a temperature above 214°C. It's elementary cell contains 1,168 atoms distributed over 1,832 atomic positions of the faced centred cubic -FCC crystal lattice (Fd-3m, no. 227 space group, a c =2.8242(1) nm). The structure's skeleton consists of 528 aluminium atoms and 351 magnesium atoms. The site occupation factor (SOF) is equal to one for all the skeleton atoms and they are arranged in a sequence of hexagonal layers [3,4]. The remaining atoms, with SOF<1, partially occupy their structural positions and many of them belong to clusters arranged in a tetrahedral lattice as it has been described in detail in [1,5,6].

The Mg 2 Al 3 compound structure transform into one of the subgroup of the Fd-3m group -the space group R3m, no. 160 below 214°C. The resulting rhombohedral β'-phase has the following lattice constants written in the hexagonal frame: a r =1.9968 (1) nm and c r =4.89114 (8) nm. One obtains the same experimental values of the lattice parameters for both phases by also using the hexagonal coordinates for the FCC structure. This means that there are only some local atomic position changes during the β-β' transition.

It is well known that many elements crystallize in the close packed structures (e.g.: the hexagonal close packed structure -HCP or the face centred cubic structure -FCC). The two components of the discussed Mg 2 Al 3 compound, aluminium and magnesium, also belong to those elements. A pure aluminium crystal lattice is FCC. In the case of Mg atoms, they crystallize in the HCP lattice. The inter-atomic distance of the nearest neighbouring atoms is about 12% larger for magnesium than the corresponding one for the aluminium atoms. Both structures are close packed, although they consist of different sequences of hexagonal layers. The sequence is ABC for the cubic (FCC) structure and AB for hexagonal (HCP) structure. Such configurations of layers are presented in the close packing of hard spheres. One should also remember that the ABC sequence of hexagonal layers is characteristic for all the cubic structures. It is present not only for FCC structure but also for other cubic lattices: simple cubic (SC) and body centred cubic (BCC). However, only every second atomic position in the main hexagonal plane is occupied for the last structures.

The structural features similar to the Samson structure have already been observed in some other complex phases, e.g.: Cu 4 Cd 3 [7] or NaCd 2 [8], where the regions corresponding to the hexagonal domains consisted of fragments of the Laves phase structures. The Laves structures are frequently described in terms of the stacking of hexagonal layers and also by commensurate modulation [9].

Sequence of hexagonal layers

The hexagonal layer sequence in the β-Mg 2 Al 3 structure has been recently published [3,4]. The atomic positions obtained by Feuerbacher et al. [2] were used. All the length values were given in relation to the lattice constancies of the hexagonal reference frame. There are rod-like domains along the main diagonal of the cubic unit cell arranged in the FCC lattice (Figure 2). The 22 layers (only half of them, namely 11, are the skeleton atoms) form the structural domain which consists of about five hundred atomic positions (many of them are only partially occupied). At the end of the main sequence of 11 layers, one can observe transitional areas where Al and Mg atoms swap places. The structure changes afterwards into two short chains of hexagonal layers (Figure 2): a three-layer chain (layers no. 13', 14' and 15') and a two-layer chain (layers no. 17' and 18'). The short chain keeps the inter-layer distances; however, they are shifted, as a whole, with respect to the 22 reference layers. Layer 17 is filled with Al atoms occupying mixed positions marked by the letter M. Such mixed positions are some commensurate positions between different A, B or C points of the hexagonal plane.

The clusters are formed in the β-phase outside the main chain of 11 hexagonal layers (Figures 2 and4). They are conglomerates of atoms which partially occupy their atomic positions in most cases (SOF<1). The spaces between the well ordered hexagonal layers and the clusters are filled with Mg atoms which partially replace the outer Al atoms (this is the case of layers no. 11 and 1). Conversely, within the neighbouring Mg layers, Al atoms partially replace the outer Mg atoms. This replacing mechanism leads from 11 ordered hexagonal layers to some crystalline disorder and a formation of the atomic clusters for the β-phase.

Coherent connection of domains

Figure 5 schematically shows a sequence of layers for the three neighbouring domains of the β'-phase (the I single domain is shown in Figure 1). The arrangement of layers within each domain is identical. The entire set consists of 22 parallel and equallyspaced layers. The distance between the two layers equals 1/22 ≈ 0.0788. The figures show only atoms which lie inside the domains: Al atoms -the thick lines and open circles; Mg atoms -the thin lines and full circles. The relative shift of the neighbouring domains along the z-axis is equal to 1/3. In the figure, three different regions located around z equal to 0.25, 0.383 and 0.647 are shaded. We used these regions for an analysis of the coherency between the different domains. Their size considerably exceeds the area of the circles which determines the maximum size of the close-packed hexagonal domains. Domain III, on the other hand, although its border is marked in the figure, is completely devoid of atoms. Such an observation leads to a conclusion that for this particular cross-section, domain III vanishes while domains I and II extend by taking over the area of domain III. It should be also emphasized that domains I and II border on each other in a coherent manner. For this cross-section, there are no atoms that would not belong to either of those two domains (this holds true for all Al and Mg atoms).

In Figure 7, one can find a cross-section (its z coordinate is equal to 0.647) within which all three domains are filled with atoms in a fairly uniform manner. This cross-section is an equivalent to β'-phase's layer 17'. The conclusion that follows

Conclusions

In this paper the stable configurations of atoms for the closed-packed structures was disccused. The FCC or HCP local configurations are mostly observed for the packing of atoms with a hard-spheres interaction. In the case of the Lennard-Jones interaction, the BCC local arrangement of atoms also becomes stable. All those three configurations play an important role in the structural analysis of Mg 2 Al 3 compound for its high-and low-temperature phases: cubic-β and rhombohedral-β'. In order to compare these two structures, the unit cells coordinates were expressed in a hexagonal lattice. There is a hexagonal superstructure in the base plane, the periodicity of which is 22/3 ≈ 7.3 times larger than the hexagonal lattice constant for Al atoms in a single domain.

The Mg 2 Al 3 structure consists of three types of rod-like hexagonal domains arranged in the sequence typical for a FCC-structure. The core of each domain contains the main chain of 11 hexagonal layers. The building element of the structure consists of hexagonal rings of aluminium atoms completed with dimmers of magnesium atoms as shown in Figure 3. The connection between complexes is carried out by hexagonal Al layers for which every second atomic position is empty (all even layers in Figure 1). The resulting structure consists of a sequence of Al hexagonal layers (with holes) separated by the appropriate Mg layers. An arrangement of such building blocks can be also described by Friauf polyhedra, which are very common for the Laves phases: C14 and C36. The set of 11 layers does not undergo any transformation between β' to β phases. The most important problem is, however, the concentration of the core of the eleven layers. The corresponding value of the concentration is MgAl 2 rather than Mg 2 Al 3 and the magnesium atoms are in deficiency. The Mg atoms are mostly located in the outer part of the domains.

The neighbouring structural domains should merge in a coherent manner. Three different mechanism of such a merging have been discussed for Mg 2 Al 3 . The first one occurs when the atoms occupy only A, B, C or M positions within two domains. The appearance of the third domain causes structural defects. This mechanism is shown in Figure 7 for layer 17' where some Al atoms occupy ~C type positions which lead to instability and, in consequence, an atomic cluster formation observed in the higher-temperature phase.

The second example of a domain's arrangement is shown in Figure 6, where two domains (domain I -layer 9B and domain II -layers 1A and 2B) enlarge at the expense of the third domain, the size pf which is reduced. Similarly to the case described in the previous paragraph, we observe a coherent merge between domains I and II whereas domain III is empty.

The third way of merging is the mutual atomic intergrowth of two domains. This is outlined in Figure 8 for domains I and III. This process leads to a structural disorder observed in the higher-temperature phase and is manifested by an atomic shift with respect to their initial positions.

The Mg atom occupies the centre of the alumina rings for β phase clusters. The resulting local structure is remniscint of the BCC rather than the FCC structure.

Fig. 1 .

 1 Fig.1. (left) Three types of stable atomic positions (A, B or C) for hexagonal lattice of close packed hard spheres. In the figure, intersection points of various lattices (a, b or c) are also presented -for instance A/B or A/C. Such intersections are denoted in the text by a letter M ('M' for Mixed). (right) The skeleton Al atoms of the inner part (r=0.22) of single domain projected onto the (111) plane for the cubic structure β-Mg 2 Al 3 (black dots) and compared to the projection of all Al atoms onto the base plane of the rhombohedral β'-phase (open circles). All the Al atoms of Mg 2 Al 3 occupy A, B or C positions of the hexagonal lattice. Three atoms of layer 17' and three of layers 20' are shifted from the perfect hexagonal positions for the β' phase only (the position of particular layer is given in Figure 2).

Figure 1 (

 1 Figure1(left) presents a hexagonal layer and positions that can be occupied by atoms within this layer. Lines denoted with symbols a, b, c determine the nodes of a hexagonal lattice. An intersection of two a lines leads to an A node. The same holds for lines b and c and the corresponding nodes B and C. Positions A, B and C correspond to the energetic minima of a rigid spheres set. In Figure1one can find other lines interceptions, as well. The resulting nodes are denoted as A/B, A/C and so on. In the further part of the paper symbol M (as 'Mixed') is assigned to all such

Figure 2 .

 2 Figure 2. The single domain of the Mg 2 Al 3 intermetallic compound. The rod-like domain is oriented along the [111] direction for the cubic β-phase and the z-direction for the rhombohedral β'-phase. Open circles represent Al atoms, and full circles -Mg atoms. The black and white symbols describe atoms with SOF=1, grey symbols correspond to positions with SOF<1. There are 11 layers of Al atoms occupying A, B or C hexagonal positions for both phases. The Al layers with the neighbouring Mg layers form the α, β or γ slabs. The two cluster positions are marked for the β-phase. They correspond to the short sequences of hexagonal layers in the β' phase, namely three 13'-15' layers and two 17'-18' layers. The main sequence of 11 hexagonal layers does not change during the transformation from β' to β phase for the Mg 2 Al 3 intermetallic compound. This part of the structure can be also constructed with the Friauf polyhedra discussed by Feuerbacher et al. (2007). If the entire space is filled by Friauf polyhedra, the ratio of Mg atoms to Al atoms would assume a 1:2 value instead of the real 2:3 ratio. The Mg atoms would be missing and the chemical balance is restored by Mg atoms scattered

Figure 3 .

 3 Figure 3. The central part of 3γ slab of the Mg 2 Al 3 structure. Al atoms hexagonal rings with Mg atoms placed symmetrically in the centres of the hexagons. All the atoms are placed in C positions of the hexagonal lattice. For α or β complexes the atoms occupy the A or B positions respectively.

Figure 4 .

 4 Figure 4. The local arrangement of atoms for the Mg 2 Al 3 β-phase clusters. A magnesium atom (dark ball) is placed in the centre of the hexagonal ring of aluminium atoms (light balls). The Mg 2 Al 3 compound structural stem is a set of stacked hexagonal layers filled interchangeably by Mg and Al atoms. In the case of the β'-phase (the lowertemperature phase) these layers are arranged into shorter and longer sequences of hexagonal layers and a transition from one sequence to another occurs by exchanging one type of atoms for another: Al for Mg and Mg for Al. This results in a very large elementary cell.

Figure 5 .

 5 Figure 5. The sequence of 22 hexagonal layers for the β' phase and their arrangement within the particular domains: I, II and III. The thick lines indicate Al layers, thin ones -Mg layers. All three domains are identical. They are shifted by 1/3 with respect to each other Along the z-axis. The structure of the inner area of the first domain is outlined in the figure in detail. The open circles are associated with Al atoms and full circles with Mg atoms. The cross-sections indicated by three shaded regions are analyzed in detail in the text.

Figure 6

 6 Figure6outlines a cross-section of the β'-phase at the level of z=0.25. The areas of particular domains are marked with continuous lines. Domain I (with layers 2B of Al and Mg) and domain II (with layers 1A and 2B) are significantly enlarged. Their size considerably exceeds the area of the circles which determines the maximum size of the close-packed hexagonal domains. Domain III, on the other hand, although its border is marked in the figure, is completely devoid of atoms. Such an observation leads to a conclusion that for this particular cross-section, domain III vanishes while domains I and II extend by taking over the area of domain III. It should be also emphasized that domains I and II border on each other in a coherent manner. For this cross-section, there are no atoms that would not belong to either of those two domains (this holds true for all Al and Mg atoms).

  is that all three domains cannot merge in a coherent way. Domains II and III are, indeed, connected in such a way and at their edges, there are atoms of M type coming from domain I. Conversely, the edge of domain I is occupied by Al atoms being placed at ~C positions, which is at the energetically unstable ones. Their presence results in the appearance of two structural clusters (of fractional SOF value) at both sides of layer 17 for the β-phase.

Figure 6 .

 6 Figure 6. A cross-section of Mg 2 Al 3 structure for z=0.25. All atoms presented in the figure occupy lattice's knots of the domains. Domains I (layer 9B) and II (layers 1A and 1B) grow at the expense of domain III. In the presented cross-section , domain III vanishes -it loses its atoms. Light symbols denote Al atoms and black ones -Mg atoms.

Figure 7 .

 7 Figure 7. A cross-section of Mg 2 Al 3 structure for z=0.647 (I-17' position). All the domains are occupied by Al (open symbols) and Mg atoms (full symbols). Domain I contains atoms positioned at ~C knots, i.e. points at which clusters form when the temperature rises and the β-phase appears. The last cross-section analyzed (z=0.383) is presented in Figure 8. Only two domains (I and III) are marked in this figure; domain II is not shown. The cross-

Figure 8 .

 8 Figure 8. At z=0.383 domains start to overlap each other. The circles (R = 0.29) indicate the borders of the areas belonging to the particular domains. Al-atoms occupying M positions penetrate the neighbouring structural domains.

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgements

This work has been partially supported by Ministry of Science and Higher Education of Poland for contract No. N N202 174135 and also by CMA-NoE activities at AGH-University of Science and Technology.


J A Hinks
email: j.a.hinks@salford.ac.uk

S E Donnelly

Copper Indium Diselenide: Crystallography and Radiation Induced Dislocation Loops

Keywords: radiation damage, defect analysis, dislocation structures, crystallography, semiconductors, TEM, electron diffraction, ion irradiation Copper indium diselenide, CuInSe2, CIS radiation damage, defect analysis, dislocation structures, crystallography, semiconductors, TEM, electron diffraction, ion irradiation, copper indium diselenide, CuInSe 2, CIS

Copper indium diselenide (CIS) is a prime candidate for the absorber layer in solar cells for use in extraterrestrial environments due to its good photovoltaic efficiency and ability to resist radiation damage. Whilst CIS-based devices have been tested extensively in the laboratory using electron and proton irradiation, there is still little understanding of the underlying mechanisms which give rise to its radiation hardness. In order to gain better insight into the response of CIS to displacing radiation, transmission electron microscope samples have been irradiated in situ with 400 keV Xe ions at the Intermediate Voltage Electron Microscope facility at Argonne National Laboratory, USA. At room temperature, dislocation loops were observed to form and to grow with increasing fluence. These loops have been investigated using g.b techniques and inside/outside contrast analysis. They were found to reside on {112} planes and to be interstitial in nature. The Burgers vector has been calculated to be b = 1/6 <221>. The compositional content of these interstitial loops was found to be indistinguishable from the surrounding matrix within the sensitivity of the techniques used. To facilitate this work, experimental electron-diffraction zone-axis pattern maps were produced and these are also presented along with analysis of the [100] zone-axis pattern.

Introduction

Copper indium diselenide (CIS) is a chalcopyrite semiconductor with space group .

Along with its derivatives (Cu(In,Ga)Se 2 ), CIS is a prime candidate for the absorber layers in photovoltaic devices for use in extraterrestrial environments due to its good photovoltaic efficiency and its ability to withstand radiation damage. Efficiencies of up to 19.9% have been achieved in the laboratory [START_REF] Repins | Characterization of 19.9% Efficient CIGS Absorbers[END_REF] and CIS-based modules are now being produced commercially with efficiencies of around 11.5% [START_REF] Powalla | [END_REF].

Investigations into radiation damage in these materials have largely concentrated on electron and proton irradiation of thin films and solar cellsparticularly to explore the effects on photovoltaic properties . Extraterrestrial testing of CIGS-based devices has also been conducted [START_REF] Schock | 2 nd World Conf. Exhib. Photovolt. Sol. Energ. Convers[END_REF][START_REF] Burgess | 24 th IEEE Photovolt. Specialists Conf[END_REF][START_REF] Curtis | [END_REF][START_REF] Guidice | 31 st Intersoc[END_REF][START_REF] Guidice | 11 th Space Photovolt[END_REF][START_REF] Imaizumi | 18 th Space Photovolt[END_REF][START_REF] Imaizumi | [END_REF][START_REF] La Roche | th Euro[END_REF][43] and work using ion irradiation either to explore lattice damage and/or doping has been performed on single crystals and thin films [7,17,[START_REF] Tomlinson | 11 th Euro. Photovolt. Sol. Energ. Conf[END_REF]32,. In particular, work by Mullan et al. using oxygen and neon ions has looked at both electrical and structural effects including transmission electron microscopy (TEM) of ion irradiated samples [17,[START_REF] Tomlinson | 11 th Euro. Photovolt. Sol. Energ. Conf[END_REF]45,[START_REF] Mullan | [END_REF]49,50].

In non-irradiated CIS, structural defects have been reported in thin films (stacking faults and microtwins on {112} planes [65][66][67][68], dislocations and dislocation loops [66][67][68][69][70])

and in Bridgman-grown single crystals (dislocations on {110} planes [65]). Ion irradiation has been found to induce microtwins, stacking faults, dislocations, dislocation loops and amorphisation [17,45,[START_REF] Mullan | [END_REF]71].

In order to gain an understanding of the atomistic mechanisms which give rise to the radiation hardness of CIS, TEM with in situ ion irradiation has been used in this study as this technique enables the internal microstructure of a sample to be imaged during irradiation thus allowing the exploration of dynamic effects (i.e. not just end-state) whilst also controlling and maintaining variables such as temperature. By using heavy inert ions, in this case Xe, at an energy such that they cause large numbers of atomic displacements throughout the thickness of the sample it is possible to monitor both the creation of structural damage and any recovery processes.

Experimental

Sample Preparation and Analysis

Copper, indium and selenium were sealed in evacuated quartz-ampoules which were then heated in a rocking furnace to form CIS via direct fusion of the elements. Single crystals of CIS were produced by refining this material using the Bridgman technique [START_REF] Bridgman | Proc. Am. Acad. Arts Sci[END_REF]. TEM samples were made by core-drilling material from the bulk, back-polishing, dimpling and then ion beam milling to electron transparency. Sample composition was examined using a JEOL JSM-6400 SEM equipped with a Link Analytical LZ-5 energy dispersive spectrometer (EDS).

Transmission Electron Microscopy

Samples were irradiated at room temperature with 400 keV Xe ions in the Intermediate Voltage Electron Microscope (IVEM) at Argonne National Laboratory, USA. The electron beam was operated at 100 kV and remained on during ion irradiation to allow video recording of the experiments. The IVEM facility consists of an Hitachi H-9000NAR coupled to an ion beamline with two sources capable of operating at up to 300 kV and 2 MeV with the ion beam forming an angle of 30° with the optical axis of the microscope [START_REF] Allen | [END_REF][74][75].

Defect analysis was performed using a JEOL JEM-200CX at the University of Salford, United Kingdom. Scanning TEM (STEM) EDS was performed using a FEI CM200 at the University of Leeds, United Kingdom.

Experimental zone-axis pattern maps (ZAPMaps) were produced by tilting along major Kikuchi lines and recording the electron diffraction patterns of major zone-axes.

Theoretical zone-axis patterns produced using JAVA Electron Microscopy Software (JEMS) [76,[START_REF] Stadelmann | Electron Microscopy Software Java Version[END_REF] were used to identify and verify the experimental patterns. JEMS was also used in the indexing of diffraction spots and Kikuchi lines. All angles were calculated assuming a = 5.789 Å and c = 11.612 Å [65].

The habit plane of the dislocation loops was identified using g.b analysis techniques.

Electron micrographs were taken using underfocused bright-field two-beam imaging conditions for both g and -g for each diffraction vector of interest and with a positive deviation parameter (s g > 0). For a sample orientation where g.b = 0 (i.e. the Burgers vector of the dislocation loop is at a right angle to the diffraction vector) the diffraction contrast caused by the strain field of the loop will be zero as the distortion of the surrounding planes will be orthogonal to the direction required to bend the planes into the Bragg condition.

Conversely for g.b = 1 conditions, the strain field around the loop contorts the matrix in the correct manner in order to strongly-excite Bragg diffraction. By establishing two non-parallel diffraction vectors for which the g.b = 0 condition exists, it is possible to identify the habit plane and thus calculate the Burgers vector of a dislocation loop. See references [START_REF] Edington | Practical Electron Microscopy In Materials Science[END_REF][START_REF] Jenkins | Characterization of Radiation Damage by Transmission Electron Microscopy[END_REF][START_REF] Williams | Transmission Electron Microscopy: a Textbook for Materials Science[END_REF] for further details.

In this study the chalcopyrite structure only featured one diffraction vector which was at 90° to the Burgers vector within the confines of sample orientation and available tilt of the In order to determine whether the loops were interstitial (extrinsic) or vacancy (intrinsic) in nature, the inside/outside contrast technique was utilised. For a given combination of diffraction vector direction and sign of deviation parameter there will be a particular sense of planar rotation required to approach the Bragg condition. Only certain regions of the strain field around a dislocation loop will bend the planes of the surrounding crystal towards the Bragg condition and this causes diffraction contrast to be generated either on the inside or the outside of the projection of the dislocation loop. This is illustrated in Figure 1 where the direction of planar distortion is indicated for four different combinations of loop nature and inclination. Also shown are two combinations of diffraction vector and deviation parameter with the direction of planar rotation required to approach the Bragg condition indicated for each situation. Comparison of the rotation required to approach the Bragg condition with the rotation due to the strain fields around the loops reveals that diffraction contrast will form inside the projection of the dislocation for the examples on the left of the diagram and outside for those on the right.

FIGURE 1

Once the habit plane of a dislocation loop is known then the direction of inclination can be determined using knowledge of the crystallography, sample orientation and the mechanical tilt performed using the goniometer and TEM sample rod. Then by reversing the diffraction vector or deviation parameter it is possible to establish the nature of a loop by comparing the images produced and noting the relative shift in the position of the diffraction contrast. See references [START_REF] Edington | Practical Electron Microscopy In Materials Science[END_REF][START_REF] Jenkins | Characterization of Radiation Damage by Transmission Electron Microscopy[END_REF][START_REF] Williams | Transmission Electron Microscopy: a Textbook for Materials Science[END_REF] for further details.

Results

Zone-Axis Pattern Maps

(1)

[100] Diffraction Pattern

Table 1 lists the allowed reflections in the chalcopyrite structure of CIS. Comparison with the indexed [100] zone-axis pattern reveals the presence of forbidden reflections as shown in

(2) Using the fractional coordinates of the atoms in the lattice basis of CIS listed in Table 2, it is possible to calculate the structure factor via Equation 8. The intensity of the 024-type reflections in the [100] zone-axis pattern are comparatively high especially considering their greater angular displacement from the 000 relative to the other reflections shown in Figure 4.

(3) (4) (5) (6) (7)
This is due to the intensity contributed by the scattering factor of selenium which is reinforcing in these reflections but subtractive in other reflections such as the 020 and 004 as demonstrated in Equations 9 to 11.

(8) 020: (9) 004: (10) 024: (11)

Dislocation Loops

Loop Development

Dislocation loops were found to form in copper-rich indium-poor selenium-rich TEM samples when irradiated at room temperature with 400 keV Xe ions as shown in Figure 5.

FIGURE 5

Three samples were irradiated to fluences of 1.3×10 13 , 7.5×10 13 and 1.3×10 14 ions.cm -2 (corresponding to approximately 0.25, 1.5 and 2.6 displacements per atom at a depth of 50 nm from the irradiated surface, respectively) resulting in typical dislocation loops sizes of 5, 40 and 50 nm, respectively. The loops were seen to form and grow under irradiation and on occasions disappear in a period of less than 1/30 th of a second (one frame of video). On occasions, loops also appeared to merge although it is hard to be definitive based on the evidence of the two-dimensional projection of the regions of the strain fields satisfying the Bragg conditionit can be difficult to conclusively distinguish between two strain fields overlapping in projection and two loops amalgamating. However, as the loops were growing it is highly likely that some would have come sufficiently close to amalgamate. The loops were stable once irradiation ceased. Figure 6 shows dislocation loop evolution in one area over several minutes as a function of fluence.

Habit Plane and Burgers Vector

The sample irradiated to the highest fluence of 1.3 × 10 14 ions.cm -2 , and hence with the largest defects, was used for determination of dislocation habit plane and Burgers vector.

Figure 7 shows eight TEM micrographs which demonstrate the habit plane of the dislocation loops to be {112}. The micrographs taken with g = and produce twin lobed contrast features consistent with end-on loops which are pure edge in character. All six of the diffraction vectors which are orthogonal (or nearly orthogonal) to the and result in invisibility of {112}-type loops. This conclusion is further illustrated in Figure 8 using annular dark field (ADF) TEM and in Figure 9 using weak-beam dark-field (WBDF) TEM;

both techniques generate contrast corresponding to the most heavily distorted planes which are located closest to the centre of a dislocation loop and thus give better information on the actual dislocation rather than about the more widely dispersed regions of the associated strain field. As can be seen in Figures 8 and9, for both ADF and WBDF the observed contrast is consistent with loops residing on the {112} planes.

Based on this habit plane, the Burgers vector can therefore be calculated as b = 1/6 <221> as illustrated in Figure 10

Loop Nature

The loops were demonstrated to be interstitial in nature. As can be seen in Figure 11, a reversal of the diffraction vector from to resulted in a shift from outside to inside contrast which is verified by the intensity-linescan data plotted in Figure 12 for one of the dislocation loops shown in Figure 11. The situation is represented schematically in Figure 13.

Loop Contents

No variation in composition between the contents of the dislocation loops and the surrounding matrix was detected by STEM EDS as demonstrated in Figure 14. The instrument used is estimated to be capable of detecting a variation of 5 at.% in a single endon interstitial loop of this size. This suggests that the interstitial planes in the loops are close to being stoichiometricas ideal {112} planes are stoichiometric (containing Cu, In and Se in a ratio of 1:1:2) this is not an unreasonable situation. No other elements were detected in either the loops or the surrounding matrix.

FIGURE 14

Additional analysis using electron energy loss spectroscopy on SuperSTEM 2 at Daresbury Laboratory, United Kingdom, also did not detect any variation in composition. As the loops grew and did not appear instantaneously, they must have formed through the agglomeration of point defects or defect clusters rather than forming as the remnants of individual collision cascades. The most likely causes for the sudden disappearance of any given dislocation loop was the proximity of a surface (a situation reached through loop growth or migration) or the disordering effects of a nearby ion induced collision cascade and thermal spike.

Discussion

Because the loops were extrinsic in nature they must have been formed through the agglomeration of mobile interstitial defects. However, this raises the question as to the fate of the vacancies whose creation must necessarily have been concurrent with that of the interstitials which went on to constitute the loops. Vacancies could be are relatively immobile in CIS and/or they could have become entangled in defect complexes as discussed below.

The migration mechanisms for copper, indium and selenium in CIS are not completely understood with the literature containing a complex picture featuring a range of proposed mechanisms and migration energies often influenced by composition . However, despite the ambiguities in this area it is clear that selenium is immobile at room temperature by any migration mechanism and therefore so too must its vacancies [START_REF] Parkes | [END_REF][105][106][107].

Zhang et al have calculated creation energies for a range of defect complexes from first principles and found the formation of u -+ n u to have a particularly low energy [116][START_REF] Wei | 11 th Int. Conf. Ternary Multinary Compd[END_REF][START_REF] Wei | [END_REF][119][120][121]. Other work by Klais et al has predicted 2V Cu +In Cu to be abundant throughout the compositions they considered and also found experimental evidence for its existence through positron annihilation [122]. Therefore it is quite possible that the copper vacancies could be become entangled in complexes such as the 2V Cu +In Cu which requires three V Cu .

Unfortunately, the 2V Cu +In Cu does not explain the fate of the indium vacancies. These could have similarly been immobilised by V In consuming complexes with relative low creation energies such as: Cu In +2Cu i [123,124]; Cu In +V Se [120]; Se Cu +Se In [125]; or any of Cu In +Cu i +V Se , Cu In +V Cu , Cu In +V Cu +V Se and V Se +Cu In [122].

As the samples in which dislocation loops were observed were slightly selenium rich it is also quite possible that the selenium vacancy population was suppressed by this excessa similar argument works for copper but not for indium due to the composition of the samples studied. However, the indium vacancy population could have been suppressed by a copper excess as the Cu In antisite defect has been calculated to have a relatively low creation energy of between 1.11 and 1.54 eV [120,[126][127][START_REF] Neumann | I-III-VI 2 -Halbleiter 2 in Verbindungshalbeiter[END_REF][START_REF] Rincón | [END_REF] (although there is one out-lying value of 1.90 eV [130]). direction in which the sample must be rotated in order for the deviation parameter (s g ) to be reduced such that these distorted planes approach the Bragg condition (i.e. diffraction vector, g, approaches the scattering wavevector, K). This causes diffraction contrast to be formed on the inside of the dislocation loop for the situations depicted on the left and on the outside for those on the right as shown in (c) (after [START_REF] Edington | Practical Electron Microscopy In Materials Science[END_REF][START_REF] Williams | Transmission Electron Microscopy: a Textbook for Materials Science[END_REF]).

Reflection Condition

hkl h + k + l = 2n hk0 h + k = 2n 0kl k + l = 2n hhl 2h + l = 4n 00l l = 4n h00 h = 2n h 0 h = 2n

F

  TEM sample rod. Therefore, diffraction vectors at 82.2° and 82.5° to the Burgers vector were used as these come close to the ideal 90° angle required for an absolute g.b = 0 condition.Although technically two g.b = 0 conditions are sufficient, one g.b = 0 and two g.b ≈ 0 conditions were used in order to ensure correct identification of the habit plane and Burgers vector.

  The experimental ZAPMaps produced to facilitate the characterisation of dislocation loops are shown in Figures2 and 3. Theoretical patterns were generated by JEMS and used for the identification and verification of the experimental zone-axis patterns. When tilting between zones in the TEM it is useful to know the angles between zones and these can be calculated using Equation1for tetragonal systems such as CIS. The experimental ZAPMaps reproduced here are complimentary to the CIS ZAPMap published previously by Kiely et al.[65] which was cornered on the [001], [021] and [221] zones.

Figure 4 .

 4 Figure 4. These can be explained by various double-diffraction events such as those listed in Equations 2 to 7. Reflections arising from double diffraction events have also been identified in the [001] and [021] zone-axis patterns of CIS by Kiely et al [65].

  FIGURE 10

  Dislocation loops were formed in CIS using 400 keV Xe ions at room temperature. These loops were interstitial in nature as shown through inside/outside contrast inversion and resided on {112} planes as demonstrated using g.b, weak beam dark field and annular dark field techniques. The Burgers vector has been calculated as b = 1/6 <221>. The loops grew through the agglomeration of mobile interstitial defects. They sometimes disappeared instantaneously either due to a local ion induced collision cascade and thermal spike or by growing/migrating into the proximity of a surface. Whilst the interstitials produced by displacing radiation clearly formed dislocation loops, the fate of the vacancies requires further study but could have been suppressed by the excess copper and selenium in the sample composition and/or entanglement in the defect complexes known to offer stable structures in defected CIS.

F

Table 2 :Figure 1 :

 21 Figure 1: Schematic diagram showing the directions in which the planes of a crystal are rotated by the strain fields of (a) interstitial loops and (b) vacancy loops; (d) illustrates the

Figure 2 :

 2 Figure 2: Quadrant of experimental [100]-centred ZAPMap.

Figure 3 :

 3 Figure 3: Experimental [221]-centred ZAPMap.

Figure 4 :

 4 Figure 4: Indexed experimental diffraction pattern of the [100]-zone with reflections forbidden by structure factor considerations indicated with dashed boxes.

Figure 5 :

 5 Figure 5: Underfocus bright-field down-zone TEM micrograph of radiation induced dislocation loops in CIS. The areal density of the defects increased with sample thickness with the thinnest areas towards the bottom of the image being too thin to accommodate loops.

Figure 6 :

 6 Figure 6: Underfocus bright-field down-zone TEM micrographs showing the development of ion beam induced dislocation loops under continued ion irradiation. Attention is drawn to examples of loop growth (follow defect highlighted with dashed box in panel (a)) and destruction (follow defect highlighted with dashed box in panel (g)). Scale marker applies to all eight micrographs.

Figure 7 :

 7 Figure 7: Underfocus bright-field two-beam TEM micrographs (s g > 0) showing determination of dislocation loop habit plane using g.b analysis. Attention is drawn to the four dislocation loops highlighted by the dashed boxes in micrographs (a) and (b) as examples of {112}-type loops which lie on the ( ) planes as demonstrated by their contrast under the various g.b conditions shownhowever, other instances are also evident in these micrographs. Note that there is no inside/outside contrast reversal for {112}-type dislocation loops between the micrographs in (a) and (b) as the loops are not inclinedthis is contrary to the situation depicted in Figure 11 where the dislocation loops have been deliberately inclined in order to produce inside/outside contrast reversal with inversion of the diffraction

Figure 8 :

 8 Figure 8: Annular dark-field TEM micrograph of {112}-type dislocation loops, in a close to edge-on orientation, indicated with white arrows (image courtesy of R. Brydson, University of Leeds, United Kingdom).

Figure 9 :

 9 Figure 9: Underfocus TEM micrographs with g = (s g > 0) and corresponding diffraction pattern. Scale marker applies to both micrographs.

Figure 10 :Figure 11 :

 1011 Figure 10: Determination of the Burgers vector using the Finish-Start/Right-Hand (FS/RH) convention looking at (112) planes down the [ ] direction. In the FS/RH convention, the Burgers vector is determined by forming a closed circuit around the dislocation (from finish to start) in a clockwise direction (right hand). The dislocation is then removed to leave a perfect crystal and the Burgers vector is the vector required to close the circuit. As shown here, the Burgers vector for the dislocation loops in CIS on {112} planes is b = 1/6 <221>.(Diagrammatic depiction of FS/RH convention after[START_REF] Jenkins | Characterization of Radiation Damage by Transmission Electron Microscopy[END_REF].)

Figure 12 :

 12 Figure 12: Linescan of contrast against distance across a dislocation loop shown in Figure 11 confirming change in position of contrast due to reversal of direction of diffraction vector from g = (─) to g = (─).

Figure 13 :

 13 Figure 13: Schematic diagram illustrating the combinations of dislocation loop inclination, nature and diffraction vectors for a positive value of the deviation parameter (s g > 0) which lead to the switching of the location of the diffraction contrast between the inside and outside

Figure 14 :

 14 Figure 14: STEM EDS results for a region containing end-on interstitial dislocation loops showing (a) an ADF image and corresponding maps for the (b) copper, (c) indium and (d) selenium signals (courtesy of R. Brydson).

Table 1 :

 1 Allowed reflections, derivable from Table2and Equation8, in the chalcopyrite structure of CIS where h, k and l are the non-zero Miller indices and n is an integer.

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Acknowledgements

The authors wish to thank: R.C. Birtcher, E. Ryan, P. Baldo and A. Liu at the IVEM at Argonne National Laboratory, USA; M.F. Beafourt of the University of Poitiers, France, for advice on dislocation loop analysis and weak beam dark field TEM; R. Brydson at the University of Leeds, United Kingdom, for help with scanning TEM; and B. Mendis and everybody at SuperSTEM, Daresbury Laboratory, United Kingdom.