CHEMPHYSCHEM COMMUNICATIONS DOI: 10.1002/cphc.201400029
Electrically Controlled Eight-Spin-Qubit Entangled-State Generation in a Molecular Break Junction Mausumi Chattopadhyaya,[a] Md. Mehboob Alam,[a] Debasis Sarkar,[b] and Swapan Chakrabarti*[a] The generation of spin-based multi-qubit entangled states in the presence of an electric field is one of the most challenging tasks in current quantum-computing research. Such examples are still elusive. By using non-equilibrium Green’s functionbased quantum-transport calculations in combination with non-collinear spin density functional theory, we report that an eight-spin-qubit entangled state can be generated with the high-spin state of a dinuclear Fe(II) complex when the system is placed in a molecular break junction. The possible gate operation scheme, gating time, and decoherence issues have been carefully addressed. Furthermore, our calculations reveal that the preservation of the high spin state of this complex is possible if the experimentalists keep the electric-field strength below 0.78 V nm1. In brief, the present study offers a unique way to realize the first example of a multi-qubit entangled state by electrical means only.
The groundbreaking technology advancements in the past three decades have finally made it possible to welcome Feynman’s proposition on the possibility of developing a computer which, in principle, should obey the weird and wonderful laws of quantum mechanics.[1] Unlike bits in a classical computer, a quantum bit or qubit in a quantum computer can exist in a coherent superposition of 0 and 1, in addition to the classical states alone.[2] Although the realization of a full-scale working quantum computer is still a fairy tale, the relentless efforts of experimentalists and theoreticians will definitely help in achieving the goal in the near future.[3–5] At present, superconductors, trapped ions, nuclear magnetic resonance in organic liquids, quantum dots,[6] molecular magnets,[7] Bose–Einstein condensates,[8] and so on are considered as the most promising candidates for generating “off” and “on” states, a prerequisite condition for the accomplishment of a qubit. Apart from these well-studied systems, certain new molecules like naphthalocyanine[9] and the TbIII-pthalocyanina-
[a] Dr. M. Chattopadhyaya, M. M. Alam, Dr. S. Chakrabarti Department of Chemistry University of Calcutta 92 A.P.C. Road, Kolkata-700009 (India) E-mail: [email protected] [b] Dr. D. Sarkar Department of Applied Physics University of Calcutta, 92 A.P.C. Road, Kolkata 700 009 (India) Supporting Information for this article is available on the WWW under http://dx.doi.org/10.1002/cphc.201400029.
2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
to complex[10] have started captivating the world of quantum computers. Both these systems have the ability to generate an “off–on” state. However, in these studies, the key component of the quantum-computing device is either in direct interaction with the external world that eventually leads to the loss of quantum information, or the “off–on” feature is achieved only in the presence of the external magnetic field. To date no single-molecular system could be found in the literature where the prime element of quantum-information processing is shielded from the external world and at the same time qubits are generated in the absence of an external magnetic field. In this communication, we show that an electric-field-induced spin-crossover (EFIS) between the high spin–high spin (HS–HS) and low spin–low spin (LS–LS) states of the [(pypzH)(NCSe)Fe(m-pypz)2Fe(NCSe)(pypzH)] complex[11] could be achieved at a field strength of Fc = 0.78 V nm1. Besides, the current obtained from the HS–HS state (“on”) of this complex is much larger than that of the LS–LS state (“off”) and surprisingly, the “on–off” behavior is originating from the spin-state-selective quantum-interference effect. Moreover, we have probed
Figure 1. Schematic representation of the dinuclear Fe(II) complex adsorbed on the Au(111) surface.
the quantum entangled state for the HS–HS state of the complex in a molecular break-junction setup (Figure 1) which ultimately offers us an unprecedented way to realize a molecular spin qubit within an electrical circuitry and this multi-partite spin-qubit feature is quite achievable if Fc < 0.78 V nm1. The main reason behind the selection of this particular complex is that at the experimental level, the single-step spin transition between the HS–HS and LS–LS states of this complex occurs at a temperature of 225 K,[11] indicating that the energy difference of the two spin states is very small. Furthermore, the absence of the hysteresis loop in the temperature-depenChemPhysChem 2014, 15, 1747 – 1751
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CHEMPHYSCHEM COMMUNICATIONS dent magnetic susceptibility variation points out that the spincrossover in this complex is a molecular property.[11] With this background, our in silico investigation begins with the geometry optimization of the HS–HS and LS–LS configurations of the selected di-nuclear Fe complex. Polarizability (a) calculations on the optimized geometries were carried out with the triple zeta valence plus polarization (TZVP) basis set in combination with the Perdew–Burke–Ernzerhof (PBE) functional.[12] It is worth recalling that at 300 K, the HS–HS configuration is more stable than the LS–LS configuration, by an amount of 5.5 meV, and it is hardly possible to achieve this milli-Hartree accuracy at any level of density functional theory (DFT) calculations and hence we have relied upon the experimental energy difference of the two spin states. Although DFT fails to provide an accurate energy difference between the two spin states, its performance in evaluating the properties, namely, the first and second derivative of the energy with respect to the electric field are quite satisfactory,[13] that is, dipole moment (p) and a values are less sensitive toward the choice of the DFT functionals. Our calculations reveal that the value of the component of a in the direction of transport (a j j ) of the LS–LS state is 738.85 a.u while the corresponding a j j of the HS–HS state is only 575.11 a.u. As a result, the second-order Stark (SOS) contribution[14–16] of the LS–LS state will definitely help overcome its initial stability deficiency at some critical field strength. Including the SOS contribution, the variation of
www.chemphyschem.org tant role in determining the Fc value needed to start the EFIS process. In our case, the applied bias required for the initiation of EFIS is 0.54 V and indeed this is quite close to that obtained (0.4 V) by Hao et al.[16] The details are presented in the Supporting Information (SI). To examine the possible electrically controlled spintronicdevice[20–24] action from the studied EFIS system, we theoretically have monitored the current–voltage (I–V) characteristics[25] of HS–HS and LS–LS states in the bias range 400 mV. The
Figure 3. I–V characteristics of the HS–HS and LS–LS states of the Fe(II) complex. The spin-polarized current of the HS–HS state (a and b spin channel) is shown in the inset.
Figure 2. Variation of the EGS for the LS–LS and HS–HS statesof the dinuclear Fe(II) complex. EGS is the ground-state energy of the respective spin states calibrated with respect to the HS state. The arrow denotes the point of electrostatic spin-crossover.
the total energy of both spin states with electric field is shown in Figure 2, where the two energy curves are crossing at a critical field strength Fc = 0.78 V nm1, and this field strength is quite attainable in a scanning tunneling microscopy (STM) setup. In this context, it is worth commenting that in the recent past, several groups reported a spin-crossover in the presence of an electric field[17] and studied the hysteresis behavior in a spin-crossover molecule.[18, 19] More recently, Hao et al.[16] showed that a molecular junction can play an impor 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
bias-dependent variations of the net current obtained from the HS–HS and LS–LS states are depicted in Figure 3 and the inset shows the spin-resolved current contribution for the a and b spin channel of the HS–HS state, separately. Figure 3 demonstrates that the current obtained from the LS–LS state is much smaller in comparison to that of the HS–HS state. Moreover, in the HS–HS state, only the b spin channel is participating in the transport process, keeping the a spin channel silent. Therefore, the LS–LS and the b spin channel of the HS–HS state will act as “off” and “on” states, respectively. This “on–off” feature is quite realizable across the Fc, equivalent to the bias = 320 mV and obviously the studied range of bias is sufficient to capture this interesting phenomenon. It is to be noted that the transport calculations have been performed without considering the multi-reference effect[26–28] explicitly. However, the transmission coefficient obtained from the non-collinear spin DFT calculation, an equivalent form of the multi-reference theory, is found to be quite similar with the simple non-equilibrium Green function DFT results (see Figure S4 in the SI). Nonetheless, at this stage, we need to address the most fundamental question of why the current obtained from the HS–HS state is so high in contrast to the LS-LS state of this material albeit eg type 3d orbitals of Fe(II) ions in the LS-LS configuration are vacant. ChemPhysChem 2014, 15, 1747 – 1751
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Figure 4. Transmission spectrum of the HS–HS and LS–LS state of the Fe(II) complex.
www.chemphyschem.org connected to the electrodes. However, in the present two probe set up, both the HO and LU b spin orbitals of the HS–HS state are lying below the EF. As a consequence, the pairwise (occupied–unoccupied) denominators of G(0) of the LS–LS state will appear with opposite sign while the corresponding sign of the HS–HS state will be the same. Therefore, the nature of the in-
To answer this strange finding, we have examined the spin-polarized transmission spectrum (SPTS) as depicted in Figure 4. The left panel of Figure 4 reveals that in the b spin channel, the number of transmission peaks with amplitude nearly equal to one is present just near the Fermi level EF and this is fully consistent with the large current obtained from the b spin channel of this material. Besides, the right panel of Figure 4 illustrates that the electrons also get transmitted near the EF of the LS-LS state but in this case, the transmission amplitude is rather very Figure 5. Dominant MPSH states of the HS–HS (a, b, c) and LS–LS (d, e) states of the Fe(II) complex. small which merely indicates that the presence of the eigen state having suitable energy in the active region of the twoterference between the propagating electronic waves in the probe set up does not necessarily help transmit the electrons HS-HS and LS-LS states will crucially be guided by the sign of from one electrode to the other. the product of the orbital coefficients, Crk Csk* . To get insight To shed light on the origin of the poor transmission near the EF of the LS–LS state, we have inspected the nature of the into the sign of Crk Csk* , we have investigated the molecular pro(0) zeroth order molecular Green function, G of both the spin jected self-consistent Hamiltonian (MPSH) states near the EF of states. The G(0) can be expressed in terms of energy and orbital both the spin states and the selective MPSH states are depicted in Figure 5. coefficients[29] as [Eq. (1)]: Interestingly, in the b spin channel of the HS–HS state, the X Crk C * sk 3d orbitals centered on the two Fe(II) ions in the given MPSH Gðrs0Þ ðE Þ ¼ ð1Þ EF ekih states are in the same phase and the product Crk Csk* for the ocK cupied and unoccupied orbitals are of the same sign which ultimately leads to the constructive interference. Besides, where Crk is the kth orbital coefficient at atom r and 2k is the Crk Csk* for the occupied and unoccupied orbitals of LS–LS state energy of the kth orbital. Considering the particle–hole sym(0) are also appeared with same sign, however, due to different metry, we can expand the G in terms of the occupied and unsign of the denominators of the G(0), the propagated electronic occupied orbitals separately and the modified expression for wave will face destructive interference and is responsible for our case will be [Eq. (2)]: the poor transmission amplitude vis--vis negligibly small curX CFe1OCC C * X CFe1UNOCC C * rent obtained from this spin state (LS–LS). This spin state deð0 Þ Fe2OCC Fe2UNOCC G ¼ þ ð2Þ E eOCC ih UNOCC EF eUNOCC ih pendent constructive/destructive interference[30] in a transition OCC F metal complex and that too in presence of electric field is unNormally, EF lies between the highest occupied (HO) and precedented. The appealing I–V characteristics of this Fe(II) complex has lowest unoccupied (LU) orbitals of a bare molecule and this raised our interest to check the potential of this system in holds good for the LS–LS state even when the Fe(II) complex is 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
ChemPhysChem 2014, 15, 1747 – 1751
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CHEMPHYSCHEM COMMUNICATIONS qubit generation, for which certain superposition/entangled state is required when the system is placed in the two-probe setup. To find out the existence of such a state, if any, we keep our trust in the non-collinear DFT technique.[31] It is worth noting that the standard Kohn–Sham equations for different spin directions, marked a and b, do not decouple and a result standard DFT fails to capture the microscopic picture of the spin spirals, canted spins and domain walls of ferromagnetic materials. In the non-collinear DFT formalism, the exchange correlation field is modified by adding an internal magnetic field contribution that helps break the spin symmetry correctly. In our approach, we have started with the HS–HS state and placed the spin on the electrodes and the two Fe atoms with two sets of initial rotation angle (q) and have noticed that the system self consistently goes into a state where the spins on the two Fe(II) ions prefer to orient themselves along the x axis (Figure 6 a). This amazing observation clearly reveals that in the HS–HS state, the Fe(II) complex has the ability to generate an eight-qubit multipartite Greenberger-Horne-Zeillinger (GHZ) entangled state. The possible existence of the GHZ state, pffiffiffi 1 2ðj00000000i þ j11111111iÞ, is a rudimentary step towards the realization of the multiqubit feature from this complex. However, to arrive at any decisive conclusion, we first need to make sure whether or not the system obeys the famous Di Vincenzo’s checklist.[32, 33] First of all, the universal Hadamard (H) gate operation on the first qubit of j00000000i pffiffiffi will generate 1 2ðj00000000i þ j10000000iÞ, and in our case the rotation gate will be Ry ðp=2ÞRz ðpÞPhðp=2Þ: On the resultant state, successive CNOT operation with the sequence of rotations Riz ðp=2ÞRjx ðp=2ÞRjy ðp=2ÞJij ðp=2ÞRjy ðp=2Þ will produce the target GHZ state. The experimental realization of the H gate operation is quite feasible since the zero field splitting (ZFS) parameter of this complex is moderately large (10.04 cm1) and is evident from Figure 6 b. It is important to mention here that both spin–spin interactions and spin–orbit coupling are incorporated in the ZFS calculation and this calculation has been done with the PBE functional[12] in combination with TZ2P basis functions. According to Levitin and Toffoli,[34] the single qubit gate operation time is limited by the expression t ¼ h=2Emax . Here, the Emax can be computed from the ZFS pattern as consistent with the energy of S = + 4 and + 3. Thus,
www.chemphyschem.org experimentally, an appropriate p=2ð y Þ microwave pulse, with an penergy gap between + 4 and + 3, will lead to the ffiffiffi 1 2ðj00000000i þ j10000000iÞ state, and the minimum time required for this operation will be 0.19 ps. Similarly, the time required for each CNOT gate operation[35] is of the order of ðp=4JÞh, where J represents the Heisenberg coupling constant. The spin-flip DFT calculations reveal that the value of J of this complex will be 0.3 meV and thus the total seven CNOT gate operations will take 1.89 ps (0.27 ps for each). At this stage, it difficult to comment on the possible Scheme for the experimental realization of the CNOT gate operations since it requires extensive investigations addressing many complicated issues and will be considered in a separate work. Nevertheless, we have given the primary evidence that CNOT gate operations are feasible and can be done with in a very short period of time. Apart from these important issues associated with qubit generation, the most crucial point of DiVincenzo’s criteria,[32, 33] is the estimation of the decoherence time. In the high-temperature limit, the Caldeira–Leggett master equation[36] for the pure decoherence problem can expressed as [Eq. (3)]: h_h ii _ d=dt 1s ðtÞ ¼ i Hs ; 1s ðtÞ 2Mg0 kB T X X 0 ; 1s ðtÞ
_
_
0
_
ð3Þ
Here, the first term of the right-hand side represents the Liouville-von Neuman term with frequency shifted Hamiltoni^ 0 s , and the second term denotes decoherence in the posian, H tion basis with squared separation, ðX X 0 Þ2 . In this term, M is the mass of the system, g0 is the effective coupling constant, T is the temperature of the bath and kB is the Boltzmann constant. In the position representation, the last term of this ex 2 0 0 pression can be re-written as g0 X X ldb 1s X; X ; t , which basically describes the spatial localization with decoherence pffiffiffiffiffiffiffiffiffiffiffi 2 time tD =tR ¼ h Dx 2mkB T . With this formula, the ratio of the decoherence (tD) and relaxation (tR) time for our system coupled with the Au electrode bath comes out as 0.1 microsecond. It is important to mention here that ldB has been calculated at temperature 300 K and in our case Dx is 0.41 nm. Theoretically, prior to decoherence, 105 gate operations could be performed. We have also checked the quantum fidelity factor, defined as (G = 1/2AB1) and the value depends on how good the real rotation matrix (B) approximates the ideal rotation matrix (A). The calculations suggests the fidelity factor in the present case will be ( ~ 1), establishing the ability of the complex in multi-qubit generation in presence of combined electric field and pulsed microwave radiation. It is worth noting that the GHZ state is obtained from the HS-HS configuration whose b spin channel would act as j1i Figure 6. a) Existence of the entangled state of the Fe(II) complex in between the molecular junction, and b) ZFS and the a as j0i. Therefore the diagram. 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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CHEMPHYSCHEM COMMUNICATIONS preservation of the HS–HS state in the presence of the applied bias is essential and hence the field strength should not exceed the value, Fc = 0.78 V nm1, otherwise the system will go to the LS–LS configuration and will no longer act as qubit generator. In summary, we conclude that EFIS at an experimentally achievable field strength can be accomplished in the studied dinuclear Fe(II) complex. Quantum-transport calculations reveal that the current obtained from the HS–HS state is much larger than that of the LS–LS state, and that the huge difference in the current of two spin states is arising out of the unprecedented spin-state-selective quantum-interference effect. To justify the prospect of qubit feature, we have identified the possible existence of a Greenberger–Horne–Zeillinger state in a break-junction setup. Our calculations reveal that approximately 105 gate operations with fidelity factor ~ 1 could be performed before the system undergoes decoherence. Details of the gate operations, including pulse shaping with optimal control theory and relaxation process, will be investigated in the next project. Finally, we believe that the present in silico findings will encourage experimentalist to build up the firstever electronic-spin-based qubit operating in the absence of an external magnetic field.
Acknowledgements M.C. and M.A. thankfully acknowledge CSIR, Govt. of India, for senior research fellowships and S.C. conveys his thanks to CRNN, University of Calcutta, and the INDNOR Project for research funding. Keywords: entanglement · non-equilibrium Green’s function · quantum interference · quantum transport · spin-qubits [1] N. Zhao, J. Wrachtrup, Nat. Mater. 2013, 12, 97 – 98. [2] J. J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. L. Morton, F. A. Zwanenburg, D. N. Jamieson, A. S. Dzurak, A. Morello, Nature 2013, 496, 334 – 338. [3] J. J. L. Morton, A. M. Tyryshkin, R. M. Brown, S. Shankar, B. W. Lovett, A. Ardavan, T. Schenkel, E. E. Haller, J. W. Ager, S. A. Lyon, Nature 2008, 455, 1085 – 1088. [4] J. Clarke, F. K. Wilhelm, Nature 2008, 453, 1031 – 1042. [5] L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, Nature 2009, 460, 240 – 244. [6] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, J. L. O’brien, Nature 2010, 464, 45 – 53.
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www.chemphyschem.org [7] S. Takahashi, I. S. Tupit syn, J. van Tol, C. C. Beedle, D. N. Hendrickson, P. C. E. Stamp, Nature 2011, 476, 76 – 79. [8] A. Vinit, E. M. Bookjans, C. A. R. S de Melo, C. Raman, Phys. Rev. Lett. 2013, 110, 165301 – 165305. [9] P. Liljeroth, J. Repp, G. Meyer, Science 2007, 317, 1203 – 1206. [10] R. Vincent, S. Klyatskaya, M. Ruben, W. Wernsdorfer, F. Balestro, Nature 2012, 488, 357 – 360. [11] B. A. Leita, B. Moubaraki, K. S. Murray, J. P. Smith, J. D. Cashion, Chem. Commun. 2004, 156 – 157. [12] J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 1996, 77, 3865 – 3868. [13] A. Droghetti, S. Sanvito, Phys. Rev. Lett. 2011, 107, 047201. [14] N. Baadji, M. Piacenza, T. Tugsuz, F. D. Sala, G. Maruccio, S. Sanvito, Nat. Mater. 2009, 8, 813 – 817. [15] M. Chattopadhyaya, M. M. Alam, S. Sen, S. Chakrabarti, Phys. Rev. Lett. 2012, 109, 257204 – 257208. [16] H. Hao, X. H. Zheng, L. L. Song, R. N. Wang, Z. Zeng, Phys. Rev. Lett. 2012, 108, 017202 – 017206. [17] M. Diefenbach, K. S. Kim, Angew. Chem. Int. Ed. 2007, 46, 7640 – 7643; Angew. Chem. 2007, 119, 7784 – 7787. [18] M. Kepenekian, B. L. Guennic, V. Robert, Phys. Rev. B 2009, 79, 094428 – 0944232. [19] M. Kepenekian, B. L. Guennic, V. Robert, J. Am. Chem. Soc. 2009, 131, 11498 – 11502. [20] R. Pati, M. McClain, A. Bandyopadhyay, Phys. Rev. Lett. 2008, 100, 46801 – 46801. [21] J. Tejada, E. M. Chudnovsky, E. d. Barco, J. M. Hernandez, T. P. Spiller, Nanotechnology 2001, 12, 181 – 186. [22] J. J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. L. Morton, D. N. Jamieson, A. S. Dzurak, A. Morello, Nature 2012, 489, 541 – 545. [23] J. Ferrer, V. M. Garca-Surez, J. Mater. Chem. 2009, 19, 1696 – 1717. [24] I. Zˇutic´, J. Fabian, S. D. Sarma, Rev. Mod. Phys. 2004, 76, 323 – 410. [25] http://www.quantumwise.com. [26] M. Vrot, S. A. Borshch, V. Robert, J. Chem. Phys. 2013, 138, 094105 – 094111. [27] M. Portais, C. Joachim, Chem. Phys. Lett. 2014, 592, 272 – 276. [28] M. Vrot, S. A. Borshch, V. Robert, Chem. Phys. Lett. 2012, 519 – 520, 125 – 129. [29] K. Yoshizawa, Acc. Chem. Res. 2012, 45, 1612 – 1621. [30] G. C. Solomon, C. Herrmann, T. Hansen, V. Mujica, M. A. Ratner, Nat. Chem. 2010, 2, 223 – 228. [31] S. Sharma, J. K. Dewhurst, C. Ambrosch-Draxl, S. Kurth, N. Helbig, S. Pittalis, S. Shallcross, L. Nordstrm, E. K. U. Gross, Phys. Rev. Lett. 2007, 98, 196405 – 196408. [32] D. P. DiVincenzo, Science 1995, 270, 255 – 261. [33] D. P. DiVincenzo, D. Loss, Phys. Rev. A 1998, 57, 120 – 126. [34] L. B. Levitin, T. Toffoli, Phys. Rev. Lett. 2009, 103, 160502 – 160505. [35] S. Ashhab, P. C. de Groot, F. Nori, Phys. Rev. A 2012, 85, 052327 – 052333. [36] M. Schlosshauer in Decohorence and The Quantum-To-Classical Transition (Ed.: A. C. Elitzur), Springer, Berlin, 2007.
Received: January 13, 2014 Revised: March 14, 2014 Published online on April 24, 2014
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Electrically Controlled Eight-Spin-Qubit Entangled-State Generation in a Molecular Break Junction Mausumi Chattopadhyaya,[a] Md. Mehboob Alam,[a] Debasis Sarkar,[b] and Swapan Chakrabarti*[a] The generation of spin-based multi-qubit entangled states in the presence of an electric field is one of the most challenging tasks in current quantum-computing research. Such examples are still elusive. By using non-equilibrium Green’s functionbased quantum-transport calculations in combination with non-collinear spin density functional theory, we report that an eight-spin-qubit entangled state can be generated with the high-spin state of a dinuclear Fe(II) complex when the system is placed in a molecular break junction. The possible gate operation scheme, gating time, and decoherence issues have been carefully addressed. Furthermore, our calculations reveal that the preservation of the high spin state of this complex is possible if the experimentalists keep the electric-field strength below 0.78 V nm1. In brief, the present study offers a unique way to realize the first example of a multi-qubit entangled state by electrical means only.
The groundbreaking technology advancements in the past three decades have finally made it possible to welcome Feynman’s proposition on the possibility of developing a computer which, in principle, should obey the weird and wonderful laws of quantum mechanics.[1] Unlike bits in a classical computer, a quantum bit or qubit in a quantum computer can exist in a coherent superposition of 0 and 1, in addition to the classical states alone.[2] Although the realization of a full-scale working quantum computer is still a fairy tale, the relentless efforts of experimentalists and theoreticians will definitely help in achieving the goal in the near future.[3–5] At present, superconductors, trapped ions, nuclear magnetic resonance in organic liquids, quantum dots,[6] molecular magnets,[7] Bose–Einstein condensates,[8] and so on are considered as the most promising candidates for generating “off” and “on” states, a prerequisite condition for the accomplishment of a qubit. Apart from these well-studied systems, certain new molecules like naphthalocyanine[9] and the TbIII-pthalocyanina-
[a] Dr. M. Chattopadhyaya, M. M. Alam, Dr. S. Chakrabarti Department of Chemistry University of Calcutta 92 A.P.C. Road, Kolkata-700009 (India) E-mail: [email protected] [b] Dr. D. Sarkar Department of Applied Physics University of Calcutta, 92 A.P.C. Road, Kolkata 700 009 (India) Supporting Information for this article is available on the WWW under http://dx.doi.org/10.1002/cphc.201400029.
2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
to complex[10] have started captivating the world of quantum computers. Both these systems have the ability to generate an “off–on” state. However, in these studies, the key component of the quantum-computing device is either in direct interaction with the external world that eventually leads to the loss of quantum information, or the “off–on” feature is achieved only in the presence of the external magnetic field. To date no single-molecular system could be found in the literature where the prime element of quantum-information processing is shielded from the external world and at the same time qubits are generated in the absence of an external magnetic field. In this communication, we show that an electric-field-induced spin-crossover (EFIS) between the high spin–high spin (HS–HS) and low spin–low spin (LS–LS) states of the [(pypzH)(NCSe)Fe(m-pypz)2Fe(NCSe)(pypzH)] complex[11] could be achieved at a field strength of Fc = 0.78 V nm1. Besides, the current obtained from the HS–HS state (“on”) of this complex is much larger than that of the LS–LS state (“off”) and surprisingly, the “on–off” behavior is originating from the spin-state-selective quantum-interference effect. Moreover, we have probed
Figure 1. Schematic representation of the dinuclear Fe(II) complex adsorbed on the Au(111) surface.
the quantum entangled state for the HS–HS state of the complex in a molecular break-junction setup (Figure 1) which ultimately offers us an unprecedented way to realize a molecular spin qubit within an electrical circuitry and this multi-partite spin-qubit feature is quite achievable if Fc < 0.78 V nm1. The main reason behind the selection of this particular complex is that at the experimental level, the single-step spin transition between the HS–HS and LS–LS states of this complex occurs at a temperature of 225 K,[11] indicating that the energy difference of the two spin states is very small. Furthermore, the absence of the hysteresis loop in the temperature-depenChemPhysChem 2014, 15, 1747 – 1751
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CHEMPHYSCHEM COMMUNICATIONS dent magnetic susceptibility variation points out that the spincrossover in this complex is a molecular property.[11] With this background, our in silico investigation begins with the geometry optimization of the HS–HS and LS–LS configurations of the selected di-nuclear Fe complex. Polarizability (a) calculations on the optimized geometries were carried out with the triple zeta valence plus polarization (TZVP) basis set in combination with the Perdew–Burke–Ernzerhof (PBE) functional.[12] It is worth recalling that at 300 K, the HS–HS configuration is more stable than the LS–LS configuration, by an amount of 5.5 meV, and it is hardly possible to achieve this milli-Hartree accuracy at any level of density functional theory (DFT) calculations and hence we have relied upon the experimental energy difference of the two spin states. Although DFT fails to provide an accurate energy difference between the two spin states, its performance in evaluating the properties, namely, the first and second derivative of the energy with respect to the electric field are quite satisfactory,[13] that is, dipole moment (p) and a values are less sensitive toward the choice of the DFT functionals. Our calculations reveal that the value of the component of a in the direction of transport (a j j ) of the LS–LS state is 738.85 a.u while the corresponding a j j of the HS–HS state is only 575.11 a.u. As a result, the second-order Stark (SOS) contribution[14–16] of the LS–LS state will definitely help overcome its initial stability deficiency at some critical field strength. Including the SOS contribution, the variation of
www.chemphyschem.org tant role in determining the Fc value needed to start the EFIS process. In our case, the applied bias required for the initiation of EFIS is 0.54 V and indeed this is quite close to that obtained (0.4 V) by Hao et al.[16] The details are presented in the Supporting Information (SI). To examine the possible electrically controlled spintronicdevice[20–24] action from the studied EFIS system, we theoretically have monitored the current–voltage (I–V) characteristics[25] of HS–HS and LS–LS states in the bias range 400 mV. The
Figure 3. I–V characteristics of the HS–HS and LS–LS states of the Fe(II) complex. The spin-polarized current of the HS–HS state (a and b spin channel) is shown in the inset.
Figure 2. Variation of the EGS for the LS–LS and HS–HS statesof the dinuclear Fe(II) complex. EGS is the ground-state energy of the respective spin states calibrated with respect to the HS state. The arrow denotes the point of electrostatic spin-crossover.
the total energy of both spin states with electric field is shown in Figure 2, where the two energy curves are crossing at a critical field strength Fc = 0.78 V nm1, and this field strength is quite attainable in a scanning tunneling microscopy (STM) setup. In this context, it is worth commenting that in the recent past, several groups reported a spin-crossover in the presence of an electric field[17] and studied the hysteresis behavior in a spin-crossover molecule.[18, 19] More recently, Hao et al.[16] showed that a molecular junction can play an impor 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
bias-dependent variations of the net current obtained from the HS–HS and LS–LS states are depicted in Figure 3 and the inset shows the spin-resolved current contribution for the a and b spin channel of the HS–HS state, separately. Figure 3 demonstrates that the current obtained from the LS–LS state is much smaller in comparison to that of the HS–HS state. Moreover, in the HS–HS state, only the b spin channel is participating in the transport process, keeping the a spin channel silent. Therefore, the LS–LS and the b spin channel of the HS–HS state will act as “off” and “on” states, respectively. This “on–off” feature is quite realizable across the Fc, equivalent to the bias = 320 mV and obviously the studied range of bias is sufficient to capture this interesting phenomenon. It is to be noted that the transport calculations have been performed without considering the multi-reference effect[26–28] explicitly. However, the transmission coefficient obtained from the non-collinear spin DFT calculation, an equivalent form of the multi-reference theory, is found to be quite similar with the simple non-equilibrium Green function DFT results (see Figure S4 in the SI). Nonetheless, at this stage, we need to address the most fundamental question of why the current obtained from the HS–HS state is so high in contrast to the LS-LS state of this material albeit eg type 3d orbitals of Fe(II) ions in the LS-LS configuration are vacant. ChemPhysChem 2014, 15, 1747 – 1751
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Figure 4. Transmission spectrum of the HS–HS and LS–LS state of the Fe(II) complex.
www.chemphyschem.org connected to the electrodes. However, in the present two probe set up, both the HO and LU b spin orbitals of the HS–HS state are lying below the EF. As a consequence, the pairwise (occupied–unoccupied) denominators of G(0) of the LS–LS state will appear with opposite sign while the corresponding sign of the HS–HS state will be the same. Therefore, the nature of the in-
To answer this strange finding, we have examined the spin-polarized transmission spectrum (SPTS) as depicted in Figure 4. The left panel of Figure 4 reveals that in the b spin channel, the number of transmission peaks with amplitude nearly equal to one is present just near the Fermi level EF and this is fully consistent with the large current obtained from the b spin channel of this material. Besides, the right panel of Figure 4 illustrates that the electrons also get transmitted near the EF of the LS-LS state but in this case, the transmission amplitude is rather very Figure 5. Dominant MPSH states of the HS–HS (a, b, c) and LS–LS (d, e) states of the Fe(II) complex. small which merely indicates that the presence of the eigen state having suitable energy in the active region of the twoterference between the propagating electronic waves in the probe set up does not necessarily help transmit the electrons HS-HS and LS-LS states will crucially be guided by the sign of from one electrode to the other. the product of the orbital coefficients, Crk Csk* . To get insight To shed light on the origin of the poor transmission near the EF of the LS–LS state, we have inspected the nature of the into the sign of Crk Csk* , we have investigated the molecular pro(0) zeroth order molecular Green function, G of both the spin jected self-consistent Hamiltonian (MPSH) states near the EF of states. The G(0) can be expressed in terms of energy and orbital both the spin states and the selective MPSH states are depicted in Figure 5. coefficients[29] as [Eq. (1)]: Interestingly, in the b spin channel of the HS–HS state, the X Crk C * sk 3d orbitals centered on the two Fe(II) ions in the given MPSH Gðrs0Þ ðE Þ ¼ ð1Þ EF ekih states are in the same phase and the product Crk Csk* for the ocK cupied and unoccupied orbitals are of the same sign which ultimately leads to the constructive interference. Besides, where Crk is the kth orbital coefficient at atom r and 2k is the Crk Csk* for the occupied and unoccupied orbitals of LS–LS state energy of the kth orbital. Considering the particle–hole sym(0) are also appeared with same sign, however, due to different metry, we can expand the G in terms of the occupied and unsign of the denominators of the G(0), the propagated electronic occupied orbitals separately and the modified expression for wave will face destructive interference and is responsible for our case will be [Eq. (2)]: the poor transmission amplitude vis--vis negligibly small curX CFe1OCC C * X CFe1UNOCC C * rent obtained from this spin state (LS–LS). This spin state deð0 Þ Fe2OCC Fe2UNOCC G ¼ þ ð2Þ E eOCC ih UNOCC EF eUNOCC ih pendent constructive/destructive interference[30] in a transition OCC F metal complex and that too in presence of electric field is unNormally, EF lies between the highest occupied (HO) and precedented. The appealing I–V characteristics of this Fe(II) complex has lowest unoccupied (LU) orbitals of a bare molecule and this raised our interest to check the potential of this system in holds good for the LS–LS state even when the Fe(II) complex is 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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CHEMPHYSCHEM COMMUNICATIONS qubit generation, for which certain superposition/entangled state is required when the system is placed in the two-probe setup. To find out the existence of such a state, if any, we keep our trust in the non-collinear DFT technique.[31] It is worth noting that the standard Kohn–Sham equations for different spin directions, marked a and b, do not decouple and a result standard DFT fails to capture the microscopic picture of the spin spirals, canted spins and domain walls of ferromagnetic materials. In the non-collinear DFT formalism, the exchange correlation field is modified by adding an internal magnetic field contribution that helps break the spin symmetry correctly. In our approach, we have started with the HS–HS state and placed the spin on the electrodes and the two Fe atoms with two sets of initial rotation angle (q) and have noticed that the system self consistently goes into a state where the spins on the two Fe(II) ions prefer to orient themselves along the x axis (Figure 6 a). This amazing observation clearly reveals that in the HS–HS state, the Fe(II) complex has the ability to generate an eight-qubit multipartite Greenberger-Horne-Zeillinger (GHZ) entangled state. The possible existence of the GHZ state, pffiffiffi 1 2ðj00000000i þ j11111111iÞ, is a rudimentary step towards the realization of the multiqubit feature from this complex. However, to arrive at any decisive conclusion, we first need to make sure whether or not the system obeys the famous Di Vincenzo’s checklist.[32, 33] First of all, the universal Hadamard (H) gate operation on the first qubit of j00000000i pffiffiffi will generate 1 2ðj00000000i þ j10000000iÞ, and in our case the rotation gate will be Ry ðp=2ÞRz ðpÞPhðp=2Þ: On the resultant state, successive CNOT operation with the sequence of rotations Riz ðp=2ÞRjx ðp=2ÞRjy ðp=2ÞJij ðp=2ÞRjy ðp=2Þ will produce the target GHZ state. The experimental realization of the H gate operation is quite feasible since the zero field splitting (ZFS) parameter of this complex is moderately large (10.04 cm1) and is evident from Figure 6 b. It is important to mention here that both spin–spin interactions and spin–orbit coupling are incorporated in the ZFS calculation and this calculation has been done with the PBE functional[12] in combination with TZ2P basis functions. According to Levitin and Toffoli,[34] the single qubit gate operation time is limited by the expression t ¼ h=2Emax . Here, the Emax can be computed from the ZFS pattern as consistent with the energy of S = + 4 and + 3. Thus,
www.chemphyschem.org experimentally, an appropriate p=2ð y Þ microwave pulse, with an penergy gap between + 4 and + 3, will lead to the ffiffiffi 1 2ðj00000000i þ j10000000iÞ state, and the minimum time required for this operation will be 0.19 ps. Similarly, the time required for each CNOT gate operation[35] is of the order of ðp=4JÞh, where J represents the Heisenberg coupling constant. The spin-flip DFT calculations reveal that the value of J of this complex will be 0.3 meV and thus the total seven CNOT gate operations will take 1.89 ps (0.27 ps for each). At this stage, it difficult to comment on the possible Scheme for the experimental realization of the CNOT gate operations since it requires extensive investigations addressing many complicated issues and will be considered in a separate work. Nevertheless, we have given the primary evidence that CNOT gate operations are feasible and can be done with in a very short period of time. Apart from these important issues associated with qubit generation, the most crucial point of DiVincenzo’s criteria,[32, 33] is the estimation of the decoherence time. In the high-temperature limit, the Caldeira–Leggett master equation[36] for the pure decoherence problem can expressed as [Eq. (3)]: h_h ii _ d=dt 1s ðtÞ ¼ i Hs ; 1s ðtÞ 2Mg0 kB T X X 0 ; 1s ðtÞ
_
_
0
_
ð3Þ
Here, the first term of the right-hand side represents the Liouville-von Neuman term with frequency shifted Hamiltoni^ 0 s , and the second term denotes decoherence in the posian, H tion basis with squared separation, ðX X 0 Þ2 . In this term, M is the mass of the system, g0 is the effective coupling constant, T is the temperature of the bath and kB is the Boltzmann constant. In the position representation, the last term of this ex 2 0 0 pression can be re-written as g0 X X ldb 1s X; X ; t , which basically describes the spatial localization with decoherence pffiffiffiffiffiffiffiffiffiffiffi 2 time tD =tR ¼ h Dx 2mkB T . With this formula, the ratio of the decoherence (tD) and relaxation (tR) time for our system coupled with the Au electrode bath comes out as 0.1 microsecond. It is important to mention here that ldB has been calculated at temperature 300 K and in our case Dx is 0.41 nm. Theoretically, prior to decoherence, 105 gate operations could be performed. We have also checked the quantum fidelity factor, defined as (G = 1/2AB1) and the value depends on how good the real rotation matrix (B) approximates the ideal rotation matrix (A). The calculations suggests the fidelity factor in the present case will be ( ~ 1), establishing the ability of the complex in multi-qubit generation in presence of combined electric field and pulsed microwave radiation. It is worth noting that the GHZ state is obtained from the HS-HS configuration whose b spin channel would act as j1i Figure 6. a) Existence of the entangled state of the Fe(II) complex in between the molecular junction, and b) ZFS and the a as j0i. Therefore the diagram. 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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CHEMPHYSCHEM COMMUNICATIONS preservation of the HS–HS state in the presence of the applied bias is essential and hence the field strength should not exceed the value, Fc = 0.78 V nm1, otherwise the system will go to the LS–LS configuration and will no longer act as qubit generator. In summary, we conclude that EFIS at an experimentally achievable field strength can be accomplished in the studied dinuclear Fe(II) complex. Quantum-transport calculations reveal that the current obtained from the HS–HS state is much larger than that of the LS–LS state, and that the huge difference in the current of two spin states is arising out of the unprecedented spin-state-selective quantum-interference effect. To justify the prospect of qubit feature, we have identified the possible existence of a Greenberger–Horne–Zeillinger state in a break-junction setup. Our calculations reveal that approximately 105 gate operations with fidelity factor ~ 1 could be performed before the system undergoes decoherence. Details of the gate operations, including pulse shaping with optimal control theory and relaxation process, will be investigated in the next project. Finally, we believe that the present in silico findings will encourage experimentalist to build up the firstever electronic-spin-based qubit operating in the absence of an external magnetic field.
Acknowledgements M.C. and M.A. thankfully acknowledge CSIR, Govt. of India, for senior research fellowships and S.C. conveys his thanks to CRNN, University of Calcutta, and the INDNOR Project for research funding. Keywords: entanglement · non-equilibrium Green’s function · quantum interference · quantum transport · spin-qubits [1] N. Zhao, J. Wrachtrup, Nat. Mater. 2013, 12, 97 – 98. [2] J. J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. L. Morton, F. A. Zwanenburg, D. N. Jamieson, A. S. Dzurak, A. Morello, Nature 2013, 496, 334 – 338. [3] J. J. L. Morton, A. M. Tyryshkin, R. M. Brown, S. Shankar, B. W. Lovett, A. Ardavan, T. Schenkel, E. E. Haller, J. W. Ager, S. A. Lyon, Nature 2008, 455, 1085 – 1088. [4] J. Clarke, F. K. Wilhelm, Nature 2008, 453, 1031 – 1042. [5] L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, Nature 2009, 460, 240 – 244. [6] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, J. L. O’brien, Nature 2010, 464, 45 – 53.
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Received: January 13, 2014 Revised: March 14, 2014 Published online on April 24, 2014
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