Evidence for Pre-formed Pairs in an Oxide Superconductor

From left to right: Mengcheng Huang, Shicheng Lu, Jeremy Levy, Guanglei Cheng, Michelle Tomczyk, Patrick Irvin. 

Authors: Guanglei Cheng, Michelle Tomczyk, Jeremy Levy

Affiliation:
Department of Physics and Astronomy, University of Pittsburgh, USA.
Pittsburgh Quantum Institute, Pittsburgh, Pennsylvania, USA.

Link to Levy Research Group >>

Strontium titanate (SrTiO₃ or simply STO) is the first and best-known superconducting semiconductor. Its many fascinating properties, especially superconductivity, motivated Georg Bednorz and Alex Müller to search for high-temperature (high-T_c) superconductors in perovskite oxides [1]. STO shares many features related to the cuprate high-T_c superconductors, including a dome-shaped phase diagram and a pseudogap phase [2]. However, STO has much lower temperature and carrier density than the high-T_c compounds. A longstanding question surrounds the nature of the pseudogap state: ‘To pair or not to pair?’ – Does the existence of a pseudogap indicate electron pairing in the absence of superconductivity?

In 1969, long before high-T_c superconductivity was discovered, D.M. Eagles predicted [3] that electrons could remain paired outside of the superconducting state in STO. Eagles predicted that electrons can form a dilute gas so that the size of pairs is very small compared to the inter-electron distance. Namely, electrons pair in real space and the superconductivity is a result of Bose-Einstein condensation (BEC), contrasting the weak pairing in momentum space described by Bardeen-Schrieffer-Cooper (BCS) theory, which is highly successful in explaining conventional superconductors. Eagles was the first to propose the concept of BEC-BCS crossover, which was later independently developed by Nobel laureate Anthony Leggett [4] and experimentally realized in an ultracold atomic gas [5]. The discovery of BEC-type superconductivity in solid state systems has been challenging.

The LaAlO₃/SrTiO₃ (LAO/STO) interface [6] has attracted numerous interests in the last decade. It hosts a two-dimensional conducting interface that possesses a wealth of strongly correlated phenomena including superconductivity, magnetism, metal-insulator transition (MIT) and spin-orbit interaction [7]. A few years ago, we developed a lithography technique that allows us to ‘write’ and ‘erase’ nanostructures at the LAO/STO interface by using a sharp conductive atomic force microscope (c-AFM) tip, thus effectively programming all the novel properties at the nanoscale [8,9]. The writing mechanism relies critically on the MIT, in which the interface becomes conducting above the critical 3 unit cell (uc) LAO thickness. While the 3uc LAO/STO interface is insulating, it is switchable by a voltage-biased c-AFM tip. Under the c-AFM tip, a series of nanoscale devices have been made available. The resulting nanowires are only a few nanometers wide, exhibit anomalously high mobility [10] and show potential for the development of new types of nanoelectronics.

To investigate electron pairing, we use a superconducting single-electron transistor (Figure 1a). The device consists of a main superconducting nanowire channel intersected with voltage leads. Two tunnel potential barriers are engineered through c-AFM ‘cutting’ procedures so that a nanoscale island is defined. Due to the nanoscale confinement, the energy levels inside the island are quantized, and carrier transport is only possible when the chemical levels of external electrodes (source and drain) are aligned with an island energy level, which is dependent on the side gate voltages.

Figure 1: (click on the figure to view with higher resolution) Device schematic and transport characteristics. (a), Device schematic written by c-AFM lithography. The nanowires are typically 5 nm wide, and the length between 2 barriers is 1 μm. (b) and (c) Differential conductance dependent on the source-drain bias and side gate voltages at B=0 T (b) and B=4 T (c). The number of diamonds is doubled in (c). Color scales are 0-80 µS. (d) Magnetic field dependence of the conductance peaks. The bifurcation of conductance peaks above B_p suggest electron pairing without superconductivity. Color scale: 0-40 µS.

Indeed, bias spectroscopy reveals [11] a series of conductance diamonds (Fig. 1b), reminiscent of so-called ‘Coulomb diamonds’ in conventional blockade physics. In the latter case, each sequential Coulomb diamond corresponds to the stability of one additional electron, and applying an external magnetic field merely changes the size of diamonds due to Zeeman Effect. Remarkably, when we increase the magnetic field, the diamonds initially remain insensitive to field, then bifurcate above a critical magnetic field B_p~2 T (Fig. 1b,c). Such behavior is clearly revealed when we track the magnetic field dependence of the zero-bias conductance peaks. We find that all of the peaks bifurcate above a ‘pairing field’ B_p (Fig. 1d), suggesting transport is dominated by electron pairs rather than single electrons below B_p. Electron pairing persists far above the critical temperature (T_c~0.3 K) and for magnetic fields far above the upper critical field (H_c2~0.2 T) for superconductivity in bulk STO.

The observed electron pairing without superconductivity is difficult to explain using BCS theory. Pair fluctuations in disordered BCS superconductor films may give signatures of pairing above T_c which is greatly suppressed by disorder. However, the corresponding pairing temperature will not exceed T_c in the clean limit. Here the pairing temperature we have observed is around several kelvin, one order of magnitude higher than the T_c in the bulk. These experimental signatures are captured by a phenomenological model that favors BEC pairing, consistent with D. M. Eagles’s theory proposed 46 years back.

References:
[1] J. Georg Bednorz and K. Alex Müller, "Perovskite-type oxides - the new approach to High Tc superconductivity", Nobel Lecture (1987).
[2] C. Richter, H. Boschker, W. Dietsche, E. Fillis-Tsirakis, R. Jany, F. Loder, L. F. Kourkoutis, D. A. Muller, J. R. Kirtley, C. W. Schneider, J. Mannhart, "Interface superconductor with gap behaviour like a high-temperature superconductor", Nature, 502, 528 (2013). Abstract.
[3] D.M. Eagles, "Possible pairing without superconductivity at low carrier concentrations in bulk and thin-film superconducting semiconductors", Physical Review, 186, 456 (1969). Abstract.
[4] Anthony J. Leggett, "A theoretical description of the new phases of liquid ³He", Reviews of Modern Physics,  47, 331 (1975). Abstract.
[5] M. W. Zwierlein, J.R. Abo-Shaeer, A. Schirotzek, C.H. Schunck, W. Ketterle, "Vortices and superfluidity in a strongly interacting Fermi gas",  Nature, 435, 1047 (2005). Abstract.
[6] A. Ohtomo,  H.Y. Hwang, "A high-mobility electron gas at the LaAlO₃/SrTiO₃ heterointerface", Nature, 427, 423 (2004). Abstract.
[7] Joseph A. Sulpizio, Shahal Ilani, Patrick Irvin, Jeremy Levy, "Nanoscale Phenomena in Oxide Heterostructures", Annual Review of Materials Research, 44, 117 (2014). Abstract.
[8] C. Cen, S. Thiel, G. Hammerl, C. W. Schneider, K. E. Andersen, C. S. Hellberg, J. Mannhart, and J. Levy, "Nanoscale control of an interfacial metal-insulator transition at room temperature", Nature Materials, 7, 298 (2008). Abstract.
[9] Cheng Cen, Stefan Thiel, Jochen Mannhart, Jeremy Levy, "Oxide nanoelectronics on demand", Science, 323, 1026 (2009). Abstract.
[10] Patrick Irvin, Joshua P. Veazey, Guanglei Cheng, Shicheng Lu, Chung-Wung Bark, Sangwoo Ryu, Chang-Beom Eom, Jeremy Levy, "Anomalous High Mobility in LaAlO₃/SrTiO₃ Nanowires", Nano Letters, 13, 364 (2013). Abstract.
[11] Guanglei Cheng, Michelle Tomczyk, Shicheng Lu, Joshua P. Veazey, Mengchen Huang, Patrick Irvin, Sangwoo Ryu, Hyungwoo Lee, Chang-Beom Eom, C. Stephen Hellberg, Jeremy Levy, "Electron pairing without superconductivity", Nature 521, 196 (2015). Abstract.

A Current Out Of Fluctuations

(From left to right) Pierre Pfeffer, Fabian Hartmann, Sven Höfling, Martin Kamp, Lukas Worschech.

Authors: Pierre Pfeffer¹, Fabian Hartmann¹, Sven Höfling^1,2, Martin Kamp¹, Lukas Worschech¹

Affiliation:
¹Technische Physik, Universität Würzburg, Physikalisches Institut and Wilhelm Conrad Röntgen Research Center for Complex Material Systems, Würzburg, Germany.
²SUPA, School of Physics and Astronomy, University of St. Andrews, United Kingdom.

In a recent experiment [1] we demonstrated the conversion of voltage fluctuations into a directed current on the nanoscale in a system consisting of two Coulomb-coupled quantum dots. The use of external gates allows a control of the current magnitude and a switching of its direction. The work was inspired by theoretical proposals by Sánchez et al. [2] and Sothmann et al. [3]. Possible future applications include energy harvesting as well as local cooling on the nanoscale, both vital to the further development of autonomous and energy-saving electronics.

Since heat and fluctuations are especially challenging factors in miniaturizing electronic circuits [4], the design of heat engines and rectifiers has become a central point of research in nanoelectronics and led to concepts and devices such as Brownian- and Büttiker-Landauer-motors [5,6] phonon rectifiers [7,8] and piezoelectric nanogenerators [9-10]. Despite being diverse concepts, they have in common an interplay of non-linearity, a certain form of symmetry breaking and the presence of fluctuations. For instance, Brownian motors enable a unidirectional particle flow even when there is no net average force in one direction, if the device’s potential is asymmetric. In this context, Sánchez et al. [2] and Sothmann et al. [3] theoretically examined rectifying devices which stand out because of their conception as three terminal devices, which allows a decoupling of the directions of energy flow and charge current.

Figure 1: Illustration of the energy transfer. Coulomb-coupling transfers energy from higher-energy electrons in the lower cavity to lower-energy electrons in the upper one.

The basic operating principle of the Coulomb coupled rectifier is illustrated in Fig. 1. Energy is transferred locally between two Coulomb-coupled cavities. If the electrons in one cavity (lower sphere in Fig. 1, in the following named QD_b ) have on average higher energies than the electrons in the other cavity (upper sphere, QD_t ), electrons in the first cavity can change into a lower-energy state and transfer the energy difference δE to electrons in the second cavity by means of Coulomb-interaction. No particle exchange takes place between the two cavities.

If QD_t is coupled to two reservoirs by means of two quantum point contacts (QPC), a rectified current through the upper cavity occurs if three essential requirements are met: The amount of fluctuations in the upper and lower cavity have to differ, the transmissions of the QPCs have to be energy-dependent, and the two QPCs have to be asymmetric (Fig. 2a). This is for instance possible by applying an external electric field and enables a switching of the current direction.

Figure 2: a) Schematic operating principle. i): QD_t is connected to the outer reservoir by two transport barriers. ii): Asymmetric configuration with charge current flow to the right. iii): Asymmetric configuration with charge current flow to the left. b) Electron microscopy image of the sample (side view) together with a schematic of the applied voltages. V_gl and V_gr allow a switching between the conditions displayed in a) ii) and a) iii).

To realize this device, we grew a GaAs/AlGaAs semiconductor heterostructure with a high mobility two-dimensional electron gas by molecular beam epitaxy. Electron beam lithography and dry chemical etching techniques were used to define the structural layout. Fig. 2b shows a side view electron microscopy image of the sample and is overlain with a schematic representation of the QDs indicating the two separate parts of the system (upper subsystem where the resulting current flows in blue and lower subsystem where the fluctuations occur in red). Moreover, the connected voltages are schematically added to the picture. V_gl and Vgr are the gate voltages applied to the upper side gates which allow us to individually manipulate the transmission coefficients of the two QPC’s. For instance, applying a negative voltage to V_gl and a positive one to V_gr opens the left but closes the right QPC and vice versa. In order to keep the central electrostatic potential of QD_t constant while asymmetrically controlling the transmission of the left and right QPC we varied the voltages in push-pull configuration with V_gl = -V_gr. The lower side gates remain unused in the presented experiments. The static gate voltage V_gb further allows to tune the transmission of the top current carrying system. A noise voltage V_noise is supplied to the lower reservoir. All experiments reported here were conducted at 4.2 K in the dark by immersing the sample in liquid helium.

Figure 3: a) Dependence of output current on the voltages applied to the two laterally defined side gates for increasing noise from 7.6 to 144 mV in steps of 15 mV. The current direction and magnitude can be altered by the gate voltage configuration and noise amplitude, respectively. b) Output powers P against the counteractive voltage Vlr for different noise amplitudes increasing from 7.6 to 150 mV in steps of 15 mV. c) Maximum output powers P max versus σ_noise.

Fig. 3a shows the rectified current through QD_t for different noise amplitudes from 7.6 to 144 mV when changing the gate voltages V_gl and V_gr. Notably, without noise, no current could be measured. Starting with a gate voltage configuration of V_gl = − V_gr = − 2 V and increasing the push-pull voltage initially increases the current from zero to a maximal value which depends on the applied noise amplitude. Thereafter, I decreases again and vanishes completely at around V_gl = 0.01 V (independently on the noise amplitude). A further increase of V_gl changes the current direction. I therefore becomes negative, reaches a minimum and finally increases back to zero again.

In order to harvest useful work from the rectifier, it has to power a load or, equivalently, the generated current has to flow against an applied voltage difference. Consequently, to measure the output power P, a voltage difference V_lr counteracting the current was applied between the channels. The output power is the product of the current and the counteracting applied voltage. Fig. 3b shows the voltage dependent output power for different noise amplitudes. The output power has a parabolic dependency on the applied voltage and vanishes at two particular points: at V_lr = 0 V (maximal current) and at the stopping voltage V_st (no current) which depends on σ_noise. Fig. 3c presents the maximum output powers, obtained at half the stopping voltage, depending on the noise amplitude. There, a quadratic dependency on the noise amplitude can be seen with a maximum output power value of P_max = 24 pW.

In conclusion, we demonstrated a rectification mechanism via Coulomb-coupled quantum dots. The presented findings are a step towards reducing the power consumption of electronic devices and may allow a further miniaturization of electronic circuits and thus pave the way towards sustainable, efficient and autonomous electronics.

References:
[1] F. Hartmann, P. Pfeffer, S. Höfling, M. Kamp, L. Worschech, "Voltage Fluctuation to Current Converter with Coulomb-Coupled Quantum Dots". Physical Review Letters, 114, 146805 (2015). Abstract.
[2] Rafael Sánchez, Markus Büttiker, "Optimal energy quanta to current conversion". Physical Review B, 83, 085428 (2011). Abstract.
[3] Björn Sothmann, Rafael Sánchez, Andrew N. Jordan, Markus Büttiker, "Rectification of thermal fluctuations in a chaotic cavity heat engine". Physical Review B, 85, 205301 (2012). Abstract.
[4] Mehdi Asghari, Ashok V. Krishnamoorthy, "Silicon photonics: Energy-efficient communication". Nature Photonics, 5, 268 (2011). Abstract.
[5] R. Dean Astumian, Peter Hänggi, "Brownian Motors". Physics Today, 55, 33 (2002). Link.
[6] M. Büttiker, "Transport as a consequence of state-dependent diffusion". Zeitschrift für Physik B, 68, 161 (1987). Abstract.
[7] Rolf Landauer, "Motion out of noisy states". Journal of Statistical Physics, 53, 233 (1988). Abstract.
[8] C. W. Chang, D. Okawa, A. Majumdar, A. Zettl, "Solid-State Thermal Rectifier". Science, 314, 1121 (2006). Abstract.
[9] Nan Zeng, Jian-Sheng Wang, "Mechanisms causing thermal rectification: The influence of phonon frequency, asymmetry, and nonlinear interactions". Physical Review B, 78, 024305 (2008). Abstract.
[10] Zhong Lin Wang, Jinhui Song, "Piezoelectric Nanogenerators Based on Zinc Oxide Nanowire Arrays", Science, 312, 242 (2006). Abstract.

Core-shell Hybrid Nanostructure Based High Performance Supercapacitor Electrode

Ashutosh Kumar Singh (left) and Kalyan Mandal

Authors: Ashutosh Kumar Singh, Kalyan Mandal

Affiliation: Department of Condensed Matter Physics and Material Sciences,
S.N. Bose National Centre for Basic Sciences, Kolkata, India.

Background:

Since last decade, energy crisis has been one of the vital problems in the society due to the excessive use of fossil-fuel resources and environmental pollution. Therefore, the development of very light-weight and environment-friendly proficient energy storage devices has become the priority of the researchers and scientists for satisfying the demand of modern consumer’s hybrid electric and portable electronics devices [1,2]. In this progression, researchers have developed new type of energy storage devices called supercapacitors, also known as electrochemical capacitors which offer high power and eneergy density, high rate capability as well as superb cycle stability as compared to conventional battery and capacitors [3,4].

The supercapacitors are classified in two groups based on their charge storage method. The first group is called pseudocapacitor which involves the redox reactions of the electrode materials at the interface of electrode and electrolyte, whereas the second group is known as electric double layer capacitor which holds charge separation at the interface of electrode and electrolyte [5,6]. By changing the morphology of the electrode materials, one can manipulate the performance of a supercapacitor. The performance and quality of the supercapacitors are very much dependent on the materials and morphology used in the preparation of their electrodes. Recently, many metal oxide based materials (like RuO₂, NiO, Fe₂O₃, MnO₂, Co₃O₄, TiO₂, etc.) have been used for the fabrication of pseudocapacitor electrodes; among them NiO and Fe₂O₃ have been widely used as redox active materials for the fabrication of supercapacitor electrodes of different morphologies, such as Fe₂O₃-nanotube, Fe₂O₃-thinfilm, electrospun Co₃O₄-nanostrtuctures, porous Fe₂O₃-nanostrtuctures, NiO–nanobelts, NiO–nanoballs, NiO–nanoflowers, NiO–nanoflakes. The reasons behind the extensive use of NiO and Fe₂O₃ as supercapacitor electrode materials are: they are very stable in nature, they are non-toxic as well as environment friendly and they are very cheap and easily available.

Aim:

After having so many of supportive properties for being used as electrode materials in supercapacitors, still their reported specific capacitance values are very low compared to their own theoretical specific capacitance value and other metal oxide based electrodes. The only problem restricts them to be used as an electrode material for high performance supercapaitor is their bad electrical conductivity and we all are aware of the fact that electrode material must have high electrical conductivity for high performance supercapacitor.

Scope:

In the recent development process of supercapacitor performance, it has been found that the electrical conductivity could be improved by introducing impurities via doping of one metal oxide material with other metal oxide material. This doping process enhances the charge movement which affects the reactions at the interface of electrode and electrolyte. So far in the literature we have not found any work based on NiO and Fe₂O₃ as mixed component transition metal oxides for supercapacitor electrodes. However, keeping all the above research facts in the mind, still there exist a plenty of remarkable opportunities to enhance the electrochemical properties of NiO and Fe₂O₃ based electrodes.

Unique features of the approach:

Therefore, we report a simple fabrication technique and unique electrochemical properties of the electrode based on core/shell Fe-Ni/Fe₂O₃-NiO hybrid nanostructures (HNs). This core-shell HNs have very high aspect ratio with a porous thin nanolayer of redox active oxides which would provide a very large surface area for redox reactions at the interface of electrode and electrolyte. This would contribute to the enhancement of the ion and electron movement and performance of the supercapacitor. In addition, the core material consists of conductive FeNi nanowires (NWs) which provides the expressway for the electrons to transport to the current collector via core material.

This would automatically improve the rate capability and power density of the supercapacitor. The electrical conductivity of the electrode could be improved by introducing impurities via doping of one metal oxide (NiO) material with another metal oxide (Fe₂O₃) material. The unique feature of this electrode fabrication technique is that it doesn’t contain any extra binder material. As a result, there would be enhancement in the charge transfer kinetics [7]. This kind of fabrication technique could be applied in the fabrication process of electrodes of all energy storage devices in general.

Significant Results:

According to our anticipations, the core/shell Fe-Ni/Fe₂O₃-NiO hybrid nanostructure shows high quality supercapacitive performance in terms of specific capacitance (1415 F/g), energy density (27.6 Wh/kg), power density (10.3 kW/kg), cycling stability (remain 95% of initial specific capacitance after 3000 charge/discharge cycle) and rate capability [7]; these profound results made it a very good and unique alternative for the next generation supercapacitor electrodes.

References:
[1] Patrice Simon, Yuri Gogotsi, "Materials for electrochemical capacitors". Nature Materials, 7, 845 (2008).  Abstract.
[2] John R. Miller, Patrice Simon, Electrochemical Capacitors for Energy Management". Science 321, 651 (2008). Abstract.
[3] Zhibin Lei, Li Lu, X.S. Zhao, "The electrocapacitive properties of graphene oxide reduced by urea". Energy & Environmental Science, 5, 6391 (2012). Abstract.
[4] Sheng Chen, Junwu Zhu, Xiaodong Wu, Qiaofeng Han, Xin Wang, "Graphene Oxide−MnO2 Nanocomposites for Supercapacitors". ACS Nano 4, 2822 (2010). Abstract.
[5] Wei Chen, R.B. Rakhi, Liangbing Hu, Xing Xie, Yi Cui, H.N. Alshareef, "High-Performance Nanostructured Supercapacitors on a Sponge". Nano Letters, 11, 5165 (2011). Abstract.
[6] Raghavan Baby Rakhi, Wei Chen, Dongkyu Cha, H. N. Alshareef, "Nanostructured Ternary Electrodes for Energy-Storage Applications". Advanced Energy Materials, 2, 381 (2012). Abstract.
[7] Ashutosh K. Singh, Kalyan Mandal, "Engineering of high performance supercapacitor electrode based on Fe-Ni/Fe2O3-NiO core/shell hybrid nanostructures". Journal of Applied Physics, 117, 105101 (2015). Abstract.

Efficient Photon Collection from a Nitrogen Vacancy Center in a Circular Bullseye Grating in Diamond

[From left to right] Luozhou Li, Edward Chen and Dirk Englund.

Authors: Luozhou Li, Edward Chen, Dirk Englund

Affiliation: Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, USA.

Link to Quantum Photonics Laboratory >>

The nitrogen-vacancy center (NV) [1] behaves much like an atom trapped in the diamond lattice. Because of the high band gap and the mostly spin-free composition of the diamond host, the NV is well isolated from the environment, so it shows well-behaved atom-like properties. Most importantly, it’s possible to optically prepare and measure the long-lived states of the associated electron and nuclear spins. NVs are potentially promising building blocks for a large-scale quantum network where the optical addressability of the NV allows flying qubits, or photons, to connect nodes of this network together. One of the fundamental bottlenecks for this to be made into a reality is the flux of photons collected from an NV, which determines how quickly the NV’s spin state can be measured and compared: the more fluorescent photons that are collected, the faster new connections can be made. The same photon collection limitation is also true for using the NV as a highly sensitive quantum sensor, where the sensitivity to electric, magnetic and temperature fields increase with increased photon collection. Thus, higher photon detection of the NV’s photoluminescence is of central importance to many NV quantum technologies, such as communication, computing, and even sensing.

In our recent work [2], we introduce a circular “bullseye” grating in diamond (Figure 1), which enables record-high photon collection from the nitrogen-vacancy (NV) color center. The bullseye grating consists of concentric slits etched into a diamond membrane [3], which itself is about half of a wavelength in thickness. The grating period satisfies the second-order Bragg condition, giving rise to the scattering of light out of the membrane. The scattered light from each grating interferes constructively out of the plane and into the far field, thereby enabling significantly higher collection efficiency. With this circular grating, we have shown that it’s possible to collect about an order of magnitude more fluorescence than is possible from an NV in un-patterned diamond.

Figure 1: (a) Illustration of an array of diamond bullseye gratings adjacent to a microwave strip line. (b) Schematic of the circular grating. ‘a’ denotes the lattice constant and ‘gap’ the air spacing between circular gratings. (c) Simulated electric field intensity (log scale) in the x = 0 plane with air above and glass below the diamond. A dipole emitter was placed in the center of the bullseye grating, and was oriented along the horizontal direction.

Achieving higher collection efficiency from the NV impacts several applications such as improved sensing of static or dynamic electromagnetic fields just outside the diamond, higher luminosity room-temperature single photon sources, and better quantum memories for quantum computing and networking. For example, NV researchers [4] have recently shown that the NV is even sensitive to changes of single proton spins, paving the way for magnetic resonance imaging of individual molecules in liquid — and this application would be improved by better fluorescence collection from the NV.

The efficient photon collection should allow for a range of new measurements, such as non-demolition measurements of NV spins — i.e., you could make a measurement and then act back on the NV spin state. We’re also using the efficient collection for medium-scale quantum registers, which would contain on the order of tens of qubits each, and for quantum sensing.

References:
[1] Marcus W. Doherty, Neil B. Manson, Paul Delaney, Fedor Jelezko, Jörg Wrachtrup, Lloyd CL Hollenberg, "The nitrogen-vacancy colour centre in diamond." Physics Reports, 528, 1-45 (2013). Abstract.
[2] Luozhou Li, Edward H. Chen, Jiabao Zheng, Sara L. Mouradian, Florian Dolde, Tim Schröder, Sinan Karaveli, Matthew L. Markham, Daniel J. Twitchen, and Dirk Englund, "Efficient photon collection from a nitrogen vacancy center in a circular bullseye grating." Nano letters, 15, 1493 (2015). Abstract.
[3] Luozhou Li, Igal Bayn, Ming Lu, Chang-Yong Nam, Tim Schröder, Aaron Stein, Nicholas C. Harris, Dirk Englund. "Nanofabrication on unconventional substrates using transferred hard masks." Scientific reports, 5, Article number 7802 (2015). Article.
[4] A. O. Sushkov, I. Lovchinsky, N. Chisholm, R. L. Walsworth, H. Park, M. D. Lukin, "Magnetic resonance detection of individual proton spins using quantum reporters." Physical Review Letters, 113, 197601 (2014). Abstract.

Novel Electromagnetic Cavities: Bound States in the Continuum

Thomas Lepetit (left) and Boubacar Kanté (right)

Authors: Thomas Lepetit and Boubacar Kanté 

Affiliation: Department of Electrical and Computer Engineering, University of California San Diego, USA. 

In the last 10 years, an intense research effort has been devoted to bringing all-optical signal generation and processing on chip to realize true photonic integrated circuits (PICs). PICs are at their core made of waveguides, which transfer signals to different devices on the circuit, and cavities, which process signals for different functionalities [1]. First, linear devices such as couplers, splitters, and add-drop filters were developed and, more recently, nonlinear devices such as frequency combs, nanolasers, and optical rams have been demonstrated [2-4]. Overall, progress has resulted in devices with increased functionalities that work at lower power and are more compact.

Cavities are an essential building block of PICs because they provide enhanced light-matter interaction. Currently, the most mature technology is based on a silicon on insulator platform and ring resonators. Typically, these dielectric resonators are several microns in diameter [5]. However, due to the difficulty of integration with much smaller electronic components, other technologies such as plasmonics have started to be investigated. One of the main advantages of plasmonic devices, which are made of noble metals such as gold and silver, is that their size is not limited by the wavelength. For example, plasmonic ring resonators of only several hundred nanometers in diameter have been demonstrated [6]. Generally, cavities are characterized by their quality factor Q, which is a measure of their capacity to store signals for a long time. At present, dielectric cavities have reached quality factors of 10⁶, which are only limited by radiation losses coming from sidewall roughness, but typically have a footprint of 80 μm². In contrast, plasmonic cavities can have a footprint as low as 1.25 μm² but their quality factors are usually below 10², being limited by thermal losses coming from conduction electrons. Therefore, there is a need for novel cavity designs that can simultaneously achieve high quality factors and low footprints.

Figure 1: Cross-section of the electric field magnitude for the coupled resonators system (Half of it). Resonator 1, whose inner radius is zero, is on top and resonator 2, with a non-zero inner radius, is at the bottom. The symmetry plane on the right of each plot denotes the symmetry plane of interest. Odd modes are mostly confined in resonator 1 and even modes mostly in resonator 2.

Recently, we have demonstrated the possibility of making electromagnetic cavities using a different concept, namely bound states in the continuum (BICs) [7]. BICs were first proposed in 1929 in the context of quantum mechanics by Von Neumann and Wigner [8]. They surprisingly showed that bound states can exist above the continuum threshold, i.e., there are states that do not decay even in the presence of open decay channels. However, due to the theoretical nature of the first proposal, BICs did not become fully appreciated until 1985 when Friedrich and Wintgen showed that they could be interpreted as resulting from the interference of two distinct resonances [9]. In this picture, one resonance traps the other and thus one quality factor decreases while the other one tends to infinity. Since BICs are essentially a wave phenomenon they also appear in electromagnetics where they translate for lossless dielectrics into an infinite quality factor. As a proof of concept, we have designed and measured a BIC in the microwave range using a periodic metasurface [10-11].

BICs are intrinsically sensitive to perturbations as they only exist at a single point in phase space. This is very useful for sensing applications but detrimental for most others. To obtain an extended BIC, we designed a system with two quasi-degenerate BICs. We achieved this by considering a unit cell with two resonators, a disk and a ring (see Figure 1). Odd modes of the disk resonator interfere and lead to one BIC and even modes of the ring resonator interfere and lead to another BIC. We use ceramic resonators of high-permittivity (ε_r=43±0.75) and they are thus only slightly coupled. Experimentally, to limit the fabrication dispersion inherent to a large array, we made the measurements in a rectangular metallic waveguide (X-band, 8.2-12.4 GHz). It is possible because such a guided setup is equivalent to an infinite array at oblique incidence as shown by image theory.

Figure 2: Modes of two dielectric resonators (ε_r=43) in a rectangular metallic waveguide (X-band). Both resonators are cylindrical (r=3.5 mm, h₁=2.25 mm, h₂=3.0 mm) and the second has a non-zero inner radius. a) Resonance frequencies vs. inner radius for even and odd modes. b) Quality factor vs. inner radius for even and odd modes for lossless and lossy resonators.

We explored phase space along a line, by varying the inner radius of the ring resonator, and showed the presence of two avoided resonance crossings (see Figure 2a), which are typical of BICs [12]. As a result, there is an extended region of phase space where the quality factor tends to infinity (see Figure 2b). BICs only serve to cancel radiation losses and in the presence of thermal losses these are the limiting factor. At present, this scheme is therefore practical only for dielectrics but it could be extended to plasmonics by introducing gain materials to achieve loss-compensation.

Beyond the fundamental interest on the limit of quality-factors given a certain volume, there is a sustained interest in reducing the footprint of many cavity-based devices for future PICs. Tailoring the optical potential further, for example by moving away from perfectly periodic structures [13], opens the possibility improving the field confinement and thus shrink devices. Our work is a first step in this promising direction.

References:
[1] L. A. Coldren, S. W. Corzine, and M. Mašanović, “Diode Lasers and Photonic Integrated Circuits”, 2nd edition, Wiley (2012).
[2] Fahmida Ferdous, Houxun Miao, Daniel E. Leaird, Kartik Srinivasan, Jian Wang, Lei Chen, Leo Tom Varghese, Andrew M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs”, Nature Photonics, 5, 770 (2011). Abstract.
[3] M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, Y. Fainman, “Thresholdless nanoscale coaxial lasers”, Nature 482, 204 (2012). Abstract.
[4] Eiichi Kuramochi, Kengo Nozaki, Akihiko Shinya, Koji Takeda, Tomonari Sato, Shinji Matsuo, Hideaki Taniyama, Hisashi Sumikura, Masaya Notomi, “Large-scale integration of wavelength-addressable all-optical memories on a photonic crystal chip”, Nature Photonics 8, 474 (2014). Abstract.
[5] W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, R. Baets, “Silicon microring resonators”, Laser Photonics Review 6, 47 (2012). Abstract.
[6] Hong-Son Chu, Yuriy Akimov, Ping Bai, Er-Ping Li, “Submicrometer radius and highly confined plasmonic ring resonator filters nased on hybrid metal-oxide-semiconductor waveguide”, Optics Letters, 37, 4564 (2012). Abstract.
[7] Thomas Lepetit, Boubacar Kanté, “Controlling multipolar radiation with symmetries for electromagnetic bound states in the continuum”, Physical Review B Rapid Communications, 90, 241103 (2014). Abstract.
[8] J. von Neumann and E. Wigner, “On unusual discrete eigenvalues”, Zeitschrift für Physik 30, 465 (1929).
[9] H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum”, Physical Review A, 32, 3231 (1985). Abstract.
[10] Boubacar Kanté, Jean-Michel Lourtioz, André de Lustrac, “Infrared metafilms on a dielectric substrate”, Physical Review B, 80, 205120 (2009). Abstract.
[11] Boubacar Kanté, André de Lustrac, Jean Michel Lourtioz, “In-plane coupling and field enhancement in infrared metamaterial surfaces”, Physical Review B, 80, 035108 (2009). Abstract.
[12] Chia Wei Hsu, Bo Zhen, Jeongwon Lee, Song-Liang Chua, Steven G. Johnson, John D. Joannopoulos, Marin Soljačić, “Observation of trapped light within the radiation continuum”, Nature, 499, 188 (2013). Abstract.
[13] Yi Yang, Chao Peng, Yong Liang, Zhengbin Li, Susumu Noda, “Analytical perspective for Bound States in the Continuum in Photonic Crystal Slabs”, Physical Review Letters, 113, 037401 (2014). Abstract.

Observation of Majorana Fermions in Ferromagnetic Atomic Chains on a Superconductor

From Left to Right: (top row) Stevan Nadj-Perge, Ilya K. Drozdov, Jian Li, Hua Chen ; (bottom row) Sangjun Jeon, Allan H. MacDonald, B. Andrei Bernevig, Ali Yazdani.

Authors: Stevan Nadj-Perge¹, Ilya K. Drozdov¹, Jian Li¹, Hua Chen², Sangjun Jeon¹, Jungpil Seo¹, Allan H. MacDonald², B. Andrei Bernevig¹, Ali Yazdani¹.

Affiliation
¹Joseph Henry Laboratories and Dept of Physics, Princeton University, USA.
²Department of Physics, University of Texas at Austin, USA.

Link to Yazdani Lab >>
Link to Allan H. MacDonald's Group >>

In 1937 Italian scientist Ettore Majorana, one of the most promising theoretical physicists at that time, proposed a hypothetical fermionic excitation, now called Majorana fermion, which has a property that it is its own anti-particle [1]. Ever since significant efforts were invested in finding an elementary particle described by Majorana. While present for many years in particle physics community, it was only in 2001 that Alexei Kitaev suggested an intriguing possibility that a type of a quasi-particle, a condensed matter analog of the Majorana fermion could exist [2]. Such quasi-particle would emerge as a zero energy excitation localized at the boundary of a one dimensional topological superconductor. Following his seminal work various systems were proposed as a potential platform for realization of Majorana quasi-particles. Apart from fundamental scientific interest, the motivation for investigating Majorana bound states partly relies on their potential for use for quantum computing [3,4].

Past 2Physics article by Ali Yazdani :
August 25, 2013: "Visualizing Nodal Heavy Fermion Superconductivity"
by Brian Zhou, Shashank Misra, and Ali Yazdani.

Previous to our experiments, the most promising experimental route to realize these elusive bound states was based on semiconductor-superconductor interfaces in which Majorana fermions would appear as conductance peaks at zero energy [5,6]. Indeed experiments in 2012 reported zero energy conductance peaks suggesting presence of Majorana modes in these type of systems [7,8], however, alternative explanations related to disorder and Kondo phenomena proposed latter could not be fully ruled out. It is worth noting that in these interfaces spatial information about localized excitations is very hard to obtain.

Fig. 1: (A) Schematic of the proposal for realization and detection of Majorana states: A ferromagnetic atomic chain is placed on the surface of strongly spin-orbit–coupled superconductor and studied using STM. (B) Band structure of a linear suspended Fe chain before introducing spin-orbit coupling or superconductivity. The majority spin-up (red) and minority spin-down (blue) d-bands labeled by azimuthal angular momentum m are split by the exchange interaction J (degeneracy of each band is noted by the number of arrows). a, interatomic distance. (C) Regimes for trivial and topological superconducting phases are identified for the band structure shown in (B) as a function of exchange interaction in presence of SO coupling. The value J for Fe chains based on density functional calculations is noted. μ is the chemical potential.

Building upon previous proposal to realize Majorana modes in an array of magnetic nanoparticles [9], we proposed to use a chain of magnetic atoms coupled to a superconductor [10]. The key advantage of this platform is that the experimentally properties of this system can easily be studied using standard scanning tunneling microscopy (STM) technique. While ours and other follow-up initial proposals [11-16] consider specific orientation of the magnetic moments, the approach works also for ferromagnetic atomic chains as long as it is coupled to a superconductor with strong spin-orbit coupling (Fig. 1A) [17]. In this case, the large exchange interaction results in a band structure of the chains such that majority spin band is fully occupied while the Fermi level is in the minority spin bands. For example the electronic structure of a linear Iron (Fe) ferromagnetic atomic chain is shown in Fig. 1B. Considering only d-orbitals which are spin-polarized it is easy to show that many of the bandstructure degeneracies are lifted and that for large range of parameters, chemical potential and the exchange energy, such chains are in topologically non-trivial regime characterized by the odd number of crossing at the Fermi level (Fig 1C). When placed on the superconducting substrate with strong-spin orbit coupling the resulting superconductivity on the chain will necessarily be topological in nature resulting in zero energy Majorana bound states located at the chain ends.

Fig. 2: (A) Topograph of the Pb(110) surface after growth of Fe, showing Fe islands and chains indicated by white arrows and atomically clean terraces of Pb (regions with the same color) with size exceeding 1000 Å. (Lower-right inset) Anisotropic atomic structure of the Pb(110) surface (Upper-left insets) images of several atomic Fe chains and the islands from which they grow (scale bars, 50 Å). (B) Topography of the chain colorized by the conductance at H= ±1 T from low (dark blue) to high conductance (dark red). (C) Difference between conductance on and off the chain showing hysteresis behavior. (D and E) Atomic structure of the zigzag chain, as calculated using density functional theory. The Fe chain structure that has the lowest energy in the calculations matches the structural features in the STM measurements, a_B is the Bohr radius.

We have developed a way to grow iron atomic chains on the surface of lead (Pb) which, due to heavy atomic mass, is expected to have strong spin-orbit coupling. For this purpose we used Pb(110) crystallographic surface orientation which has characteristic anisotropy (Fig. 2(A) lower left inset). When a sub-monolayer of Fe is evaporated and slight annealing, the anisotropy of the substrate could trigger growth of one-dimensional atomic chains. We investigated the resulting structures by using STM at cryogenic temperatures (temperature was 1.4K in the experiment). On relatively large atomically ordered regions of the Pb(110) surface we observed self-assembled islands as well as single atom wide chains of Fe. Depending on growth conditions, we find Fe chains as long as 500 Å with ordered regions approaching 200 Å. In order to confirm ferromagnetic order on the chain we have performed spin-polarized measurements using bulk antiferromagnetic Chromium STM tips. Tunneling conductance (dI/dV) at a low bias voltage as a function of the out-of-plane magnetic field shows contrast for opposite fields (Fig. 2B) and a hysteresis behavior (Fig. 2C, note that no hysteresis is observed on the Pb substrate). The observed hysteresis loop corresponds to the tunneling of electrons between two magnets with the field switching only one of them at around 0.25 T.

We also observed the variation of the spin-polarized STM signal along the chain which is likely due to its electronic and structural properties. Indeed, both topographic features and periodicity of signal variation could be very well explained by our theory collaborators who performed density functional theory modeling. Their calculations suggested that our chains have zig-zag structure which explains both topographic information obtained using STM and matches well with our spin-polarized measurements (Fig. 2D and 2E).

Fig. 3: (A) STM spectra measured on the atomic chain at locations corresponding to those indicated in (B) and (C). For clarity, the spectra are offset by 100 nS. The red spectrum shows the zero-bias peak at one end of the chain. The gray trace measured on the Pb substrate can be fitted using thermally broadened BCS superconducting density of states (dashed gray line, fit parameters Δs = 1.36 meV, T = 1.45 K). (B and C) Zoom-in topography of the upper (B) and lower end (C) of the chain and corresponding locations for spectra marked (1 to 7). Scale bars, 25 Å. (D and E) Spectra measured at marked locations, as in (B) and (C). (F) Spatial and energy-resolved conductance maps of another atomic chain close to its end, which shows similar features in point spectra as in (A). The conductance map at zero bias (middle panel) shows increased conductance close to the end of the chain. Scale bar, 10 Å.

After establishing basic properties of our chains we investigated low-energy excitations using spatial spectroscopic mapping, see Fig 3. While on the surface of bare Pb(110), there is clear structure of the superconducting gap on the Fe atomic chain, the presence of the in-gap states is predominant (Fig. 3A). Most notably a peak close to zero bias voltage is observed near the chain end together with asymmetric less-developed gap-like structure in the middle of the chain (Fig. 3D and Fig. 3E). Both spatially resolved spectra and the spectroscopic maps at low bias voltage show signatures expected from Majorana bound states (Fig. 3F). The ability to correlate the location of the zero bias conductance peak with the end of the atomic chains is one of the main experimental results of our work. This is one of the basic requirements for interpreting that this feature is associated with the predicted Majorana bound state of a topological superconductor. In addition to robust observation of the zero bias peaks in many chains, we have performed several control experiments to eliminate other potential effects which may give similar looking signatures. For example, when superconductivity is suppressed by applying small magnetic field, the spectrum on the chain becomes featureless in contrast to what would be expected for Kondo effect. Also for very short chains zero biased peaks were not observed, ruling out trivial effects related to the chain ends. Furthermore, in order to increase experimental resolution we took measurements with superconducting tip which also confirm the over picture consistent with Majorana bound states in this system.

The observed spectroscopic signatures are consistent with the existence of Majorana bound states in our system. An obvious extension of our experiments is to create two dimensional islands and search for propagating Majorana modes or, for example, investigate other systems with both even and odd number of band crossing at Fermi level in order to further test the concept behind our study. Ultimately the future experiments will focus on manipulation of Majorana bound states in this system [18].

References: 
[1] Ettore Majorana, "Teoria simmetrica dell’elettrone e del positrone", Il Nuovo Cimento, 171 (1937). Abstract.
[2] A. Yu. Kitaev, "Unpaired Majorana fermions in quantum wires". Physics Uspekhi, 44, 131 (2001). Full Article.
[3] A. Yu. Kitaev, "Fault-tolerant quantum computation by anyons". Annals of Physics. 303, 2 (2003). Abstract.
[4] Jason Alicea, Yuval Oreg, Gil Refael, Felix von Oppen, Matthew P. A. Fisher, "Non-Abelian statistics and topological quantum information processing in 1D wire networks". Nature Physics, 7, 412 (2011). Abstract.
[5] Yuval Oreg, Gil Refael, and Felix von Oppen, "Helical Liquids and Majorana Bound States in Quantum Wires", Physical Review Letters, 105, 177002 (2010). Abstract.
[6] Roman M. Lutchyn, Jay D. Sau, S. Das Sarma, "Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures". Physical Review Letters, 105, 077001 (2010). Abstract.
[7] V. Mourik, K. Zuo, S.M. Frolov, S.R. Plissard, E.P.A.M. Bakkers, L.P. Kouwenhoven, "Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices", Science 336, 1003 (2012). Abstract. 2Physics Article.
[8] Anindya Das, Yuval Ronen, Yonatan Most, Yuval Oreg, Moty Heiblum, Hadas Shtrikman, "Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of Majorana fermions". Nature Physics, 8, 887 (2012). Abstract. 2Physics Article.
[9] T. P. Choy, J. M. Edge, A. R. Akhmerov, C. W. J. Beenakker, "Majorana fermions emerging from magnetic nanoparticles on a superconductor without spin-orbit coupling". Physical Review B, 84, 195442 (2011). Abstract.
[10] S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, Ali Yazdani, "Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor". Physical Review B, 88, 020407 (2013). Abstract.
[11] Falko Pientka, Leonid I. Glazman, Felix von Oppen, "Topological superconducting phase in helical Shiba chains". Physical Review B, 88, 155420 (2013). Abstract.
[12] Jelena Klinovaja, Peter Stano, Ali Yazdani, Daniel Loss, "Topological Superconductivity and Majorana Fermions in RKKY Systems". Physical Review Letters, 111, 186805 (2013). Abstract.
[13] Bernd Braunecker, Pascal Simon, "Interplay between Classical Magnetic Moments and Superconductivity in Quantum One-Dimensional Conductors: Toward a Self-Sustained Topological Majorana Phase". Physical Review Letters, 111, 147202 (2013). Abstract.
[14] M. M. Vazifeh, M. Franz, "Self-Organized Topological State with Majorana Fermions". Physical Review Letters, 111, 206802 (2013). Abstract.
[15] Sho Nakosai, Yukio Tanaka, Naoto Nagaosa, "Two-dimensional superconducting states with magnetic moments on a conventional superconductor". Physical Review B, 88, 180503 (2013). Abstract.
[16] Younghyun Kim, Meng Cheng, Bela Bauer, Roman M. Lutchyn, S. Das Sarma, "Helical order in one-dimensional magnetic atom chains and possible emergence of Majorana bound states". Physical Review B, 90, 060401 (2014). Abstract.
[17] Stevan Nadj-Perge, Ilya K. Drozdov, Jian Li, Hua Chen, Sangjun Jeon, Jungpil Seo, Allan H. MacDonald, B. Andrei Bernevig, Ali Yazdani, "Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor". Science 346, 602-607 (2014). Abstract.
[18] Jian Li, Titus Neupert, B. Andrei Bernevig, Ali Yazdani, "Majorana zero modes on a necklace". arXiv:1404.4058 [cond-mat] (2014).

Enhancement of Long-Range Correlations in a 2D Vortex Lattice by an Incommensurate 1D Disorder Potential

(Left to Right) Top Row: Isabel Guillamón, Rosa Córdoba; Middle Row: Javier Sesé, José María De Teresa, M. Ricardo Ibarra; Bottom Row: Sebastián Vieira, Hermann Suderow.

Authors:
Isabel Guillamón^1,2, Rosa Córdoba^3,*, Javier Sesé³, José María De Teresa^3,4, M. Ricardo Ibarra³, Sebastián Vieira¹, Hermann Suderow¹

Affiliations:
¹Laboratorio de Bajas Temperaturas, Departamento de Física de la Materia Condensada, Instituto de Ciencia de Materiales Nicolás Cabrera, Condensed Matter Physics Center, Unidad Asociada de Bajas Temperaturas y Altos Campos Magnéticos, Universidad Autónoma de Madrid, Spain
²H.H. Wills Physics Laboratory, University of Bristol, UK
³Laboratorio de Microscopías Avanzadas (LMA), Instituto de Nanociencia de Aragón (INA), Universidad de Zaragoza, Spain
⁴Instituto de Ciencia de Materiales de Aragón (ICMA), CSIC-Universidad de Zaragoza, Spain
^*Present address: Department of Applied Physics, Eindhoven University of Technology, The Netherlands.

Recently we studied the effect of random disorder in a particularly simple two dimensional system and found a new recipe to enhance order. We used a thin superconductor under magnetic fields[1]. The Cooper pairs in the superconductor turn and form quantum vortices. Each one is like a small tornado, but they are all very tiny and, what is most important, they all look exactly the same and repel each other. As a result, they form a perfect hexagonal lattice, which can be imaged with a surprising accuracy.

Past 2Physics article by this group:
March 10, 2013: "Nanostructuring Improves Vortex Pinning in Superconductors at Elevated Temperatures and Magnetic Fields" by R. Córdoba, T. I. Baturina, J. Sesé, A. Yu. Mironov, J. M. De Teresa, M. R. Ibarra, D. A. Nasimov, A. K. Gutakovskii, A.V. Latyshev, I. Guillamón, H. Suderow, S.Vieira, M. R. Baklanov, J. J. Palacios, V.M.Vinokur.

To introduce random disorder in a controlled way, we made a one-dimensional nanostructure and showed that, under certain conditions, the hexagonal lattice floats over the nanostructure (Fig.1 right panel). The floating lattice disorders when increasing the vortex density. Random disorder appears because both lattices are incommensurate. We find that the transition into the disordered state occurs far above present expectations from random field theory. Our conclusion is that the weak one-dimensional correlations inhibit the effect of random disorder.

Figure 1: Vortices in a hexagonal lattice are shown as red points. A linear modulation acts on the vortex positions. When the vortex density is low, vortices are confined to the minima of the linear modulation, and the interaction among them is essentially confined to the lines along the potential minima, as schematically shown in the left panel. However, when the vortex density increases, one can accommodate many vortices within one modulation’s wavelength. In the right panel we schematically show this situation (for clarity, we only draw a small amount of vortices). In that case, the lattice and 1D potential modulation have incommensurate spatial periods and the orientation of the lattice no longer follows the linear modulation. The resulting vortex landscape is quasi-random. The vortex positions are slightly displaced with respect to the ordered lattice. In this work, we show that the displacement grows logarithmically with distance, a feature which demonstrates that the disorder created by this situation is scale-invariant.

To see this in more detail, let us think of many identical particles that repel each other forming a single layer on a corrugated surface and at zero temperature. Let the position of the particles depend on the size of the corrugation. If the surface shows a periodic one-dimensional pattern, as the wrinkles you create on a carpet when you lay it down on your floor, you may obtain hexagonal order commensurate to the wrinkles (left panel of Fig.1, [2]). The question is, what happens if you produce a fully random pattern which looks the same at all scales and increase the size of the corrugation? To answer this question (posed in several theory papers some time ago, see e.g.Refs. [3,4]) we first need to create such a random pattern. Think of a carpet with many small (much smaller than the particle size) and stiff yarns of many different lengths. To make such a carpet, you have to prune it yarn by yarn choosing each time a completely random height. This looks difficult.

Instead, we preferred to use the one-dimensional modulation [5]. The wavelength of the modulation is incommensurate to the wavelength of the ordered particle lattice. Disorder in the particle positions appears automatically, simply because both lattices do not match.

Then, as the second step, we decrease the interaction strength among particles. This is easy for vortices in superconductors [6]. We just have to increase the magnetic field, and we effectively increase the size of the corrugation, with respect to the strength of the interaction among particles.

The experiment directly showing the disorder transition in the vortex lattice was made using a scanning tunneling microscope effectively at absolute zero in the low temperature laboratory of the Universidad Autonóma de Madrid [7]. The thin superconductor with a weak one-dimensional modulation was fabricated in the Laboratory of advanced Microscopies & Nanoscience Institute of Aragón [8]. We eliminated the effect of temperature altogether and produced a very well controlled linear nanostructure. We imaged up to several thousands of vortices in Madrid. This allowed us to characterize the critical exponents of the transition very accurately.

Transitions are often measured in macroscopic experiments, where the obtained information is the result of the average of particles’ positions and the interaction among them. Even microscopic experiments, as neutron scattering, provide an average over many particles. In our work, we were able to show the macroscopic behavior by imaging lots of vortices one by one. This allowed us to characterize precisely the transition. The order-disorder transition goes on by particle displacements producing topological defects (dislocations and disclinations) in an otherwise ordered lattice. The experiment shows how random quenched disorder progresses in the lattice when increasing the magnetic field. We find a logarithmic increase of displacements with respect to the ordered lattice at low fields and density fluctuations in the high field disordered phase. Both effects nicely agree with random field theories.

But we also find that the size of the random potential at which the transition takes places does not agree with random field theory. On rather universal arguments, theory finds that those topological defects should arise at a critical disorder strength of 1/8 [3,4]. The experiment gives a much higher critical value, of 1/2. The lattice stays ordered over a significantly larger range of magnetic fields than expected. What is favoring order? The answer lies in the correlations produced by the one-dimensional modulation. This was discussed theoretically for spin systems [9], but had not been fully addressed by experiments before. Although the one-dimensional modulation does not influence the orientation of the lattice nor directly determines the precise positions of the particles in the lattice, correlations give consistency to the two dimensional hexagonal order.

Figure 2: Image of the vortex lattice in the high field disordered phase at 5.5 T taken with our microscope. Background color represents the intensity of vortex density fluctuations. The lattice is shown by its Delaunay triangulation. Disclinations –five or seven coordinated vortices, green and orange dots, respectively– and disclination pairs –dislocations– are present over the whole surface producing a very disordered vortex lattice. The 1D thickness modulation generating the symmetry breaking disorder is also shown.

The result that you can help supporting an ordered hexagonal two-dimensional lattice in a random medium by introducing a tiny amount of symmetry breaking correlations is both useful and unexpected. It adds now a new looking-glass for vortex physics in nanostructured superconductors.

Our to-do list includes studying vortex dynamics in presence of the disorder potential. By applying a current, we want to see when vortices start moving and study the stability of superconductivity to disorder. The careful observation of vortex lattices in superconductors will be useful to understand more about the influence of disorder in macroscopic quantum coherence.

References:
[1] I. Guillamón, R. Córdoba, J. Sesé, J. M. De Teresa, M. R. Ibarra, S. Vieira, H. Suderow, "Enhancement of long-range correlations in a 2D vortex lattice by an incommensurate 1D disorder potential". Nature Physics, 10, 851 (2014). Abstract.
[2] Piero Martinoli, "Static and dynamic interaction of superconducting vortices with a periodic pinning potential". Physical Review B, 17, 1175–1194 (1978). Abstract.
[3] Thomas Nattermann, Stefan Scheidl, Sergey E. Korshunov, Mai Suan Li, "Absence of reentrance in the two-dimensional XY-model with random phase shift". Journal of Physics I France, 5, 565–572 (1995). Abstract.
[4] David Carpentier, Pierre Le Doussal, "Melting of two dimensional solids on disordered substrates". Physical Review Letters, 81, 1881 (1998). Abstract.
[5] Laurent Sanchez-Palencia, Maciej Lewenstein, "Disordered quantum gases under control". Nature Physics, 6, 87 (2010). Abstract.
[6] Ernst Helmut Brandt, "Vortex-vortex interaction in thin superconducting films". Physical Review B, 79, 134529 (2009). Abstract.
[7] http://lbtuam.es
[8] http://ina.unizar.es/, http://www.icma.unizar-csic.es/ICMAportal/, http://www.unizar.es 
[9] J. Wehr, A. Niederberger, L. Sanchez-Palencia, M. Lewenstein, "Disorder versus the Mermin–Wagner–Hohenberg effect: From classical spin systems to ultracold atomic gases". Physical Review B, 74, 224448 (2006). Abstract.

Imaging Spin-Valley-Layer Locking in a Transition-Metal Dichalcogenide

Phil King

Author:  Phil King 

Affiliation: SUPA, School of Physics and Astronomy, University of St Andrews, UK

Link to King Group >>

Transition-metal dichalcogenides (TMDs) of the form MCh₂, where M is a transition-metal and Ch a chalcogen, have received much attention in recent years. They can be stabilised as single Ch-M-Ch monolayers, which display a host of attractive materials properties including direct band gaps in the visible region and ambipolar conduction, suggesting a range of applications in electronics and optoelectronics [1]. Moreover, they host degenerate band extrema at the corners of the hexagonal Brillouin zone, which gives rise to a so-called valley degree of freedom. A combination of strong spin-orbit interactions with broken inversion symmetry in the monolayer causes this valley pseudospin to become strongly coupled to the real spin [2]. Valley-dependent optical selection rules combined with suppressed inter-valley scattering resulting from their coupled spin-valley texture has opened new possibilities for optical control of spin and valley pseudospins [3-5].

This suggests unique potential to exploit TMDs in novel schemes of electronics exploiting the spin (a.k.a. spintronics) or valley (a.k.a. valleytronics) degrees of freedom, with the ultimate potential for faster, smaller, and more energy-efficient devices. One might naturally expect this potential to be lost for their bulk counterparts, where the most common structure (the 2H polymorph) is formed by stacking single TMD monolayers together with a 180° rotation between neighbouring layers (other stacking sequences host their own interesting properties [6], but we don’t consider these here). The bulk unit cell therefore contains two such monolayers, having a centre of inversion. It is well established that such inversion symmetry, together with time-reversal symmetry, enforces all electronic states in solids to be spin-degenerate.

In our recent work published in Nature Physics [7], performed in a collaboration between my group in St Andrews (UK) and researchers at the Norwegian University of Science and Technology, the universities of Tokyo (Japan), Aarhus (Denmark), and Suranaree (Thailand), the Max-Planck Institute in Stuttgart (Germany), MAX-IV Laboratory (Sweden) and Diamond Light Source (UK), we have instead observed spin-polarised states persisting in centrosymmetric bulk WSe₂.

We used angle-resolved photoemission spectroscopy (ARPES) to probe the electronic structure of bulk crystals of 2H-WSe₂. Through a process of Mott scattering, we also measured the spin polarisation of the emitted photoelectrons, and discovered that the electronic states around the corners of the Brillouin zone were almost 100% spin polarised. Surely this would seem to contradict the inversion symmetry that this material possesses?

Figure 1: Valence band dispersions of WSe₂ measured by angle-resolved photoemission, showing excellent agreement with theoretical calculations of the k_z-dependent bulk electronic structure (coloured lines). The spin texture measured at the K and K’ points of the Brillouin zone is shown schematically by coloured arrows.

Through a combination of photon-energy dependent ARPES experiments and first-principles theoretical calculations, we observed how, for the electronic states close to the Brillouin zone corners, their wavefunctions are spatially localised within single monolayers of the bulk crystal structure where locally, inversion symmetry is not present. The combination of this inversion symmetry breaking together with strong spin-orbit coupling drives these states to develop huge spin polarisations, leading to spin-valley locking as for isolated monolayers.

Figure 2: Angle-resolved photoemission measurements of WSe₂ throughout the Brillouin zone, schematically showing the intertwined layer- and momentum-dependent spin texture uncovered here.

The 180° rotation between neighbouring monolayers in the bulk crystal structure, however, imposes an additional layer-dependent sign change of the spin polarisation for a given valley, as found in previous theoretical calculation [8], and recently also suggested from polarisation-resolved optical experiments [9]. By exploiting photon energy-dependent interference between photoelectrons emitted from different crystal layers, we could tune the measured photoelectron spin-polarisation nearly to zero, effectively averaging over neighbouring layers, or could selectively probe just the top monolayer of the crystal. These measurements together provide the first direct observation of the entangling of the spin with valley and layer pseudospins in a bulk transition-metal dichalcogenide.

Moreover, our study provides an experimental observation that local, rather than global, inversion symmetry breaking is sufficient to stabilise spin-polarised states in solids [10], contrary to conventional wisdom. This is exciting because it reveals that a whole new class of materials which we previously thought must have only spin-degenerate energy bands can in fact locally host spin-polarised states. Controlling this could bring fantastic new opportunities for spin- and valleytronics, and a whole arsenal of new materials in which we can achieve this.

References:
[1] Qing Hua Wang, Kourosh Kalantar-Zadeh, Andras Kis, Jonathan N. Coleman, Michael S. Strano, "Electronics and optoelectronics of two-dimensional transition metal dichalcogenides". Nature Nanotechnology, 7, 699 (2012). Abstract.
[2] Di Xiao, Gui-Bin Liu, Wanxiang Feng, Xiaodong Xu, Wang Yao, "Coupled Spin and Valley Physics in Monolayers of MoS₂ and Other Group-VI Dichalcogenides". Physical Review Letters, 108, 196802 (2012). Abstract.
[3] Hualing Zeng, Junfeng Dai, Wang Yao, Di Xiao, Xiaodong Cui, "Valley polarization in MoS₂ monolayers by optical pumping". Nature Nanotechnology, 7, 490 (2012). Abstract.
[4] Kin Fai Mak, Keliang He, Jie Shan, Tony F. Heinz, "Control of valley polarization in monolayer MoS₂ by optical helicity". Nature Nanotechnology, 7, 494 (2012). Abstract.
[5] Xiaodong Xu, Wang Yao, Di Xiao, Tony F. Heinz, "Spin and pseudospins in layered transition metal dichalcogenides". Nature Physics, 10, 343 (2014). Abstract.
[6] R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, Y. Iwasa, "Valley-dependent spin polarization in bulk MoS₂ with broken inversion symmetry". Nature Nanotechnology, 9, 611 (2014). Abstract.
[7] J.M. Riley, F. Mazzola, M. Dendzik, M. Michiardi, T. Takayama, L. Bawden, C. Granerød, M. Leandersson, T. Balasubramanian, M. Hoesch, T.K. Kim, H. Takagi, W. Meevasana, Ph. Hofmann, M.S. Bahramy, J.W. Wells, P. D. C. King, "Direct observation of spin-polarized bulk bands in an inversion-symmetric semiconductor". Nature Physics, 10, 835 (2014). Abstract.
[8] Zhirui Gong, Gui-Bin Liu, Hongyi Yu, Di Xiao, Xiaodong Cui, Xiaodong Xu, Wang Yao, "Magnetoelectric effects and valley-controlled spin quantum gates in transition metal dichalcogenide bilayers". Nature Communications, 4, 2053 (2013). Abstract.
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