# Research and collaboration

#### Spin transfer torque and stochastic magnetization dynamics in the quantum limit

P. Virtanen and T.T. Heikkilä, Phys. Rev. Lett.

The state of small magnets containing a single magnetic domain can be described via their magnetisation vector. Typically the size of the magnetisation is fixed to some particular value, but its direction may vary. One way of changing the magnetisation is via the spin transfer torque that is exerted on the magnet by a spin-polarised current injected into the magnet from a nearby ferromagnetic electrode. Under proper conditions, this spin transfer torque may drive the small magnet into a stable precession. As a result, the precession drives a reciprocal effect to the spin transfer torque: spin pumping, i.e., a spin current is injected from the magnet, and converted into a real current in the ferromagnetic electrode. These phenomena have been intensively studied in the past two decades, and they form a basis in the search of new types of methods to create magnetic memories. However, the stochasticity of the spin transfer torque or spin pumping have not been thoroughly studied in the past. We have devised a means to calculate the full probability distribution of magnetisation dynamics and spin pumping, valid in the quantum limit where the precession rate (times the Planck's constant) exceeds the temperature (times Boltzmann constant). Our approach allows for example calculation of transition probabilities, current noise, and other stochastic phenomena in the magnet.

**118**, 237701 (2017)The state of small magnets containing a single magnetic domain can be described via their magnetisation vector. Typically the size of the magnetisation is fixed to some particular value, but its direction may vary. One way of changing the magnetisation is via the spin transfer torque that is exerted on the magnet by a spin-polarised current injected into the magnet from a nearby ferromagnetic electrode. Under proper conditions, this spin transfer torque may drive the small magnet into a stable precession. As a result, the precession drives a reciprocal effect to the spin transfer torque: spin pumping, i.e., a spin current is injected from the magnet, and converted into a real current in the ferromagnetic electrode. These phenomena have been intensively studied in the past two decades, and they form a basis in the search of new types of methods to create magnetic memories. However, the stochasticity of the spin transfer torque or spin pumping have not been thoroughly studied in the past. We have devised a means to calculate the full probability distribution of magnetisation dynamics and spin pumping, valid in the quantum limit where the precession rate (times the Planck's constant) exceeds the temperature (times Boltzmann constant). Our approach allows for example calculation of transition probabilities, current noise, and other stochastic phenomena in the magnet.

#### Amplification close to a quantum limit in optomechanical systems

C. F. Ockeloen-Korppi, T. T. Heikkilä, M. A. Sillanpää, F. Massel:

*Theory of noiseless phase-mixing amplification in a cavity optomechanical system, Quantum Sci. Technol.*.**2**, 035002 (2017) [arXiv:1610.07579]C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, M. A. Sillanpää,

Francesco Massel, T.T. Heikkilä, J.-M. Pirkkalainen, S.U. Cho, H. Saloniemi, P.J. Hakonen, and Mika A. Sillanpää,

The quality of amplifiers can be characterised by the amount of noise they add to the amplified signal. Quantum mechanics poses a lower limit to this added noise: for large gain (amplification), the added noise to the input signal is always more than half a quantum, corresponding an added energy of

*Noiseless quantum measurement and squeezing of microwave fields utilizing mechanical vibrations,*Phys. Rev. Lett.**118**, 103601 (2017).**C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, and M. A. Sillanpää,**

*Low-noise amplification and frequency conversion with a multiport microwave optomechanical device,*Phys. Rev. X**6**, 041024 (2016).Francesco Massel, T.T. Heikkilä, J.-M. Pirkkalainen, S.U. Cho, H. Saloniemi, P.J. Hakonen, and Mika A. Sillanpää,

*Microwave amplification with nanomechanical resonators,*Nature**480**, 351-354 (2011)The quality of amplifiers can be characterised by the amount of noise they add to the amplified signal. Quantum mechanics poses a lower limit to this added noise: for large gain (amplification), the added noise to the input signal is always more than half a quantum, corresponding an added energy of

*hf/2,*where*h*is Planck's constant and*f*is the signal frequency. The question is how to reach this limit. For optical frequencies (visible light) systems with noise performance very close to the quantum limit have been constructed since the 1990's, but the task has been much harder for setups involving signals in the microwave (GHz) regime. We have shown, both theoretically and experimentally (M. Sillanpää's group in Aalto University performing the experiments) that a system comprising a microwave resonator (antenna) coupled to a high-quality micro mechanical resonator can be tuned so that amplification very close to the quantum limit can be obtained. In our recent work a system consisting of two microwave resonators coupled to the same mechanical resonator allows for reaching close to quantum limited performance, and simultaneously allowing for a frequency conversion of microwave signals. This type of setup could also be used to convert quantum information between microwave and optical regimes.#### Graphite and its electronic structure

T. Hyart, R. Ojajärvi and T.T. Heikkilä, J. Low Temp. Phys. (2018)

T. Hyart and T.T. Heikkilä, Phys. Rev. B 93, 235147 (2016)

T.T. Heikkilä and G.E. Volovik Nexus and Dirac lines in topological materials, New J. Phys.

Regular graphite that can be found for example in pencils has an intriguing electronic structure. We have shown in our recent work that graphite in its pristine form is neither a regular metal, nor an insulator, but forms a semimetal structure where electronic bands cross each other. In physics such crossings are typically forbidden due to the coupling between such bands, and only allowed in the presence of some symmetry forbidding the coupling. We have shown that the crossing of the bands in graphite happens within a line of momenta, so that graphite is an example of a nodal line semimetals. In fact, there are two major types of graphite, related to how the graphene layers are stacked. In the more common Bernal (AB) graphite, the nodal line "protected" by the mirror symmetries of the graphite structure, whereas in Rhombohedral (or ABC) stacked graphite the protection arises from inversion symmetry, along with the more usual spin rotation and time-reversal symmetries. What is more, we have also shown that both types of graphite also contain a more rich topological structure, exhibiting in AB graphite a transition between type I and type II Dirac lines, and a nexus point where Dirac lines merge and annihilate each other. This nexus point is an example of the structure dubbed "new fermions" in the recent physics literature, as it host a point of a triple degeneracy of electronic states. Another way to distinguish different types of nodal lines is to count how many times they go around the first Brillouin zone. Typical A-type nodal line materials host nodal lines within the Brillouin zone, whereas type B nodal lines go at least once around the Brillouin zone. These are topologically distinct possibilities, as type A-lines can exist and be annihilated alone, whereas the B-type nodal lines have a monopole-type topological charge, which means that they have to appear and disappear in pairs of opposite topological charge. In the above-cited works we have shown that AB graphite hosts type A lines, whereas ABC graphite hosts type B lines. Moreover, we showed how the type B lines can be destroyed in a process where the pairs of lines meet, cut each other to form type A lines, and finally vanish.

T.T. Heikkilä and G.E. Volovik Nexus and Dirac lines in topological materials, New J. Phys.

**17**, 093019 (2015)Regular graphite that can be found for example in pencils has an intriguing electronic structure. We have shown in our recent work that graphite in its pristine form is neither a regular metal, nor an insulator, but forms a semimetal structure where electronic bands cross each other. In physics such crossings are typically forbidden due to the coupling between such bands, and only allowed in the presence of some symmetry forbidding the coupling. We have shown that the crossing of the bands in graphite happens within a line of momenta, so that graphite is an example of a nodal line semimetals. In fact, there are two major types of graphite, related to how the graphene layers are stacked. In the more common Bernal (AB) graphite, the nodal line "protected" by the mirror symmetries of the graphite structure, whereas in Rhombohedral (or ABC) stacked graphite the protection arises from inversion symmetry, along with the more usual spin rotation and time-reversal symmetries. What is more, we have also shown that both types of graphite also contain a more rich topological structure, exhibiting in AB graphite a transition between type I and type II Dirac lines, and a nexus point where Dirac lines merge and annihilate each other. This nexus point is an example of the structure dubbed "new fermions" in the recent physics literature, as it host a point of a triple degeneracy of electronic states. Another way to distinguish different types of nodal lines is to count how many times they go around the first Brillouin zone. Typical A-type nodal line materials host nodal lines within the Brillouin zone, whereas type B nodal lines go at least once around the Brillouin zone. These are topologically distinct possibilities, as type A-lines can exist and be annihilated alone, whereas the B-type nodal lines have a monopole-type topological charge, which means that they have to appear and disappear in pairs of opposite topological charge. In the above-cited works we have shown that AB graphite hosts type A lines, whereas ABC graphite hosts type B lines. Moreover, we showed how the type B lines can be destroyed in a process where the pairs of lines meet, cut each other to form type A lines, and finally vanish.

#### High-temperature superconductivity in flat band systems

Risto Ojajärvi, Timo Hyart, M.A. Silaev, and Tero T. Heikkilä,

*Competition of electron-phonon mediated superconductivity and Stoner magnetism on a flat band*[arXiv:1801.01794]T.T. Heikkilä and G.E. Volovik,

V. Kauppila, T. Hyart, and T.T. Heikkilä,

*Flat bands as a route to high-temperature superconductivity in graphite*, arXiv:1504.05824**V.J. Kauppila, F. Aikebaier, and T.T. Heikkilä,**

*Flat band superconductivity in strained Dirac materials,*Phys. Rev. B**93**, 214505 (2016)V. Kauppila, T. Hyart, and T.T. Heikkilä,

*Collective amplitude mode fluctuations in a flat band superconductor*, Phys. Rev. B**93**, 024505*(2016)***Traditionally, superconductivity is viewed as a low-temperature property of materials. Within the BCS theory this is understood to result from the fact that the pairing of electrons takes place only close to the usually two-dimensional Fermi surface residing at a finite chemical potential. Because of this, the critical temperature is exponentially suppressed compared to the microscopic energy scales. On the other hand, pairing electrons around a dispersionless (flat) energy band leads to very strong superconductivity, with a mean-field critical temperature linearly proportional to the microscopic coupling constant. We have studied the microscopic theory of superconductivity for different types of (approximate) flat band models, and infer the characteristics of such models, deviating from that of regular superconductors. Our studies may pave the way to construction and detailed understanding of very high-temperature superconductivity in systems exhibiting such flat bands.**

#### Nonequilibrium and thermoelectric effects in superconductor-ferromagnet hybrid structures

Faluke Aikebaier, M.A. Silaev, and Tero T. Heikkilä,

P. Virtanen, T.T. Heikkilä and F.S. Bergeret,

M. Silaev, P. Virtanen, F.S. Bergeret, and T.T. Heikkilä,

F. Giazotto, T.T. Heikkilä, and F.S. Bergeret,

M. Silaev, P. Virtanen, T.T. Heikkilä, and F.S. Bergeret,

A. Ozaeta, P. Virtanen, F.S. Bergeret, and Tero T. Heikkilä,

Superconductors are characterised by extreme properties: lack of electrical resistance, perfect diamagnetism, poor electronic heat conduction, and typically at most tiny thermoelectric response. We have shown that a hybrid structure consisting of a ferromagnetic insulator and a superconducting film exhibits giant thermoelectric effects, and a very high-efficiency thermoelectric device could be constructed from such a hybrid structure. Our finding also indirectly explains seemingly unrelated and previously unexplained observations, such as the long-range spin accumulation observed in such structures. Some of our predictions have been confirmed experimentally (group of D. Beckmann in the University of Karlsruhe).

*Supercurrent Induced Charge-Spin Conversion in Spin-Split Superconductors*arXiv:1712.08653P. Virtanen, T.T. Heikkilä and F.S. Bergeret,

*Stimulated quasiparticles in spin-split superconductors,*Phys Rev. B**93**, 014512 (2016)M. Silaev, P. Virtanen, F.S. Bergeret, and T.T. Heikkilä,

*Long-Range Spin Accumulation from Heat Injection in Mesoscopic Superconductors with Zeeman Splitting,*Phys. Rev. Lett.**114**, 167002 (2015)F. Giazotto, T.T. Heikkilä, and F.S. Bergeret,

*Very large**thermophase in ferromagnetic Josephson junctions,*Phys. Rev. Lett.**114**, 067001 (2015)M. Silaev, P. Virtanen, T.T. Heikkilä, and F.S. Bergeret,

*Spin Hanle effect in mesoscopic superconductors,*Phys. Rev. B 91, 024506 (2015)A. Ozaeta, P. Virtanen, F.S. Bergeret, and Tero T. Heikkilä,

*Predicted very large thermoelectric effects in ferromagnet-superconductor junctions in the presence of a spin-splitting field,*Phys. Rev. Lett.**112**, 057001 (2014)Superconductors are characterised by extreme properties: lack of electrical resistance, perfect diamagnetism, poor electronic heat conduction, and typically at most tiny thermoelectric response. We have shown that a hybrid structure consisting of a ferromagnetic insulator and a superconducting film exhibits giant thermoelectric effects, and a very high-efficiency thermoelectric device could be constructed from such a hybrid structure. Our finding also indirectly explains seemingly unrelated and previously unexplained observations, such as the long-range spin accumulation observed in such structures. Some of our predictions have been confirmed experimentally (group of D. Beckmann in the University of Karlsruhe).

#### Approaching single-photon strong coupling regime in optomechanics

J.-M. Pirkkalainen, S.U. Cho, F. Massel, J. Tuorila, T.T. Heikkilä, P.J. Hakonen, and M.A. Sillanpää,

Optomechanics studies hybrid systems of microwave or visible light quantum optical setups together with vibrating wires. The field has been driven by the analysis of gravitational wave detection in such setups. In typical realisations the coupling between the two parts of the system has been very weak so that the electromagnetic system needs to be heavily pumped so that the physics of this non-linear dispersive coupling can be studied. We have theoretically shown and experimentally (in M. Sillanpää's group in Aalto University) demonstrated that utilising a superconducting contact and its Josephson effect, this coupling can be greatly enhanced towards the single-photon strong coupling regime, where a single photon in the electromagnetic cavity would lead to an observable optomechanical frequency shift, for example visible as a photon blockade effect.

*Cavity optomechanics mediated by a quantum two-level system,*Nature Commun. 6, 6981 (2015)**T.T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M.A. Sillanpää,**

*Enhancing optomechanical coupling via the Josephson effect,*Phys. Rev. Lett. 112, 203603 (2014)Optomechanics studies hybrid systems of microwave or visible light quantum optical setups together with vibrating wires. The field has been driven by the analysis of gravitational wave detection in such setups. In typical realisations the coupling between the two parts of the system has been very weak so that the electromagnetic system needs to be heavily pumped so that the physics of this non-linear dispersive coupling can be studied. We have theoretically shown and experimentally (in M. Sillanpää's group in Aalto University) demonstrated that utilising a superconducting contact and its Josephson effect, this coupling can be greatly enhanced towards the single-photon strong coupling regime, where a single photon in the electromagnetic cavity would lead to an observable optomechanical frequency shift, for example visible as a photon blockade effect.