Unitary Scattering Protected by Pseudo-Hermiticity

L. Jin

doi:10.1088/0256-307X/39/3/037302

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Express Letter

Unitary Scattering Protected by Pseudo-Hermiticity

L. Jin^*

School of Physics, Nankai University, Tianjin 300071, China

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Received Date: December 06, 2021

Published Date: February 28, 2022

Abstract

Abstract

Hermitian systems possess unitary scattering. However, the Hermiticity is unnecessary for a unitary scattering although the scattering under the influence of non-Hermiticity is mostly non-unitary. Here we prove that the unitary scattering is protected by certain type of pseudo-Hermiticity and unaffected by the degree of non-Hermiticity. The energy conservation is violated in the scattering process and recovers after scattering. The subsystem of the pseudo-Hermitian scattering center including only the connection sites is Hermitian. These findings provide fundamental insights on the unitary scattering, pseudo-Hermiticity, and energy conservation, and are promising for light propagation, mesoscopic electron transport, and quantum interference in non-Hermitian systems.

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Recent years, two-dimensional (2D) magnetism in layered van der Waals (vdW) materials has attracted considerable attention since potential applications in 2D spintronics.^[1–7] Directly measuring magnetization is crucial to revealing the magnetic structures underlying nontrivial behaviors,^[8–10] such as abnormal magnetizing,^[11] magnetic phase transitions,^[12–14] and hidden orders.^[15,16] However, conventional magnetic measurement techniques encounter difficulties when probing the magnetic properties of nano-scale vdW magnets. For example, vibrating sample magnetometry has a sensitivity of 10⁻⁹ A⋅m², which is insufficient for such studies.

Dynamic cantilever magnetometry (DCM) has emerged as an ultra-sensitive technique,^[17–19] the method enables the detection of magnetic moment signals at the magnitude of 10⁻¹⁷ A⋅m².^[20] When a nanometer-sized sample is attached onto the free end of a cantilever, the effective spring constant of cantilever alters with external magnetic field due to magnetic restoring force exerting on the sample. This additional magnetic spring constant Δk is associated with the magnetization of the sample.^[21] Based on this relationship, the magnetic properties of sample can be derived from eigen frequency shifts Δf of the cantilever sensor. So far, DCM has been employed in measurements of individual nanometer-scaled particles such as magnetic nanotube and magnetic mesocrystals,^[18,22–26] which can be mounted onto cantilever by using micro-manipulator. However, due to the lack of appropriate transfer technique, the thickness of vdW materials studied by DCM has been limited to micrometer scale.^[27]

In this work, we develop a two-step fabricating method for transferring exfoliated atomically thin flakes onto cantilever sensor, enabling us to extend DCM measurements to 2D magnets. Employing this technique, we investigate the magnetic properties in thin CrPS₄ flake with a thickness down to 8 nm by measuring the Δk. In the case of H ∥ c, the spin flop transition leads to a dip in Δk, followed by a gradual increase and then reaches next region with a change in slope. For the case of H ⊥ c, the low-temperature results indicate a canted antiferromagnetic state. As T increases, a bump followed by a hump in Δk is observed until Δk is completely negative above 30 K, which can be attributed to the presence of spin reorientation. The H–T phase diagrams of 2D CrPS₄ revealed by DCM experiment are consistent with the previous studies on bulk samples,^[28] which demonstrates the reliability of DCM in investigating magnetic properties of 2D magnets.

CrPS₄, a ternary transition metal chalcogenide (TTMC), has monoclinic symmetry with space group C2/m.^[29,30] Its closely packed surface is constituted by S atoms in monolayers as presented in Fig. 1(a). Magnetically, CrPS₄ exhibits intralayer ferromagnetism and interlayer antiferro-coupling,^[31,32] which contribute to a complex metamagnetic phase transition. The weak spin interactions in CrPS₄ leads to little dependence of transition temperature on thickness.^[32] Thus, CrPS₄ is an appropriate candidate for demonstrating the capability of DCM in examining the magnetism in 2D magnets. The zero-field-cooled (ZFC) magnetization characterizations of bulk CrPS₄ were measured with external field μ₀H = 0.1 T applied both parallel to the c direction [Fig. 1(b)] and in the ab plane [Fig. 1(c)]. The Néel temperature of our sample is around 38 K, which is consistent with previous studies.^[29]

Fig. 1. Characterization of CrPS₄ and fabrication of the DCM device. (a) The crystal structure of CrPS₄. [(b), (c)] The c-direction and ab-plane magnetic susceptibilities, respectively, measured under a magnetic field of μ₀H = 0.1 T. (d) Schematic representation of the transferring method of the 2D CrPS₄ flakes. Gray needles represent the micro manipulators for transfer. The main steps have been arranged in sequential order using numbered circles. (1) The polycarbonate (PC) film was affixed onto the polydimethylsiloxane (PDMS) and pressed against the flake, followed by heat melting to enhance adhesion. (2) Cool the substrate to solidify the PC film and peel off the flake. (3) The flake was aligned on top of the target silicon lamella and pressed against the lamella. (4) Heat the substrate and peel off the PDMS. (5) Soak the substrate in chloroform and dissolve the PC film. (6) Dry off the substrate and transfer lamella to the cantilever sensor. (e) Schematic drawing of the DCM devices S1. The inset shows the photograph of the silicon lamella with sample attached on (scale bar: 5 μm). (f) Atomic force microscopy image of the flake on S1 with a thickness of 8 nm. The thickness of the flake is shown in (g).

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We designed and fabricated sensitive cantilever beam for DCM measurements. Firstly, a thin film of SiO₂ with thickness of 1 μm was grown on top Si layer of a silicon-on-insulator (SOI) substrate by plasma-enhanced chemical vapor deposition (PECVD) to balance stress beneath. Patterns of cantilevers were micro-fabricated by ultra-violet lithography on top surface of SOI. An inductively coupled plasma (ICP) deep reactive ion etch (DRIE) was adopted to etch SOI to buried oxide (BOX) layer. On bottom surface of SOI substrate, the patterned 500 μm silicon was removed by DRIE (Bosch Process) on a deep silicon etcher system. Finally, sacrificial SiO₂ layer was released by dry release etch using vapor HF (hydrogen fluoride) chemistries (sacrificial vapor release etching).

Conventionally, a focused ion beam (FIB) is used to structure micro/nano-sized samples and transfer them onto the cantilever,^[18] especially for thin films.^[19] For 2D flakes, FIB fabricating might induce damage in samples. In our study, a two-step transfer method is schematically illustrated in Fig. 1(d), whose process comprises two essential steps. Initially, the exfoliated 2D flake was mechanically peeled off and transferred onto a silicon lamella. Subsequently, by using the micro-manipulators, the 2D flake together with the lamella were mounted onto the free end of a cantilever. This technique provides two advantages in DCM experiment: (1) a major reduction in thickness, (2) a major boost in surface area for thin film to enhance the magnetic signal. The lamella attached to the extremity of cantilever can play a role as a mass-loaded end to enhance the sensitivity. Moreover, the use of lamella facilitates the mounting process. By utilizing this technique, we have successfully fabricated CrPS₄ sample S1 with a thickness down to 8 nm and an area of 30 μm². The silicon lamella was cut to a size of 12 μm × 8 μm × 1 μm by using FIB. Its magnetic signal can be negligible in DCM experiments. The single-crystal Si cantilever used in our study is 200 μm long, 10 μm wide and 200 nm thick. The morphological characterization depicted in Fig. 1(e) demonstrates that our method is a clean transfer technique.

The DCM insert was sealed in a vacuum sleeve, maintaining a pressure below 10⁻⁶ mbar. By using a rotatable sample stage, a magnetic field up to 9 T can be applied parallel or perpendicular to the cantilever. The cantilever used in our DCM has a low spring constant of 0.14 mN/m. The eigen frequency f₀ of 2.7 kHz can be measured by using a home-made laser interferometer. The frequency responses were precisely monitored by employing the ring-down method. When the signal is strong, due to the mechanical properties of the cantilever beam, the antisymmetric effect in magnetic fields is obvious. In order to exclude the antisymmetric effect in magnetic fields, the Δf has been symmetrized in positive and negative magnetic fields.

Figure 2 displays the DCM experiment results in S1 for H ∥ c. The measured Δf as a function of H at low temperatures could be classified into three regimes. In the low-field range below a critical field of around 0.8 T, Δf rapidly decreases to a minimum. In the intermediate regime, Δf gradually increases and then enters next region at around 7.6 T. The similar behavior has typically been observed in magnetic torque signals of bulk CrPS₄,^[29] however, its quantitative interpretation of magnetism is still missing.

Fig. 2. The experimental and calculated results for the magnetic field dependence of Δf in the case of H ∥ c. (a) The field dependence of Δf in S1 with H ∥ c from 5 K to 40 K. Inset schematically illustrates the experimental setup. The cantilever is securely fixed along the y-axis and vibrates within the y–z plane. The laser is directed along the z-axis. (b) The calculated Δf for an antiferromagnetic thin film when H is applied along the easy axis. The red line illustrates the numerical results. The left inset exhibits the schematic view of the relative orientations of the cantilever axis, applied magnetic field H, and the magnetic moments in sample. Other insets depict the deduced spin alignments of CrPS₄ as black and green arrows.

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In order to analyze the magnetic phase transitions at low temperatures, we treated the sample as a uniaxial particle by following the Stoner–Wohlfarth-type model presented by Weber et al.^[18] For CrPS₄ samples, as reported previously, there are two inequivalent magnetic structures that induce complicated anisotropies.^[30] Nevertheless, at low temperature, the out-of-plane anisotropy dominates. Thereby we considered a projected anisotropy constant K (K > 0) to simplify the analysis. In this model, two sets of macrospins (m_a and m_b) with interactions have been presented. The spin alignments have been depicted as green and black arrows in Fig. 2(b). The ϕ_a and ϕ_b denote the deviations of m_a and m_b from the easy axis. The two sets of macrospins remain in equilibrium between the inherent anisotropy, magnetic torque induced by H and the antiferromagnetic interactions. An antiferromagnetic interaction constant A has been introduced. Am_a can be regarded as an effective magnetic field applied to m_b. Thus, the free energy of the system can be written as the sum of the cantilever energy, the Zeeman energy, an effective anisotropy energy and an antiferromagnetic interaction energy

$\begin{array}{l} E = & \frac{1}{2} k_{0} l_{e}^{2} θ^{2} - (M_{a} V H \cos (ϕ_{a} - θ) \\ + M_{b} V H \cos (ϕ_{b} - θ)) \\ + \frac{1}{2} K V (\sin^{2} ϕ_{a} + \sin^{2} ϕ_{b}) \\ + A M_{a} M_{b} V^{2} \cos (ϕ_{a} - ϕ_{b}), \end{array}$

(1)

where k₀ and l_e are the inherent spring constant and effective length of the cantilever, respectively. V is the volume of the CrPS₄ sample. The magnetizations of two macrospins are M_a and M_b with M_a = M_b = M₀. As shown in inset of Fig. 2(b), θ depicts the vibration deviation. Considering the system in an equilibrium state, the energy of the system must satisfy $\frac{\partial E}{\partial ϕ_{a}} = \frac{\partial E}{\partial ϕ_{b}} = 0$ , giving

$\begin{array}{l} \frac{\partial E}{\partial ϕ_{a}} = M_{a} V H \sin (ϕ_{a} - θ) + \frac{1}{2} K V \sin 2 ϕ_{a} \\ - A M_{a} M_{b} V^{2} \sin (ϕ_{a} - ϕ_{b}) = 0, \\ \frac{\partial E}{\partial ϕ_{b}} = M_{b} V H \sin (ϕ_{b} - θ) + \frac{1}{2} K V \sin 2 ϕ_{b} \\ + A M_{a} M_{b} V^{2} \sin (ϕ_{a} - ϕ_{b}) = 0. \end{array}$

(2)

The solutions of the equation reveal the possible spin alignments in samples. One solution for θ = 0 is ϕ_a₀ = −ϕ_b₀ = 0, which indicates an A-type antiferromagnetic (A-AFM) spin arrangement. When H reaches a critical value H_sf, which satisfies $M_{0} H_{sf} + (K - A V M_{0}^{2}) (\frac{2 M_{0} H_{sf}}{- K + 2 A V M_{0}^{2}} - \frac{- K + 2 A V M_{0}^{2}}{M_{0} H_{sf}^{2}}) = 0$ , the macrospins prefer to stay in plane with lower total energy and the H_sf was denoted as spin-flop field. Correspondingly, the macrospins align as $ϕ_{a_{0}} = - ϕ_{b_{0}} = \arccos (\frac{M_{0} H}{- K + 2 A M_{0}^{2} V})$ , revealing the canted spin alignments. With increasing H, the macrospins gradually rotate to the external field and orient towards the same direction at $H_{s} = \frac{- K + 2 A M_{0}^{2} V}{M_{0}}$ . The connection between the spin alignment and the mechanical response of the cantilever sensor was established by the formula $Δ k = \frac{1}{l_{e}^{2}} \frac{\partial^{2} E}{\partial θ^{2}} - k_{0}$ (see chapter S4 in the Supporting Information for more details). According to $Δ f = f_{0} \frac{Δ k}{2 k_{0}}$ , we can obtain the calculated Δf as a function of H,

$\begin{array}{l} Δ f = & \frac{f_{0} V}{2 k_{0} l_{e}^{2}} (\frac{H K M_{0}}{K + H M_{0} + A M_{0}^{2} V} \\ - \frac{H K M_{0}}{K - H M_{0} + A M_{0}^{2} V}), for | H | < H_{sf}, \\ Δ f = & \frac{f_{0} V}{2 k_{0} l_{e}^{2}} \cdot 2 M_{0} H K (\frac{2 H^{2} M_{0}^{2}}{{(- K + 2 A V M_{0}^{2})}^{2}} - 1) \\ [M_{0} H + (K - A V M_{0}^{2}) (\frac{2 M_{0} H}{- K + 2 A V M_{0}^{2}} \\ - \frac{- K + 2 A V M_{0}^{2}}{M_{0} H^{2}})]^{- 1}, for H_{sf} < | H | < H_{s}, \\ Δ f = & \frac{f_{0} V}{2 k_{0} l_{e}^{2}} \frac{2 M_{0} | H | K}{M_{0} | H | + K - A V M_{0}^{2}}, for | H | > H_{s} . \end{array}$

(3)

The numerical results have been shown in Fig. 2(b), which are consistent with the experiment data. The singularities at H = ±H_sf reflects breakdown in Δf–H curve. The corresponding ϕ_a₀ and ϕ_b₀ indicate that the macrospins, initially aligning as an A-AFM spin alignment, flop down into the ab plane. By combining the calculated Δf and the experiment results, the dips in measured Δf shown in Fig. 2(a) can be attributed to the occurrence of spin flop. Theoretically, the flopping behavior was a sudden action, which reflects a sharp jump in calculated results. Experimentally, the Néel vector deviated from the c axis in CrPS₄ for H ∥ c^[30] and the measured results show the Δf gradually decreases below H_sf. It should be noticed that the amplitude of the dip is dependent on the relative orientation of the crystal axes in CrPS₄ to the direction of cantilever $\hat{r}$ . It becomes more pronounced when a axis is mounted closer to $\hat{r}$ (as detailed in the Supporting Information). Nonetheless, the H_sf derived from the different sample is consistent. Above H_sf, the calculated curve agrees well with the measurement results in S1 at 5 K. The best fits give parameters of M₀ = 1.65 kA/m, AV = 0.0047 and K = 12.9 kJ/m³. As shown in Fig. 2(b), the calculated Δf rapidly increases with increasing H. A gradual decrease in its slope indicates that the canted AFM state undergoes a transition from initially metastable state to a steady state. Correspondingly, the spins rotate towards the external field, resulting in an increase in the total magnetic moment. Δf shows a kink at H = H_s, then eventually reaches saturation. For H > H_s, the calculated Δf shows a positive high-field asymptote behavior, where all the spins align in one direction as a field-polarized paramagnetic (FPP) type spin alignment and total magnetization satisfies M_s = 2M₀.

Figure 3 summarized the H–T diagram for H ∥ c in CrPS₄ flake with different thickness. The diagram has been delineated into three regions. Below H_sf, spins align as an A-AFM ground state. The decrease in H_sf was observed as the temperature increases, indicating the reduction in anisotropy. The region of H_sf < H < H_s, corresponding to the canted spin alignments, was denoted as canted AFM. For H > H_s, the FPP region indicates a field-polarized paramagnetic-type spin alignments.

Fig. 3. Magnetic phase diagram of CrPS₄ determined from the DCM measurements with H applied along c axis. The spin flop boundaries are represented as triangles (circles for the saturation states) and four patterns denote four different samples.

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To further investigate the temperature dependence of anisotropy in 2D CrPS₄, the DCM experiments have been conducted in S1 for H ⊥ c. Figure 4(a) shows field dependence of Δf for H ⊥ c in a temperature range from 5 K to 40 K. We firstly focus on the data at temperatures below 20 K, where the Δf at low field is positive. With further increasing H, the Δf goes through a maximum, and reaches saturation with a change in slope for large H. The uniaxial model has been extended for the case of H ⊥ c and K > 0 (see the Supporting Information for more details). The numerical results, indicating two macrospins gradually rotate to the external field with $ϕ_{a_{0}} = π - ϕ_{b_{0}} = \arcsin (\frac{M_{0} H}{K + 2 A M_{0}^{2} V})$ from the ac plane, agree well with our experiment data.

Fig. 4. Magnetic field dependence of Δf in the case of H ⊥ c. (a) The magnetic field dependence of Δf in S1 with H ⊥ c from 5 K to 40 K. (b) The zoom-in Δf–H curve at selected temperatures. The data has been normalized and shifted for clarify. The positive M shape and the negative W shape have been highlighted. The frequency shift at zero field is always zero. The dashed lines indicate the value Δf = 0 at different temperatures. Black triangles mark the position of the valley.

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Now we turn to the evolution of metamagnetic phases above 20 K. In Fig. 4(b), the curves have been normalized and shifted for clarity. As temperature increases, the shape of the Δf–H curve gradually deviates from the M shape. Remarkably, a hump emerges right after the bump in the Δf at around H = 1.3 T when T reaches 25 K. This change is induced by a relatively larger in-plane magnetization. In this case, the spins aligns close to ab plane. The low-field Δf–H curve becomes negative above 30 K and maintains a W shape at low field until 35 K. The negative signal indicates the dominance of in-plane anisotropy and the reorientation of the easy axis. The evolution of the curve shape shows that the spin reorientation is a continuous transition. Above 38 K, the Δf shows a quadratic field dependence. The upward parabola indicates that CrPS₄ enters a paramagnetic state where the in-plane shape anisotropy dominates.

The H–T phase diagram for H ⊥ c has been summarized from different CrPS₄ samples. As shown in Fig. 5, the phase diagram is roughly delineated as four regions. The FPP region represents the saturated magnetic phase with two macrospins aligning one direction. PM denots the paramagnetic phase of CrPS₄. The FPP-PM phase boundary is more distinct than the discussion before. AFM1 and AFM2 represent different cases for canted spin alignments, respectively. In AFM2, the spins align closer to the a axis. With increasing magnetic field, the spins prefer to align closer to the b axis (AFM1). The boundary between them is determined by the reversal field at the valley of Δf. Due to the different relative directions of crystal axes to the cantilever, the reversal fields vary for different samples. Nonetheless, the evolution of magnetic anisotropy induced by spin reorientation can be directly distinguished by performing DCM experiments.

Fig. 5. Magnetic phase diagram of CrPS₄ with H applied perpendicular to c axis. The saturation fields (circles) and the reversal fields (triangles) are extracted from the Δf–H curves in Figs. 4(a) and 4(b), Figs. S2 and Fig. S4. The insets depict the corresponding spin alignments in different phases.

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The magnetic phase diagrams for H ∥ c and H ⊥ c, constructed by DCM measurements in 2D CrPS₄, are basically consistent with that reported in bulk samples. Note that the alteration of spin orientation identified by commercial magnetometry is subtle and indirect. Specifically, the canted spin alignments and the evolution of spin reorientation in CrPS₄ can be only revealed by neutron scattering measurements, which is, however, unavailable for individual 2D particles. As discussed above, the effective magnetic spring constant Δk is directly related to anisotropic magnetic susceptibility. The inherent dynamic property of DCM, on the other hand, makes it sensitive to quasi-static states. Moreover, the smallest detectable magnetotropic coefficient in our study can be equivalent to a magnetic moment 4.5 × 10⁶μ_B at 1 T (see Section S10 in the Supporting Information), which enables effective characterization of ultra weak magnetism even for single-layer samples. In this sense, DCM technique has the potential to investigate the exotic properties in 2D system, including skyrmions,^[33–35] hidden orders,^[36,37] and quantum critical phenomena.^[38]

In summary, we have extended the high-precision DCM measurements to 2D antiferromagnets through a clean transfer technique. The effective magnetic spring constant Δk has been measured in atomically thin CrPS₄ flake. Based on the field dependence of magnetic spring constant, the connection between the spin alignments and the Δk of cantilever sensor is established to reveal the metamagnetic phase transitions for both H ∥ c and H ⊥ c. The temperature evolution of the Δf–H curve has been analyzed, from which we reveal the occurrence of spin reorientation. Our work pioneers magnetism measurement of a single atomically thin sample and opens a route to effectively explore novel phenomenon in 2D materials.

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