Local quantum Fisher information and quantum correlation in the mixed-spin Heisenberg XXZ chain

Abstract

We study the local quantum Fisher information (LQFI) in the mixed-spin Heisenberg XXZ chain. Both the maximal and minimal LQFI are studied and the former is essential to determine the accuracy of the quantum parameter estimation, the latter can be well used to characterize the discord-type quantum correlations. We investigate the effects of the temperature and the anisotropy parameter on the maximal LQFI and thus on the accuracy of the parameter estimation. Then we make use of the minimal LQFI to study the discord-type correlations of different site pairs. Different dimensions of the subsystems cause different values of the minimal LQFI which reflects the asymmetry of the discord-type correlation. In addition, the site pairs at different positions of the spin chains have different minimal LQFI, which reveals the influence of the surrounding spins on the bipartite quantum correlation. Our results show that the LQFI obtained through a simple calculation process provides a convenient way to investigate the discord-type correlation in high-dimensional systems.

Graphical abstract

Keywords

local quantum Fisher information / quantum correlation / mixed-spin Heisenberg XXZ chain

1 Introduction

With the trend of miniaturization and integration of modern electronic devices, how to manage heat effectively becomes a critical issue to improve the performance of electronic devices. Thermoelectric (TE) materials, which can convert heat energy into electricity to reuse waste heat, become an ideal material for thermal management and have been studied extensively [1–3]. The performance of TE materials is determined by the figure of merit (ZT), which is defined as $ZT=S^2\sigma T/\kappa$, where $S$, $\sigma$ and $\kappa$ represents the Seebeck coefficient, electrical conductivity and thermal conductivity, respectively. Obviously, the properties of a high Seebeck coefficient, high electrical conductivity and low thermal conductivity aid to a good ZT value. Due to its tunable bandgaps, special quantum confinement effect, superior electrical conductivity and carrier mobility as well as thickness-dependent thermal conductivity, two-dimensional (2D) materials provide promising candidates to search for excellent TE properties [4–7]. Among the various emerging 2D materials, transitional metal dichalcogenides, TMDCs (MX₂, M = Mo, Pd, W, Ti; X = S, Se) is an important element of van der Waals layered materials [8,9]. PdSe₂ has an unusual quadruple coordination of metal atoms and the two sulfur atoms in the plane are connected by a covalent bond. In a single layer structure, each Pd atom binds to four Se atoms in the same layer and two Pd atoms and three S atoms can form a wrinkled pentagon, which is rather rare in the current materials. As one of the most air-stable TMDCs, PdSe₂ shows a high charge carrier mobility and tunable thermal property by external fields, making extensive applications in field-effect transistors, optoelectronic devices [8,10–13], thermal switches and TE devices [14,15]. Moreover, the bandgap of PdSe₂ can be changed by engineering its thickness and from bulk to monolayer, its bandgap changes from 0 eV to 1.3 eV [13]. Although PdSe₂ has shown these interesting physical properties, its phonon thermal transport was studied until recently.

Due to its strong interlayer coupling, the thermal conductivity of PdSe₂ shows a thickness-dependent behavior [16–19]. The thermal conductivity of bilayer of PdSe₂ grown by CVD is 36.8 W·m⁻¹·K⁻¹ at room temperature measured by Raman characterization, while the thermal conductivity value of few-layer PdSe₂ is only 10.1 W·m⁻¹·K⁻¹ [18]. Zhang et al. [17] used first-principles calculation to study the acoustic velocity, bandgap and density of states of PdSe₂ and demonstrated that the optical modes with low frequencies are increasingly involved in the scattering of acoustic phonons as the number of layers decreases, leading to the decrease of thermal conductivity. Qin et al. [19] discovered that monolayer PdSe₂ has a low lattice thermal conductivity around 3 W·m⁻¹·K⁻¹ along the x direction at room temperature by the first-principles calculation. Wang et al. [20] proposed that PdSe₂ can achieve a continuously switchable thermal conductivity through strain-driven structural phase transition. Moreover, due to the low symmetrical pentagonal fold structure, PdSe₂ has an anisotropy in lattice structure and thus an anisotropic transport properties [21], and the thermal transport anisotropy ratio as high as 1.42 is demonstrated by micro-Raman thermometry [16].

In semiconductor materials, phonon is the main carriers of heat and its behaviors have a great effect on the thermal conductivity [22–25]. Defects, like vacancies, impurities and dislocations are inevitably introduced in the sample preparation process, and the defect engineering could be further used to modulate the phonon transport. Many studies have used “defect engineering” to regulate the thermal conductivity of 2D materials and investigated the influence of defects on heat transport [26–28]. By controlling the oxygen plasma exposure time, Aiyiti et al. [29] tuned the thermal conductivity of MoS₂ to a desired value and observed the transition from crystal to amorphous. Zhao et al. [30] studied the change of thermal conductivity of MoS₂ flakes with different defect concentrations under helium ion (He⁺) irradiation, and it showed that the Mo vacancy could significantly impede phonon transport compared to that of S vacancy. Wu et al. [27] used extensive molecular dynamics simulations to calculate the effects of different types and doses of defects on thermal conductivity of BN and the relationship between phonon-phonon scattering term and phonon-defect scattering term was proposed. For PdSe₂, there exist a wide range of binary phases like Pd₁₇Se₁₅, Pd₇Se₄ and Pd₄Se, and by laser irradiation, electron beam irradiation, electric field and so on, PdSe₂ is expected to undergo phase transition [14,15]. Among these phases, it has been shown that Pd₁₇Se₁₅ is metallic phase and has superconducting behavior, which could be obtained by defect engineering and therefore provides an appropriate approach for engineering phonon transport in PdSe₂ [13,31].

In this work, we study the phonon thermal transport in both intrinsic and defected PdSe₂. By wet transfer method, few-layer PdSe₂ flake was placed onto a micro-electro-thermal systems (METS) device and the thermal bridge method was employed to measure the thermal conductivity of PdSe₂. The Ar⁺ was further employed to introduce defects into PdSe₂. With increasing the Ar⁺ dose, the thermal conductance of PdSe₂ kept decreasing until to a critical dose, where the thermal conductance dropped dramatically by 50%. During the Ar⁺ irradiation, Raman characterization was carried out as well to record the lattice structure evolution of defected PdSe₂, where a possible phase change at the critical dose was observed, corresponding to the change observed in the thermal conductivity. Our work provides a novel approach to tuning the thermal transport in novel 2D materials, enabling further thermal management at nanoscale.

2 Methods

2.1 Preparation of suspended-pad micro-devices

Bulk PdSe₂ single crystals were synthesized by a self-flux method as previous reported [32]. Unlike graphene and boron nitride, whose layers are connected by weak van der Waals forces [33,34], it is challenging to acquire few-layer PdSe₂ by mechanical exfoliation due to its strong layer to layer coupling [35]. By using Ar⁺ or O²⁺ to bombard clean SiO₂/Si substrates, the interaction force between PdSe₂ flakes and substrates was enhanced, and the flake with desired thickness was further obtained. Before transferring onto the pre-patterned suspended thermal bridge device, the atomic force microscope (AFM) is employed to measure the thickness of samples. The thickness of measured PdSe₂ sample is 9.6 nm, as shown in Fig.1(a) and (b).

Fig.1 (a) Optical image and (b) AFM image of PdSe2. The thickness of the sample is 9.6 nm. (c) Optical image of transferred METS sample. Scale bars have been shown in each figure.

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For the thermal bridge method measurement, there are typically three methods to transfer samples to the devices, including drop-cast method [36], dry/wet transfer method [37] and nano-manipulation method [38]. Considering the operative difficulty and yield, the wet transfer method was used in this study. The substrates with targeted samples were spin-coated with PMMA at the rate of 2000 raps/s with duration of 30 s. Then, the substrates were heated on the hot plates at 100 °C to accelerate the solidification process. The substrates coated with PMMA were then immersed into 20% KOH solution. Under an optical microscope, the marked sample on the suspended PMMA film region was placed on top of the middle of the thermal bridge. Before using acetone to dissolve PMMA, the devices were heated to 120 °C for twenty minutes to well stick the flake with the thermal bridge electrodes. Finally, the critical point dryer (CPD) was used to remove the acetone and the final METS device is shown in Fig.1(c).

2.2 Thermal bridge measurement

In 2001, the thermal bridge method was firstly used to measure the thermal conductivity of micro- and nano-scale samples [39]. This method can measure not only the thermal conductivity, but also the electric conductivity of nanosheets and nanowires [40–42]. A METS device has two suspended SiN_x membranes with integrated Pt heaters/resistance-thermometers and extra electrodes. One pad is treated as heater and the other sensor. Each pad is connected with six suspended cantilevers to avoid the influence of substrate. During the measurement, a direct current (DC) produced by Keithely 6221 current source meter flows into heater pad to act as a heat source. One part of the heat ($Q_1$) flows through the suspended cantilevers to the substrate, and the other part ($Q_2$) flows through the sample to the sensor. $Q_1$ and $Q_2$ can be expressed as

(1)$Q_1=G_{\mathrm{b}}\times {\mathrm{\Delta }T}_{\mathrm{h}},$

(2)$Q_2=G_{\mathrm{s}}\times \left(\mathrm{\Delta }{T}_{\mathrm{h}}-\mathrm{\Delta }{T}_{\mathrm{s}}\right)=G_{\mathrm{b}}\times \mathrm{\Delta }{T}_{\mathrm{s}},$

where $G_{\mathrm{b}}$ is the thermal conductance of heater membrane. $G_{\mathrm{s}}$ is the thermal conductance of the sample. $\mathrm{\Delta }{T}_{\mathrm{h}}$ and $\mathrm{\Delta }{T}_{\mathrm{s}}$ is temperature rise of heater and sensor. Under a thermal steady state, $G_{\mathrm{b}}$ and $G_{\mathrm{s}}$ are derived as

(3)$G_{\mathrm{b}}={\displaystyle \frac{I^2\times R_{\mathrm{h}}+{\displaystyle \frac{1}{2}}{I}^2\times R_{\mathrm{L}}}{{\mathrm{\Delta }T}_{\mathrm{h}}+\mathrm{\Delta }{T}_{\mathrm{s}}}},$

(4)$G_{\mathrm{s}}={\displaystyle \frac{\left(I^2\times R_{\mathrm{h}}+{\displaystyle \frac{1}{2}}{I}^2\times R_{\mathrm{L}}\right)\times \mathrm{\Delta }{T}_{\mathrm{s}}}{\mathrm{\Delta }{T_{\mathrm{h}}}^2-\mathrm{\Delta }{T_{\mathrm{s}}}^2}},$

where $R_{\mathrm{h}}$ and $R_{\mathrm{L}}$ respectively represent the resistance of heater pad and suspended cantilevers. When applying a direct current (DC), the current source meter Keithely 6221 produces a tiny alternating current (AC) at the same time and the lock-in amplifier SR 830 is employed to measure the change of resistance of heater and sensor. Thus, the temperature change is obtained and the thermal conductance of the sample is calculated. During measurement, the thermal bridge device is placed in a high vacuum thermostat with a pressure less than 10⁻⁵ mbar, which avoids the effect of thermal convection on the measurement. Two Cu covers were employed to reduce the possible thermal radiation inside the thermostat. As shown in Fig.2(a), the resistance of Pt loops and the temperature coefficient of resistance (TCR) of the measured device can be acquired. Combining Eqs. (3) and (4), $G_{\mathrm{s}}$ and $G_{\mathrm{b}}$ can be obtained, and the thermal conductivity of PdSe₂ is further calculated by the following equation:

Fig.2 (a) Measured temperature-dependent resistance of heater and sensor and the calculated TCR. (b) Intrinsic thermal conductivity of PdSe2. The extracted thermal conductivity ranges from (9.4±1.1) W·m−1·K−1 at room temperature.

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(5)$\kappa =G_{\mathrm{s}}{\displaystyle \frac{L}{S}}=G_{\mathrm{s}}{\displaystyle \frac{L}{w\times t}},$

where L, w, and t are the length, width, and thickness of the suspended part of sample, which are 2 µm, 6.5 µm, and 9.6 nm, respectively.

The measurement errors include the temperature drift in the thermostat, the error of sample dimensions, the error in the contact thermal resistance and so on, which can be expressed as

(6)${\displaystyle \frac{\delta \kappa }{\kappa }}=\sqrt{{\left({\displaystyle \frac{\delta G}{G}}\right)}^2+({{\displaystyle \frac{\delta L}{L}})}^2+{\left({\displaystyle \frac{\delta W}{W}}\right)}^2+{\left({\displaystyle \frac{\delta t}{t}}\right)}^2+{\left({\displaystyle \frac{\mathrm{\Delta }{R}_c}{R_c}}\right)}^2},$

where R_c is the thermal resistance between samples and thermal bridge devices. Combining the above factors, the uncertainty of the thermal conductivity is obtained accordingly.

3 Results and discussion

By using the thermal bridge method, the temperature dependent thermal conductance was obtained as shown in Fig.2(b). The thermal conductivity of PdSe₂ at room temperature is 9.4±1.1 W·m⁻¹·K⁻¹, consistent with the theoretical calculation [17]. At low temperature, the phonon Umklapp scattering is weak, the mean free path of phonons (MFP) of PdSe₂ is mainly limited by defects, boundaries, and so on [43,44]. Impurities like PMMA residues during the transfer process could contribute as well. As the temperature rises, more and more phonons are excited and the thermal conductivity of PdSe₂ rises. With further increasing the temperature, the thermal conductivity of PdSe₂ reaches a maximum value, which is (10.8±0.9) W·m⁻¹·K⁻¹ at a temperature of 170 K as shown in Fig.2(b). At even high temperature, the phonon Umklapp scattering is enhanced and MFP of phonons is reduced, resulting in decreasing thermal conductivity [37,45].

In order to tune the thermal conductivity of few-layer PdSe₂, the Plasma etching machine was used to introduce defects into PdSe₂. Here, we chose Ar⁺ with a radio-frequency (RF) power ~ 4 W to well control the concentration of defects. We followed the assumption that per unit time generates the same amount of plasma and the ionized ions uniformly bombard the sample [29]. The measured thermal conductance change with the irradiation time is summarized in Fig.3, which can be divided into the following parts, as discussed below. Here, we focus on the thermal conductance measured at room temperature.

Fig.3 The thermal conductance of PdSe2 versus the Ar+ irradiation time at 300 K and 310 K. The thermal conductance of PdSe2 decreases with an increase in the defect concentration.

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1) t < 4 s: Due to the low Se-Se intralayer bonding energy, PdSe₂ is sensitive to defects [13]. As shown in Fig.3, when the irradiation time was less than 4 s, there was a slight fluctuation in the thermal conductance with increasing the dose. In the meantime, the morphology of the samples did not change under the optical microscope, implying that the thickness of the sample keeps unchanged. The fluctuation of thermal conductance was possibly attributed to both the introduced defects and the removal of impurities. During irradiation, the enhanced phonon-defect scattering resulted in the reduction of thermal conductivity. While some organic impurities may remain on the sample in the transfer process, which can be removed by ionized particle during the bombardment. Thus, it is reasonable that the thermal conductance shows a tiny fluctuation.

2) t > 4 s: When the irradiation time was increased to 6 s, the thermal conductance of the sample had a sharp drop around 50%. With further increasing the irradiation time, more Se vacancies were created and got concentrated. In this process, the defected PdSe₂ would collapse into the other stable phase. According to paper [13], for PdSe₂ under Ar⁺ irradiation, there exists a phase transition from PdSe₂ to Pd₁₇Se₁₅. For the metallic phase Pd₁₇Se₁₅, the thermal conductance would reach a saturated value as the irradiation time increased further.

In order to further understand the effect of defects on phonon transport, the Raman spectrometer is employed to explore the lattice structure changes of PdSe₂ under the introduction of defects. After each thermal measurement under different Ar⁺ irradiation, the Raman characterization was performed. The evolutions of Raman spectrum are summarized in Fig.4. The PdSe₂ with a thickness of 9.6 nm showed two sensitive peaks, ${\mathrm{A}}_{\mathrm{g}}^1$ and ${\mathrm{A}}_{\mathrm{g}}^3$, which were ~145 and ~255 cm⁻¹, respectively, consistent with the previous experiments [46]. ${\mathrm{A}}_{\mathrm{g}}^1$ is connected with the in-plane vibration of covalent bond between Pd and Se atoms, and ${\mathrm{A}}_{\mathrm{g}}^3$ is related to the in-plane vibration of two neighboring covalently bonded Se atoms. As the irradiation time increased on a time scale of 0 to 4 seconds, the intensity of these two Raman peaks gradually decreased. The positions of the two modes did not change, indicating that the crystal structure of the sample kept constant. This change coincided with a slight fluctuation in the thermal conductivity. When the time exceeds 4 s, the modes of ${\mathrm{A}}_{\mathrm{g}}^1$ and ${\mathrm{A}}_{\mathrm{g}}^3$ showed a slight red-shift and with further increasing the irridiation time, these two peaks almost vanished and at the same time, the thermal conductance reaches a saturated value. The disappearance of Raman peaks suggested the possible phase transition from PdSe₂ to Pd₁₇Se₁₅. As shown in the work conducted by Akinola et al. [13], under the Ar⁺ irridiation, semiconductor PdSe₂ could change into stable metallic Pd₁₇Se₁₅, which could be characterized by low-frequency anti-Stokes peaks (below 50 cm⁻¹). Futher study would be carried out to perform this Raman characterizations for the defected PdSe₂.

Fig.4 The evolution of Raman sensitive peaks, ${\mathrm{A}}_{\mathrm{g}}^1$ and ${\mathrm{A}}_{\mathrm{g}}^3$, for few layer PdSe2 flakes under various irradiation doses.

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4 Conclusion

In summary, we experimentally studied the effect of Ar⁺ on phonon transport in few-layer PdSe₂. By using a wet transfer method, the exfoliated few-layer PdSe₂ flakes were transferred onto the pre-patterned suspended thermal bridge devices and the intrinsic thermal conductivity of PdSe₂ was measured by the thermal bridge method. By Ar⁺ irradiation, defects with different concentrations were introduced into the PdSe₂ sample. When the irradiation time is 6s, the thermal conductance of the sample had a sharp drop. Raman spectroscopy was employed to characterize the structure of defected PdSe₂, where a phase transition was observed. Our work provides a convenient approach to engineering the thermal transport in 2D layered materials by defects engineering, which can be applied to a broad class of emerging 2D materials for engineering their thermal properties. By introducing defects, the thermal conductivity of PdSe₂ is modified, shedding light on the applications of PdSe₂ in thermal management, TE energy conversion and so on.

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Declarations

The authors declare that they have no competing interests and there are no conflicts.

Acknowledgements

The work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 12175052 and the Postdoctoral Science Foundation of China (No. 2022M722794).

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