Introduction

Positive thermal expansion (PTE) is a common property of materials, which is caused by anharmonic thermal vibrations between atoms. Differently, negative thermal expansion (NTE) exhibiting as an unusual property has been received wide attention in the last two decades. Various interesting NTE materials are known to date, such as oxides1,2,3,4,5, intermetallic compounds6,7, fluorides8,9, cyanides10,11,12, and nitrides13,14. The discovery of NTE materials opens up the possibility to control the thermal expansion for high precision applications such as microelectronic devices, optical mirrors, micromechanics, and so on.

In the study of NTE materials, various kinds of mechanisms have been found out. In general, the NTE mechanisms can be divided into two categories: one is electron-driven type, such as valence transition15,16,17,18, magnetovolume effect (MVE)6,7,13,19,20 and spontaneous volume ferroelectrostriction (SVFS)21; the second is phonon-mediated type, such as rigid unit modes (RUMs)22,23 and guitar-string effect9,24. On the basis of these mechanisms, the methods of chemical substitution2,25, nanocrystallization26,27, guest ions or molecular intercalation12,28, and other methods have been tried to regulate NTE. Actually, most of these methods weaken the NTE properties, scilicet the magnitude and the temperature range of the NTE. As a result, it is not easy to enhance NTE properties. Considering the fact that there is no inherent contradiction among the above NTE mechanisms, the combination of two of these mechanisms could help to improve NTE in magnitude and temperature range through a dual effect.

As a ternary rare-earth RT2X2 compounds (where R is a rare earth metal, T is a 3d or 4d transition metal, and X is silicon or germanium), it is interesting to observe that the a-axis of YbMn2Ge2 exhibits the NTE phenomenon29,30, which is rare in the RT2X2 compounds. In such temperature region, the magnetic structure undergoes a transition from antiferromagnetism to paramagnetism31, and the valence state of Yb shows a temperature-dependent behavior29. As we know, the magnetic transition may bring about the magnetovolume effect6,7,13,19,20, and the valence transition can also give a volume change15,16,17,18. Therefore, YbMn2Ge2 is a good candidate for studying the dual effect on thermal expansion. In the present study, the NTE behavior of YbMn2Ge2 has been systematically investigated by means of synchrotron X-ray diffraction (SXRD). The combined analysis of neutron powder diffraction (NPD) and X-ray absorption near edge structure (XANES) reveals the clear relationship between the dual effect and the NTE in YbMn2Ge2. The dual effect, due to the spontaneous volume magnetostriction induced by Mn moment and to the valence transition by Yb atoms, gives rise to enhanced NTE, both in magnitude and temperature range. This work enables us to have a deeper understanding of the NTE mechanism in the specific case of YbMn2Ge2 and, above all, provides an idea to obtain interesting NTE materials by exploiting the dual effect from the well-known NTE mechanisms.

Results and discussion

Enhanced negative thermal expansion

Temperature dependent SXRD measurements of YbMn2Ge2 were performed to obtain accurate crystal structure information and lattice parameters (Supplementary Fig. 1). The Rietveld method was adopted to refine the crystal structure, which revealed a good agreement with the I4/mmm space group (Supplementary Fig. 2a). As shown in Fig. 1a, the Yb, Mn, and Ge atoms occupy the Wyckoff sites 2a (0, 0, 0), 4d (0, 0.5, 0.25), and 4e (0, 0, z), respectively. They stack with the sequence of Yb-Ge-Mn-Ge-Yb along the c-axis. The temperature dependence of the lattice parameters is plotted in Supplementary Fig. 2b. It can be observed that a-axis shows a strong NTE while c-axis shows a normal positive thermal expansion, which is consistent with the previous literature studies30. Combining the lattice parameters of a and c, the unit cell volume can be calculated (Fig. 1b). Interestingly, a NTE is observed in the unit cell volume over a relatively wide temperature (αv = −32.9 × 10−6 K−1, 400–575 K, ΔT = 175 K).

Fig. 1: Crystal structure and thermal expansion characteristics.
figure 1

a Crystal structure of YbMn2Ge2, and b temperature dependence of the unit cell volume.

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Magnetic structure

Note that such NTE over a relative wide temperature range is rare in ternary rare-earth RT2X2 compounds32,33. Accordingly, it is of great scientific significance to study the NTE mechanism in the present YbMn2Ge2. It is well known that MVE is the key factor for the NTE behavior happening in the magnetic NTE compounds such as the Invar alloys, the prototype magnetic NTE materials discovered by C. É Guillaume34. Herein, the detailed magnetic information has been extracted from the NPD measurements collected in the temperature range 35–615 K (Supplementary Fig. 3). Considering the influence of magnetism on the diffraction patterns, the pattern at 600 K was used as the refinement for the non-magnetic state. As mentioned in the experimental part, an aluminum foil was wrapped outside the vanadium cans for a better temperature management. Therefore, two phases of YbMn2Ge2 and Al were used to refine the pattern at 600 K. The calculated pattern is in a good agreement with the predominant reflections. A few fractions of impurities can be discerned (Supplementary Fig. 4), as often happens in the preparation of Yb-containing compounds35,36.

As observed from the temperature-dependent NPD patterns, some diffraction peaks change significantly with temperature (Supplementary Fig. 5). Based on comprehensive research on the magnetic structure of the RT2X2 series compounds, it is possible to evaluate the magnetic structure from the temperature dependence of some key indicators in the NPD patterns, such as the (101) and (103) reflections37,38,39. It can be seen that the integrated intensities of the two reflections have the same trend (Fig. 2a), which decreases with increasing temperature, reaching the minimum value at 525 K. Such threshold temperature is well consistent with the in-plane antiferromagnetic structure (AFl) transition point TN (~510 K)31,40,41. Furthermore, the (111) reflection is observed below 300 K, which means that the Mn sublattice has an interlayer antiferromagnetic component superimposed on the in-plane antiferromagnetism, prompting the formation of another antiferromagnetic structure called AFmc41. The above two magnetic structures were respectively applied to the refinement of NPD data in different temperature regions (Supplementary Fig. 6). It can be seen that the calculated results and the experimental data are well matched regardless of peak position or intensity.

Fig. 2: Evolution of magnetic structure.
figure 2

Temperature dependence of a the integrated intensity of (101), (103), and (111) reflections, and b the magnetic moment and the canting angle, θ, of Mn atoms. The insets show the two antiferromagnetic structures of AFmc and AFl.

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The schematic diagrams of two magnetic structures (AFmc and AFl) are shown in Fig. 2b. The angle (θ) between the direction of the Mn magnetic moment and the c-axis increases from 62° at 35 K to 90° at 300 K, which represents the transition of magnetic structure from AFmc to AFl. From the dependence on temperature of Mn magnetic moment (Fig. 2b), it can be observed that the magnetic moment slightly decreases with increasing temperature at the beginning, with a small effect on the unit cell volume. With further increasing temperature, the magnetic moment decreases more rapidly when approaching the magnetic transition temperature. Such dramatic change of the magnetic moment brings a negative effect on the thermal expansion, namely MVE, which can be quantitatively described by the spontaneous volume magnetostriction. However, from the comparison between Figs. 1b and 2b, it is interesting to observe that there is a large discrepancy between the temperatures where NTE and antiferromagnetic state disappear. NTE still persists in the paramagnetic state. Furthermore, the coefficient of thermal expansion between 575 K and 700 K is obviously lower than that above 700 K (Fig. 1b), which means that something else has to contribute to the NTE besides the MVE.

Temperature dependence of valence state

It is known that Yb can show two electronic configurations, Yb2+ (4f145d06s2) and Yb3+ (4f135d16s2)42. Because the energy difference between these two valence states is very small, external perturbations can easily cause changes between the two valence states43. As a result, the lattice parameters change since the atomic volume of Yb2+ (radius 1.16 Å) is larger than that of the Yb3+ (radius 1.01 Å)44. Therefore, the change of Yb valence state with temperature can be the phenomenon that further enhances the NTE behavior of YbMn2Ge2. To verify this, a XANES study as a function of temperature has been performed to determine the valence state of Yb in YbMn2Ge2. The collected Yb LIII-edge XANES spectra are shown in Fig. 3a. The first peak on the left at ~8943 eV is related to the presence of divalent Yb, while the second peak at ~8949 eV is related to the trivalent Yb45,46. It can be clearly seen that the peak intensity of divalent Yb is reduced by increasing temperature, while that of trivalent is enhanced. Two sets of functions are used to fit the data, which include a Lorentz and an arctangent function (Supplementary Fig. 7). The valence state can be so estimated by the integrated intensities of divalent and trivalent peaks45. As shown in Fig. 3b, the valence state of Yb is ~2.40 at low temperature, and remains almost unchanged between 200 K and 300 K, which is similar to the results in the literature29. Above 300 K, the valence state gradually increases with increasing temperature. Since there is a volume difference between Yb2+ and Yb3+ as mentioned above, not only the a-axis29,30 but also the unit cell volume shows NTE due to the change in the valence state. At 700 K, the value rises to ~2.82. One needs to note the fact that the valence state of Yb in YbMn2Ge2 can quickly rise to 2.78 under a relatively small compressive pressure of 1.4 GPa, but only a small increase of 0.13 in the valence can be reached when the pressure is further enhanced to 10 GPa47. Therefore, it is reasonable to believe that the valence state of 2.82 at 700 K is close to its maximum limit in the present temperature dependence of valence transition. The Tvalen. at ~700 K can be assumed as the end temperature of valence transition.

Fig. 3: The valence state characters of Yb.
figure 3

Temperature dependence of a the Yb LIII-edge XANES spectra, and b the valence state of Yb in YbMn2Ge2. Error bars represent confidence intervals of valence state.

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The dual effect

To understand more in detail the origin of the enhanced NTE in YbMn2Ge2, one needs firstly to subtract the contribution of phonon vibrations to thermal expansion. Using the data of unit cell volume between 750–800 K which is far away from the effects of magnetism and valence transition, the contribution from phonon vibrations to the nominal thermal expansion (Vnm) is tentatively estimated according to the Debye-Grüneisen model48,49. As shown in Fig. 4a, it is interesting to find that the difference between the experimental volumes (Vexp) and the Vnm is still pronounced above the temperature of magnetic transition (TN = 525 K). It is worth noting that such anomaly exists until 700 K, which is well consistent with the temperature for the end of valence transition. Therefore, not only magnetism but also valence transition contributes to the NTE of YbMn2Ge2, which can be called the dual effect. In order to try to isolate the contribution of valence transition from that of MVE, the difference (ΔVtol) between Vexp and Vnm is plotted as a function of valence state of Yb (Fig. 4b). Since the TN is at 525 K, we can use the valence value in the temperature range of 575–700 K, which is well above the TN, to safely fit the contribution of valence transition to the NTE without the contribution from the MVE. As expected, the relationship between valence state and volume change is linear50, and the line shown in Fig. 4b represents the contribution from the valence change, namely ΔVvalen.. Then the contribution from MVE can be determined by the spontaneous volume magnetostriction using the equation of ωmagn. = (ΔVtol.−ΔVvalen.)/Vnm. As shown in Fig. 4c, it is interesting to find a good correlation between ωmagn. and the square of Mn magnetic moment (M2), which generally takes place in magnetic NTE materials51,52. Therefore, through the above analysis, the contributions of MVE and valence transition to NTE can be separated. As shown in Fig. 4d, the purple area represents the contribution from MVE, while the orange one stands for the contribution from valence change. Considering the ωmagn. in YbMn2Ge2, the value of 1.3% has the same order of magnitude of the values in magnetic NTE materials, such as 0.7–1.4% for Mn3AN53. For the value of ωvalen. related to the valence transition, this is equal to 0.5%, comparable to the values of 0.9% in Yb0.4In0.6Cu254 and 0.6% in YbIn0.75Ag0.25Cu450. Figure 4d gives a vision of the enhanced NTE in YbMn2Ge2 due to the dual effect of MVE and valence transition.

Fig. 4: The enhanced NTE induced by the dual effect.
figure 4

a Experimental unit cell volume (Vexp) determined by SXRD and nominal volume (Vnm) estimated from the phonon vibrations contriution. b Volume difference (ΔVtol = Vexp−Vnm) plotted as function of Yb valence and valence transition contribution (ΔVvalen.) to NTE. c Correlation between M2 and ωmagn.. d The dual effect of magnetism and valence transition to the NTE of YbMn2Ge2.

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In the RT2X2 compounds, no magnetic moment has been found in the T site except for Mn atom55,56,57. Therefore, to better understand the enhanced NTE resulted from the dual effect, the thermal expansion of YbFe2Ge2 without magnetovolume effect has been studied for comparison. As shown in Supplementary Fig. 8, according to the fitting of phonon vibrations by the Debye-Grüneisen model, it can be found that even though YbFe2Ge2 does not show NTE, but its thermal expansion is weakened by the single effect of valence transition. The single effect of valence transition contributes less to NTE compared with the dual effect. It needs to mention that there exists a large room to study the detailed difference between the existing single effect and the dual effect.

The present enhanced NTE in YbMn2Ge2 induced by the dual effect suggests a way to find other NTE materials, that is, to construct a material containing both variable valence atoms and magnetic atoms. In addition, the dual effect is not limited to the present combination of MVE and valence transition, which can be combined in different ways through the available NTE mechanisms, such as valence transition, MVE, ferroelectricity, and low-frequency phonons. An interesting example has been found in the solid solutions of BiNi1−xFexO3 (0.25 ≤ x ≤ 0.5), where NTE can be enhanced due to the simultaneous presence of valence transition and polar-to-nonpolar transition4. Other possible dual effect can be the combination between MVE and ferroelectricity in some multiferroics, or by combining low-frequency phonons and valence transition in some open framework structure materials. Enhanced or strong NTE can be expected, when the dual effect is exploited.

In summary, an enhanced NTE has been found out in the YbMn2Ge2 compound over a wide temperature range (αv = −32.9 × 10−6 K−1, ΔT = 175 K). The origin of such enhanced NTE has been investigated by combining NPD and XANES measurements. As the temperature increases, the Mn moment decreases rapidly as it approaches to TN, which leads to NTE due to the MVE. Meanwhile, the NTE is enhanced by the simultaneous valence transition of Yb from 2.40 to 2.82. These two concurrent contributions to NTE give rise to an enhanced NTE with a wide temperature range. Such dual effect, here related to magnetism and valence transition, represents an interesting opportunity to develop NTE materials.

Methods

Sample preparation

The polycrystalline sample of YbMn2Ge2 was prepared by an arc melting under a high purity argon atmosphere. The constituent elements with 99.9% purity were used. The button sample was melted at least four times to ensure good homogenization. Then, the sample was annealed at 1223 K for five days in a quartz tube in argon atmosphere (200 mbar), and quenched quickly into cold water.

Experiment methods

The temperature dependence of SXRD data was collected at the beamline 11-ID-C of Advanced Photon Source (APS) of Argonne National Laboratory with a wavelength of λ = 0.1173 Å from 200 K to 800 K. Temperature dependence of XANEX spectra was taken at the Yb LIII-edge from 200 K to 700 K at the 20-BM-B beamline of APS. NPD measurements were performed on the time-of-flight (TOF) diffractometer GPPD (General Purpose Powder Diffractometer) at China Spallation Neutron Source (CSNS), Dongguan, China. The sample was loaded in the vanadium cans during the test and the NPD patterns were collected with wavelength band from 0.2 to 4.2 Å. For the NPD measurements, additional aluminum foil was wrapped outside the vanadium tube for a better temperature management.