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Advances in Condensed Matter Physics

Volume 2012 (2012), Article ID 913168, 7 pages

http://dx.doi.org/10.1155/2012/913168

## Origin of the Giant Negative Thermal Expansion in

^{1}Department of Physics, University of Science and Technology of China, Anhui, Hefei 230026, China^{2}Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Anhui, Hefei 230026, China

Received 16 June 2012; Accepted 11 September 2012

Academic Editor: Koshi Takenaka

Copyright © 2012 B. Y. Qu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The giant negative thermal expansion in the Ge-doped antiperovskite Mn_{3}CuN compound is theoretically studied by using the first principles calculations. We propose that such a negative thermal expansion property is essentially attributed to the magnetic phase transition, rather than to the lattice vibration of the Ge-doped compound. Furthermore, we found that the doped Ge atoms in the compound significantly enhance the antiferromagnetic couplings between the nearest neighboring Mn ions, which effectively stabilizes the magnetic ground states. In addition, the nature of the temperature-dependent changes in the volume of the Ge-doped compound was revealed.

#### 1. Introduction

Controlling the thermal expansion behavior of materials is eagerly required in modern advanced industries [1, 2]. Commonly, such control can be realized through mixing a kind of positive thermal expansion (PTE) materials with a kind of negative thermal expansion (NTE) materials whose volumes contract on heating. As we know, there are many PTE materials in the world, but the NTE ones rarely exist yet. Therefore, a challenging issue in controlling the thermal expansion behavior of the materials is to synthesize materials possessing NTE property.

Mn_{3}CuN, a member of the anti-perovskite manganese nitride Mn_{3}MN (M = Zn, Cu, etc.) [3–5], does not possess the negative thermal expansion property. However, recent experiments [6] reported that, with partial substitution of Ge for Cu in the Mn_{3}CuN compound, the synthesized compounds exhibited isotropic negative thermal expansion. Typically, the Ge-doped compound, , showed the NTE property in a wide temperature window of 316–386 K, together with a giant negative thermal expansion coefficient (The linear thermal expansion coefficient of a material is definited as . Here, is the length of the material at temperature , and is the change of the length when temperature has a change of ) of K^{−1}. More importantly, the working temperature and the width of the temperature window as well as the negative thermal expansion coefficient of the Ge-doped compound were modulated through the introduction of Ge dopant.

Prompted by the pioneering research [6], many other groups have studied the doping effect on the NTE property for some counterparts of , typically like [7], [8], [9], and so on [10, 11]. All the efforts reported in these papers showed that it was the doped Ge that brought the NTE property into these anti-perovskite compounds, just like that of the compound. Therefore, the doped Ge was regarded as a “magic element” for the observed NTE property in the class of the anti-perovskite manganese nitride compounds [6].

Recently, revealing the nature of such a “magic element” in the elegant performance of the NTE property has aroused many attractions, Matsuno and coworkers [12] proposed that the Ge dopant caused a local structural distortion in the compound, leading to the negative thermal expansion of the system in a broad temperature range. This was supported by the neutron powder diffraction and NMR measurements on [13]. On the other hand, through fitting the possible magnetic configurations to their experimental data, Kodama et al. [14] found that the ordering moment of the ground state known as the anti-ferromagnetic (seen in Figure 1(a)) of the compound gradually increased with decreasing temperature. This might also be responsible for the NTE property of the concerned compound. Even so, it is still unclear why the doped Ge atoms make the compound be of NTE property.

Usually, the thermal expansion property of a compound arises from different mechanisms, such as the lattice vibration, the temperature-excited magnetic phase transition, and the diffusion of ions/atoms between interstitial sites. Among these different mechanisms, both the lattice vibration and the phase transition are mainly responsible for the thermal expansion property of many materials. However, for our concerned compound, the relation between its thermal expansion behavior and the lattice vibration is still puzzling.

In this theoretical study, the compound is selected as a representative, since it exhibited a striking NTE property in experiment. Our calculations indicate that the lattice vibration contributes to the positive thermal expansion, and the magnetic phase transition from anti-ferromagnetic phase to paramagnetic phase contributes to the negative thermal expansion. Both of the PTE and NTE are compensative to each other, resulting in the observed thermal expansion behavior of the compound. Furthermore, a novel mechanism is proposed to explain the role of the doped Ge for the broad working temperature range of NTE in the compound.

#### 2. Computational Details

In our calculations, a supercell consisting of 20 atoms is used to simulate the magnetic structures of compound, where half of Cu atoms were replaced by Ge atoms. Based on the symmetry of the supercell, there are three kinds of distributions of Ge in it. These configurations are all optimized by using the VASP code [15, 16], in which the projector augmented wave (PAW) pseudopotentials and the Perdew-Burke-Ernzerhof (PBE) parameterizations of the generalized gradient approximation [17] are adopted. The cut-off energy is 400 eV and a gamma-centered k-point mesh of is used for the geometry optimization. Optimizations are performed until the residual forces acting on the atoms are smaller than 0.01 eV/Å.

By calculations, the energies of the supercell with three kinds of distributions of Ge in the compound are achieved, respectively. We find that the energy differences between the Ge distributions are less than 18 meV per unit cell. Such very small energy differences imply that these different distributions of Ge in compound may exist at finite temperatures. Even so, we select the lowest-energy one to serve our following calculations.

#### 3. Results and Discussions

##### 3.1. Vibrational Effect on the Variation of Volume

Structurally, is composed of the corner-shared Mn_{6}N octahedron, somewhat like the structural feature in the ZrW_{2}O_{8} compound [18], where an open-framework structure consisting of corner-linked rigid polyhedra associates with the transverse vibrational modes that cause NTE. Thus, it is necessary to examine the thermal expansion behavior induced by the vibrational modes. Within the quasiharmonic approximation, the thermal expansion coefficient is related to the vibrational modes of a material, which is calculated through the formula [19]
where is the th frequency, the bulk modulus, the volume of the system, and the Grüneisen parameter relevant to the th vibrational mode. Here, the Grüneisen parameter is defined as

To evaluate the coefficient of the thermal expansion, we first compute the bulk modulus and all vibrational frequencies at point for the supercell. Based on the obtained energy as a function of the volume of the system the bulk modulus of the concerned system is predicted to be about 128.2 GPa (to accurately achieve the vibrational modes, a larger supercell containing chemical unit cells is selected in our calculations relevant to the vibrational modes.) Meanwhile, the dynamic matrix of the considered system is generated through the forces acting on each atom, with using the frozen phonon approximation. By solving the dynamic matrix, we obtain all of the vibrational frequencies and the related eigenmodes. According to the obtained eigenmodes and the bulk modulus, the thermal expansion coefficient is obtained, as shown in Figure 2. Apparently, the value of is very small at low temperature and then increases rapidly with increasing temperature. When the temperature reaches 100 K or so, almost converges to the value of K^{−1}. This value is in good agreement with that ( K^{−1}) from experiment [6]. Such agreement indicates that the widely used frozen phonon approximation is also reliable to evaluate the dynamical property of the compound. More importantly, the calculated values of shown in Figure 2 are positive in the whole considered temperature range. This strongly predicts that the transverse motion from the rigid octahedron has no contribution to the observed NTE property in the compound.

##### 3.2. Magnetic Phase Transition versus the Variation of Volume

Experiments found the observed NTE behavior in the compound accompanied with a magnetic transition from the low temperature -type anti-ferromagnetic (AFM) phase to the high temperature paramagnetic (PM) phase [6, 20]. However, such NTE behavior was not found in the Mn_{3}CuN compound yet [6, 21]. Clearly, the observed NTE property of does tightly couple with the doped Ge atoms. To understand the role of the doped Ge atoms in the concerned NTE behavior, it is valuable to study the Mn_{3}CuN compound firstly.

For the undoped MnCuN compound, where the Mn ions contribute to local magnetic moments, showing non-collinear magnetic configuration. Its ground state was experimentally demonstrated to be ferromagnetic. Furthermore, a magnetic configuration (notated as C0, seen in Figure 1(b)) was suggested to explain the ground state [21]. Our calculations reproduce the magnetic configuration C0, with the ferromagnetic components of local magnetic moments of 0.18 and 0.30 for the Mn ions locating at two unequal planes, respectively, which are close to the values of 0.20 and 0.65 reported in experiment [21]. However, such agreement does not directly support C0 configuration as the ground state of the Mn_{3}CuN compound. In fact, a ground state configuration should have the lowest energy among all of the possible magnetic configurations. To examine whether the configuration C0 is the ground state or not, we generate many kinds of magnetic configurations, followed by full relaxations. Among these considered magnetic configurations, a lowest-energy configuration (notated as C1, seen in Figure 1(c)) is found, whose energy is lower by about 0.3 eV per primitive unit cell than that of the C0 configuration. Furthermore, we find that the configuration C1 characterizes the FM feature, and the ferromagnetic component averaged on Mn ions in this configuration is about 0.1 , being smaller than the experimental value of 0.35 .

Moreover, for this undoped compound, we find that these magnetic configurations have different energies, meaning that these magnetic configurations may be thermally excited at different temperatures. On the other hand, the equilibrium volumes of all considered magnetic configurations are very close to each other. Hence, the thermally excited transformation between different magnetic configurations does not change the volume of the compound nearly. In other words, the Mn_{3}CuN compound does not show NTE on heating. This is consistent with the observation in experiment very well [6].

Then, we turn to the compound. For this compound, the magnetic state at low temperature was demonstated to be AFM in experiment. Moreover, experiments suggested that a special AFM configuration named as just corresponded to the ground state of the compound [13]. On the theoretical side, it is necessary to clarify its magnetic ground state. To do this, we randomly generate about eighty magnetic configurations plus the configuration in this compound. After full relaxations of the spins and atomic positions for each concerned case, we only have twenty four inequivalent magnetic configurations. Figure 3(a) displays the energy as a function of the lattice constant for each magnetic configuration. From Figure 3(a), one can find that the lowest-energy magnetic configuration is , and its equilibrium lattice constant is evaluated to be 3.864 Å, in agreement with the experimental value of 3.910 Å [13]. Furthermore, our calculations indicate that the local magnetic moments of the Mn ions in the configuration are 2.63 , being in good consistent with the experimental value of [21]. Such agreement supports that is really the magnetic ground state for compound.

The experiments revealed that with raising temperature, the compound exhibited a volume contraction. Basically, the temperature effect not only excites more magnetic configurations of the system, but also distorts each magnetic configuration to some extent. To qualitatively evaluate the distorted magnetic configurations, we randomly deviate the orientations of the local moments of Mn ions from their initial orientations for each magnetic configuration, and thus a series of the distorted configurations can be generated through controlling the deviating extent. The volume of the compound at each distorted configuration is optimized with fixing the orientations of the local moments. Our calculations show that the volume of the distorted magnetic configurations almost does not change, except for the configuration. As shown in Figure 3(b), with distorting the configuration seriously, the energy of the system increases significantly, and the volume of the lattice decreases largely. This predicts that the volume contraction on heating is actually contributed from the distorted configuration together with the thermal excited metastable configurations.

On the other hand, when the compound was heated up to the Neel temperature, it transfers from the AFM state to the PM state [13]. As we know, the PM state is a magnetic disorder state, which cannot be simply approximated to be either a non-magnetic state or a magnetic ordering state with lowmagnetic moment in a theoretical calculation. Since there are many different magnetic configurations in the compound, the PM state may be approximatively regarded as a mixture of many magnetic structures that can be thermally excited, when the compound is heated up above the Neel temperature. According to this, the volume of the compound at the PM state is expressed as the statistically averaged volume of many different magnetic structures including , by using the Boltzmann distribution function. Combining this consideration with the energy curves shown in Figure 3, we statistically compute the lattice constant at each considered temperature. Of these temperature-dependent lattice constants, the lattice constant at 500 K is selected as a reference, and thus we obtain the ratio of the lattice constant contraction, . As displayed in Figure 4(a), the lattice constant exhibits a gradual decrement with increasing temperature. This implies that the considered magnetic interactions really cause the NTE property in the whole temperature range. To go further, combining this NTE behavior with the PTE behavior from the lattice dynamics shown in Figure 2, we find that the compound exhibits the NTE behavior only in a temperature range of 250–340 K, as seen in Figure 4(b), in consistent with the results reported in experiment [6]. Our calculations above reveal that the volume contraction from the magnetic couplings is actually compensated by the lattice vibrations to some extent. The effect of such compensation is strongly relevant to the temperature. Namely, when the temperature is either below 250 K or over 340 K, the volume change of the compound from the lattice vibration overpasses that from the magnetic interaction, showing the PTE behavior. And in the temperature range between 250 and 340 K, the NTE effect is much stronger than the PTE effect, so that the compound exhibits a net effect of NTE in such a temperature window.

##### 3.3. The Role of the Doped Ge Atoms in the NTE Property

As mentioned above, it is the doped Ge atoms that cause the compound to be of NTE property. Meanwhile, the observed NTE property is stemmed from the magnetic interaction between ions in the compound. So, the nature of the observed NTE property is essentially associated with how the doped Ge atoms affect the magnetic couplings between ions. Physically, the magnetic couplings between ions in a compound can be expressed by the exchange parameters between different ions within the framework of the classical isotropic Heisenberg theory [22–25]. In our concerned compound, three typical kinds of exchange parameters, , , and are considered. Here, is the coupling between an Mn ion and its nearest neighbor Mn; is the super exchange between an Mn ion and its second neighbor Mn ion, separated by an N atom; is the coupling between an Mn ion and its second neighbor Mn ion, without any atom between them. Based on the classical isotropic Heisenberg theory [22–25], the energy gain from the magnetic couplings is expressed as for and for Mn_{3}CuN.

Table 1 lists the calculated exchange parameters for both compounds. Among the three kinds of exchange parameters, the amplitude of for the compound is considerably bigger than that for the Mn_{3}CuN compound. This indicates that the doped Ge strongly enhances the magnetic interaction between the nearest-neighboring Mn ions in the compound. It is worth noting that in our concerned anti-perovskite manganese nitride compound, the couplings between the nearest-neighboring Mn ions form a triangular structure in space. Such a triangular feature is just responsible for the noncollinear anti-ferromagnetic configuration, which is the basic characteristic of the configuration. So, the doped Ge atoms effectively enhance the AFM configuration much more than the other configurations in the compound. This is just corresponding to the fact that has the ground state of , whereas Mn_{3}CuN does not have yet [6, 21].

As reported in experiments, the doped Ge atoms not only caused the NTE property, but also made the NTE occurring in a broad temperature range for . However, these features did not occur in the (M = Zn, Si, and Ga) compounds. Basically, the different behaviors in NTE of the compounds caused by the different dopants tightly couple with the different electronic structures of the dopants. For the isolated atoms of Cu, Ge, Zn, and Mn, the energies of their highest occupied orbitals are in the order of . Clearly, the energy differences of the orbitals between Mn and M (M = Cu, Zn and Ge) are quite big. This means that the 4s orbital of either Cu or Zn and the 4p orbital of Ge do not significantly couple with the valence orbital of Mn in the compound. So, in the compound, most electrons on these atomic orbitals are unpaired and thus locate at the unsaturated states which are almost at the Fermi level. Especially, is much higher than the others, and the number of electrons on Ge 4p is larger than the other orbitals considered above. Thus the doped Ge in the compound contribute more states at the Fermi level than the other cases. Meanwhile, the local density of states (LDOS) at the Fermi level from the Mn ions drops down largely when the content of Ge is 50 [26] in the compound. With rising temperature, more electrons from the Ge 4p orbital are thermally excited into the conduction band in such metallic systems. These conduction electrons may polarize some of local electrons in the compound, altering the magnetic couplings between Mn ions and the local magnetic moments accordingly. Namely, some of magnetic states in the Ge-doped compound is easily excited by temperature, even temperature rises slightly. As a result, the Ge-doped compound shows a gradually temperature-dependent change in volume. This is corresponding to the wide temperature range for the NTE working, as reported in experiments [6].

#### 4. Conclusion

We have investigated the lattice vibrations and the magnetic phase transition for the Ge-doped metallic anti-perovskite compound, based on first-principles calculations. We find that the lattice vibrations contribute to the PTE property rather than the NTE property. Our calculations indicate that the observed NTE is actually resulted from the magnetic phase transition from the phase to the PM phase. Furthermore, we suggest that the thermal effect excites more valence electrons from the doped Ge atoms to the conduction bands in the compound. These donated conduction electrons polarize the local electrons and change the local magnetic moments of Mn ions, leading to the contraction of the compound in volume.

#### Acknowledgments

This work is supported by the National Basic Research Program of China (2009CB939901) and the National Science Foundation of China with Grant no. NSFC10974184. The HP-LHPC of USTC is acknowledged for computational support. The authors gratefully acknowledge the valuable comments from Professor Koshi Takenaka.

#### References

- A. W. Sleight, “Compounds that contract on heating,”
*Inorganic Chemistry*, vol. 37, no. 12, pp. 2854–2860, 1998. View at Scopus - G. D. Barrera, J. A. O. Bruno, T. H. K. Barron, and N. L. Allan, “Negative thermal expansion,”
*Journal of Physics Condensed Matter*, vol. 17, no. 4, pp. R217–R252, 2005. View at Publisher · View at Google Scholar · View at Scopus - D. Fruchart and E. F. Bertaut, “Magnetic studies of the metallic perovskite-type compounds of manganese,”
*Journal of the Physical Society of Japan*, vol. 44, no. 3, pp. 781–791, 1978. View at Scopus - W. S. Kim, E. O. Chi, J. C. Kim, N. H. Hur, K. W. Lee, and Y. N. Choi, “Cracks induced by magnetic ordering in the antiperovskite ZnNMn
_{3},”*Physical Review B*, vol. 68, no. 17, Article ID 172402, 4 pages, 2003. View at Scopus - D. Fruchart, E. F. Bertaut, R. Madar, G. Lorthioir, and R. Fruchart, “Structure magnetique et rotation de spin de Mn
_{3}NiN,”*Solid State Communications*, vol. 9, no. 21, pp. 1793–1797, 1971. View at Scopus - K. Takenaka and H. Takagi, “Giant negative thermal expansion in Ge-doped anti-perovskite manganese nitrides,”
*Applied Physics Letters*, vol. 87, no. 26, Article ID 261902, 3 pages, 2005. View at Publisher · View at Google Scholar · View at Scopus - K. Takenaka, K. Asano, M. Misawa, and H. Takagi, “Negative thermal expansion in Ge-free antiperovskite manganese nitrides: tin-doping effect,”
*Applied Physics Letters*, vol. 92, no. 1, Article ID 011927, 3 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus - Y. Sun, C. Wang, Y. Wen, K. Zhu, and J. Zhao, “Lattice contraction and magnetic and electronic transport properties of Mn
_{3}Zn_{1-x}Ge_{x}N,”*Applied Physics Letters*, vol. 91, no. 23, Article ID 231913, 3 pages, 2007. View at Publisher · View at Google Scholar · View at Scopus - R. Huang, L. Li, F. Cai, X. Xu, and L. Qian, “Low-temperature negative thermal expansion of the antiperovskite manganese nitride Mn
_{3}CuN codoped with Ge and Si,”*Applied Physics Letters*, vol. 93, no. 8, Article ID 081902, 3 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus - B. S. Wang, P. Tong, Y. P. Sun et al., “Enhanced giant magnetoresistance in Ni-doped antipervoskite compounds GaCMn
_{3-x}Ni_{x}(*x*= 0.05, 0.10),”*Applied Physics Letters*, vol. 95, no. 22, Article ID 222509, 3 pages, 2009. View at Publisher · View at Google Scholar - R. J. Huang, W. Xu, X. D. Xu, L. F. Li, X. Q. Pan, and D. Evans, “Negative thermal expansion and electrical properties of Mn
_{3}(Cu_{0.6}Nb_{x}Ge_{0.4 - x})N ($x=0.05-0.25$) compounds,”*Materials Letters*, vol. 62, no. 16, pp. 2381–2384, 2008. View at Publisher · View at Google Scholar · View at Scopus - J. Matsuno, K. Takenaka, H. Takagi, D. Matsumura, Y. Nishihata, and J. Mizuki, “Local structure anomaly around Ge dopantsin Mn
_{3}Cu_{0.7}Ge_{0.3}N with negative thermal expansion,”*Applied Physics Letters*, vol. 94, no. 18, Article ID 181904, 3 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus - S. Iikubo, K. Kodama, K. Takenaka, H. Takagi, M. Takigawa, and S. Shamoto, “Local lattice distortion in the giant negative thermal expansion material Mn
_{3}Cu_{1-x}Ge_{x}N,”*Physical Review Letters*, vol. 101, no. 20, Article ID 205901, 4 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus - K. Kodama, S. Iikubo, K. Takenaka, M. Takigawa, H. Takagi, and S. Shamoto, “Gradual development of Γ
^{5g}antiferromagnetic moment in the giant negative thermal expansion material Mn_{3}Cu_{1-x}Ge_{x}N (x*~*0.5),”*Physical Review B*, vol. 81, no. 22, Article ID 224419, 8 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus - G. Kresse and J. Hafner, “Ab initio molecular dynamics for liquid metals,”
*Physical Review B*, vol. 47, no. 1, pp. 558–561, 1993. View at Publisher · View at Google Scholar · View at Scopus - G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,”
*Physical Review B*, vol. 54, no. 16, pp. 11169–11186, 1996. View at Scopus - J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,”
*Physical Review Letters*, vol. 77, no. 18, pp. 3865–3868, 1996. View at Scopus - G. Ernst, C. Broholm, G. R. Kowach, and A. P. Ramirez, “Phonon density of states and negative thermal expansion in ZrW
_{2}O_{8},”*Nature*, vol. 396, no. 6707, pp. 147–149, 1998. View at Publisher · View at Google Scholar · View at Scopus - K. Wang and R. R. Reeber, “Mode Grüneisen parameters and negative thermal expansion of cubic ZrW
_{2}O_{8}and ZrMo_{2}O_{8},”*Applied Physics Letters*, vol. 76, no. 16, pp. 2203–2204, 2000. View at Scopus - K. Takenaka and H. Takagi, “Magnetovolume effect and negative thermalexpansion in Mn
_{3}(Cu_{1-x}Ge_{x})N,”*Materials Transactions*, vol. 47, no. 3, pp. 471–474, 2006. View at Publisher · View at Google Scholar · View at Scopus - S. Iikubo, K. Kodama, K. Takenaka, H. Takagi, and S. Shamoto, “Magnetovolume effect in Mn
_{3}Cu_{1-x}Ge_{x}N related to the magnetic structure: neutron powder diffraction measurements,”*Physical Review B*, vol. 77, no. 2, Article ID 020409, 4 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus - N. M. Rosengaard and B. Johansson, “Finite-temperature study of itinerant ferromagnetism in Fe, Co, and Ni,”
*Physical Review B*, vol. 55, no. 22, pp. 14975–14986, 1997. View at Scopus - Y. M. Zhou, D. S. Wang, and Y. Kawazoe, “Effective ab initio exchange integrals and magnetic phase transition in fcc Fe and Mn antiferromagnets,”
*Physical Review B*, vol. 59, no. 13, pp. 8387–8390, 1999. View at Scopus - J. T. Wang, L. Zhou, D. S. Wang, and Y. Kawazoe, “Exchange interaction and magnetic phase transition in layered Fe/Au(001) superlattices,”
*Physical Review B*, vol. 62, no. 5, pp. 3354–3360, 2000. View at Scopus - X. Gao, Y. M. Zhou, and D. S. Wang, “Analysis of the determination of magnetic phase transition temperatures by the first-principles calculations,”
*Journal of Magnetism and Magnetic Materials*, vol. 251, no. 1, pp. 29–37, 2002. View at Publisher · View at Google Scholar · View at Scopus - B. Y. Qu, H. Y. He, and B. C. Pan, “The elastic properties of Mn
_{3}(Cu_{1-x}Ge_{x})N compounds,”*AIP Advances*, vol. 1, no. 4, Article ID 042125, 8 pages, 2011. View at Publisher · View at Google Scholar