Giant-diamagnetic and magnetization-step effects in HgMnTe monocrystal
Liangqing Zhu, Tie Lin, Jun Shao, Zheng Tang, Junyu Zhu, Xiaodong Tang, Junhao Chu
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A p r Giant-diamagnetic and magnetization-step effects in HgMnTemonocrystal
Liangqing Zhu, ∗ Tie Lin, Jun Shao, Zheng Tang, Junyu Zhu, Xiaodong Tang, and Junhao Chu
1, 2, † National Laboratory for Infrared Physics,Shanghai Institute of Technical Physics,Chinese Academy of Sciences, 200083 Shanghai, China Key Laboratory of Polar materials and Devices, Ministry of Education,East China Normal University, 200062 Shanghai, China
Abstract
In Hg − x Mn x Te (x ≥ ions in Hg − x Mn x Te, a quasi-static spin wave forms and produces the GDM phenomenon belowthe critical temperature and magnetic field. Meanwhile, this theory is proved by Monte Carlosimulations in a two-dimensional AF cluster based on XY model. Hence, it is possible to emergelong-range magnetic order structure in SG state.
PACS numbers: 75.10.Nr, 75.20.-g, 75.30.Ds, 78.66.Hf, 75.50.Pp
Typeset by REVTEX 1t low temperature, it is well known that the combined effects of randomness and frus-tration may lead to spin-glass (SG) behavior in disordered spins system, such as magneticalloy, magnetic oxides and semimagnetic semiconductor (SMSC or DMS). [1, 2] For the SGof metallic alloy or magnetic metal oxides, it is difficult to separate the contribution of theconduction electrons from that of the localized spins , liking RKKY mechanism. Therefore,for a better understanding of SG, Mn-based SMSCs are appropriate candidates for studyingSG in experiment, due to pure antiferromagnetic (AF) exchange interaction and very lowcarriers concentration. In SG state, the global ground state of system always is to a ma-jor concern problem and not be resolved until now. Generally, mostly classical SG theoriesbased on mean-field theory and short-range AF exchange (e.g., Ising model and Sherrington-Kirkpatrick model[3]) suppose that the spins have no long-range magnetic order but insteadhave frozen or quasi-static orientations which vary randomly over macroscopic distances atlow temperatures.[1] Meanwhile, it predicts the limit concentration of the SG transition isabout 17% in SMSC with fcc structure.[4]Whereas, in Hg − x Mn x Te (fcc structure), the existing results of magnetic and specific heatexperiments have proved that: (i) the AF exchange interaction between Mn ions containslong-range exchange mechanisms, such as Bloembergen-Rowland exchange;[5–7] (ii) Mn ions are not exactly random uniform distribution but random fluctuation distribution inspace.[8–10] Both features go against the basic hypothesis of classical SG theories. Can itproduce some new effects on the spin arrangement of SG state, liking long-range magneticordered structures? From a fundamental perspective, this is a very important issue in SGtheory. For this purpose, we investigate the magnetic properties of Hg − x Mn x Te with variantMn concentrations, particularly near the SG regime.In this work, the DC field susceptibility (2 −
300 K) and magnetization (2 −
10 K) mea-surements with physical property measurement system (PPMS) of Quantum Design werecarried out on four Hg − x Mn x Te monocrystal samples grown by modified Bridgman methodand annealed in Hg vapor. Four samples denoted as NO1 (x ≃ ≃ ≃ ≃ cm − below 10 K.Figure 1 presents the magnetization curves of four samples at different temperatures.2 .................................................................................................................................... -0.02-0.0100.010.02 . . ... ............................................. ........................... -0.06-0.04-0.0200.020.040.06 . . ............................................................................... -400 -200 0 200 400 H (Oe) -0.1-0.0500.050.1 . . .......................................................................................................... -400 -200 0 200 400 H (Oe) -0.0100.01 . . M ( e m u / g ) x Mn ≈ Mn ≈ Mn ≈ Mn ≈ PM Magnetization Step
GDMGDM χ d ≈ -6.3 × -3 χ d ≈ -3.0 × -3 (a) (b)(c) (d) NO1 NO2NO3 NO4 H c ≈ c ≈
150 H c ≈ FIG. 1: The temperature-dependent magnetization curves of Hg − x Mn x Te monocrystals with dif-ferent Mn concentrations. These results show that when Mn concentration approaches or exceedsthe limit of SG transition (17%) in fcc structure, GDM and magnetization-step emerge belowthe critical temperature (T c ) and magnetic field (H c ). The susceptibility of GDM is about 100-1000 times than that of classic diamagnetic. In addition, the values of T c and H c go up as Mnconcentration increases. From 2 K to 5 K, the magnetization curves of NO1 are simple straight lines indicating goodparamagnetic (PM) state, as shown in Figure 1 (a). However, the remaining samples (NO2,NO3 and NO4) emerge novel and interesting magnetization phenomena at low temperatures,as illustrated in Figure 1(b), (c) and (d). For NO2 and NO3, both of magnetization curvesshow giant-diamagnetic (GDM) phenomenon below 4 K (called the critical temperature T c ).The following are the master features of GDM: (1) The absolute value of GDM susceptibilityis very large and depends on Mn concentration ( χ d =-3.0 × − for NO2 and χ d =-6.3 × − for NO3), which is about 100 − χ d =-10 − ∼ -10 − ). (2) There is a critical magnetic field (H c ) for the existence of GDM at firmedtemperature, e.g., H c ≃
130 Oe for NO2 and H c ≃
150 Oe for NO3 at 2 K. As magneticfield exceeds H c , GDM state rapidly changes into paramagnetic state. (3) H c decreases withtemperature rising. As regards NO4, when temperature lower than 5 K (about the T c ),its magnetization curves emerge magnetization-step instead of GDM. Meanwhile, this stepalso only exists under a critical magnetic field (e.g., H c ≃
190 Oe at 2 K), and graduallydisappears as temperature rises. 3
T (K)
T (K)
234 100Oe200Oe-73-72-71 SG
ZFC T f ≈ GDM T f ≈ χ ( × - e m u / ( g ⋅ O e )) ZFC ≈ x Mn ≈ Mn ≈ (a) (b) NO3 NO4
FIG. 2: The susceptibility curves ( χ -T) of NO3 and NO4 measured by ZFC method. (a) showsthe χ -T curves of NO3 at 100 Oe and 200 Oe. Both curves emerge a cusp structure at 3.5 K,which means SG transition. Meanwhile, at 100 Oe, the curve appears large negative value (GDM)below 3 K. (b) is the χ -T curve of NO4 at 100 Oe which emerges both SG transition (5 K) andmagnetization-step (below 3 K). Comparing the results of magnetization measurements in four Hg − x Mn x Te samples, itis clear that: (i) When Mn concentration approaches or exceeds the limit of spin-glasstransition (17%) in fcc structure, GDM and magnetization -step will appear below thecritical temperature (T c ) and magnetic field (H c ); (ii) As Mn concentration increases, theT c and H c of GDM and magnetization -step also slowly go up.In order to making a further justification for GDM and magnetization-step, the suscep-tibility measurements with the ZFC method were carried out on NO3 and NO4. Figure 2shows the susceptibility curves ( χ -T) of NO3 and NO4 from 2 K to 30 K at weak magneticfields (H=100 Oe and 200Oe). For NO3, its χ -T curves emerge a cusp structure at about3.5 K (exactly T c ), which means the occurrence of SG transition, as shown in Figure 2(a).In the SG regime, the value of susceptibility markedly changes with magnetic field, whichare positive value at H=200 Oe (greater than H c at 2 K), but appears large negative value(corresponding to GDM) at H=100 Oe (less than H c at 2 K). As well, the χ -T curve of NO4at 200 Oe magnetic field also emerges SG transition at 5 K (T c ) and the rapidly reduction ofsusceptibility below 5 K (corresponding to magnetization-step), as illustrated in Figure 2(b).Hence, the results of susceptibility curves prove the existence of GDM and magnetization-step again. More importantly, the GDM and magnetization-step are associated with the SGtransition.Generally, the classical diamagnetic comes from electron orbit precession and electro-4agnetic induction, the value of which is independent of temperature and magnetic field.But the GDM of Hg − x Mn x Te is related to temperature and magnetic field. In addition,the relaxation magnetization model of SG proposed by Lundgren, which assumes a uniformrandom distribution and short-range AF interactions of magnetic ions, only leads to posi-tive susceptibility (paramagnetic) in DC magnetic measurements.[11, 12] So, what are thereasonable physical mechanisms of GDM and magnetization-step in Hg − x Mn x Te?We think these novel magnetization phenomena should come from the effects of stronglong-range AF interactions between Mn ions in Hg − x Mn x Te. The following are our rea-sons in details. In the magnetization process, whether a spin of Mn can be flipped bymagnetic field depends on the competition of three factors: thermal fluctuation, magneticfield and AF interactions between Mn ions. Usually, thermal fluctuation leads to spinschaotic flipping, magnetic field causes the orderly arrangement of spins and AF interactionsmake spins frozen. The condition of whether a spin is free or magnetic frozen is[13] X j J nn S ˆ s i · ˆ s j ≥ k B T + g Mn µ B S Mn H (1)where P j J nn S Mn ˆ s i · ˆ s j is the AF exchange energy with the nearest neighbors, k B T is thethermal kinetic energy and g Mn µ B S Mn H is the magnetization energy.At high temperatures, due to strong thermal fluctuations, Eq.1 can not be satisfied evenat zero magnetic field. Thus, the spins of Mn ions freely rotate and can be overturnedeasily by magnetic field. However, at low temperatures, thermal fluctuations become weakand Eq.1 is easy to satisfied, especially for high Mn concentration and strong long-range AFinteractions between Mn ions. In this case, the spins of Mn ions are mostly frozen andform AF clusters, as shown in Figure 3(a).Then, according to Eq.1, when Mn concentration with uniform random distribution ap-proaches or exceeds a critical value, it needs the same nonzero magnetic field for all Mn ions to break down Eq.1 and generate magnetization below a critical temperature (T c ). Inother words, magnetization can not appear when magnetic field and temperature are bothless than the critical values (H c and T c ) in high Mn concentration area, as shown in NO4sample. This is the physical mechanism of magnetization-step in Hg − x Mn x Te.As to the physical mechanism of GDM, it involves two key factors. One stems fromthe effect of non-uniform random distribution of Mn ions in Hg − x Mn x Te. When thedistribution of Mn is inhomogeneous in space, the magnetized conditions (Eq.1) of Mn ree AF - Clusters Spin - Glass T ¯ H a L B AF - ClusterSpinWave M H b L M ¦ B H c L Spin wave in Spin Glass
FIG. 3: (a) shows the formation of AF clusters and spin-glass as temperature decreases. (b) showsthe structure of quasi-static spin wave in a 2D AF cluster with long-range AF interactions andnon-uniform random distribution of Mn ions. (c) shows the possible interconnect structure ofquasi-static spin waves in SG state. ions will be different from each other. It leads to, at the same temperature and magnetic field,the spins in high concentration region are easy to frozen, but in low concentration region areeasily magnetized. As a result, the spin arrangement of Mn ions is also inhomogeneousin space, especially in AF cluster which can be taken as the unit to compute the spinarrangement of ground state in non-uniform random distributing spin system.The other is due to the strong long-range AF interactions between Mn ions inHg − x Mn x Te. In high Mn concentration area, this will cause that the AF exchange en-ergy with further neighbors (J fn (R)S Mn ) is stronger than thermal kinetic energy of Mn ion when temperature below a critical value. Consequently, the spins of Mn ions emergemulti-frustration effect, which exist not only between the nearest neighbors but also withfurther neighbors. strongly correlatedCombining the effects of two factors, a spin flip of Mn ion occurred at the edge of AFcluster can impact the spin arrangement of both the nearest neighbors and further neighbors,and then produce a series of chain reactions in AF cluster, liking ”dominoes” effect. At theright temperature and magnetic field, this effect can create a quasi-static spin wave withoscillating spin arrangement, as presented in Figure 3(b). More importantly, the amplitudeof spin waves also depends on the Mn concentration, which may lead to a net negative6agnetic moment (opposite to magnetic field direction) in non-uniform random distributingAF clusters. Meanwhile, in the SG state, AF clusters are interconnect with each otherforming a AF ”super-cluster”. Hence, the quasi-static spin waves can propagate in space,causing the GDM phenomenon in Hg − x Mn x Te, as shown in Figure 3(c). This quasi-staticspin wave caused by local spins is similar to the spin-density wave (SDW) caused by electronsin chromium alloys.[14]It is hard to strictly confirmed the model of quasi-static spin waves by analytical method.However, it is possible to verify the rationality of this model by numerical simulation. Forthis purpose, Monte Carlo (MC) simulation was employed to compute the spin configurationof a two dimensional (2D) AF cluster based on XY model at different temperatures andmagnetic fields. Moreover, in order to overcoming the influence of metastable as much aspossible, simulated annealing was applied to seek the ground state of 2D AF cluster.[15]As an example, the simulation results of a 2D AF cluster referring to the propertiesof Hg Mn . Te (NO3 sample) are discussed in this paper.[16, 17] This 2D AF clusteris taken as circular shape (the diameter is about 32.52 nm) and contains 399 Mn ionswith non-uniform random distribution along the radial direction. In the central region,the average distance between the nearest Mn ions (¯ a nn ) is about 1.163 nm. From thecenter to the edge, ¯ a nn decreases linearly, which is about 1.221 nm in the edge region.In Hg − x Mn x Te, the exchange function of long-range AF interactions between Mn ions istaken as J ( R ij ) = J (¯ a nn /R ij ) , where R ij is the distance between two Mn ions and J (=-7.0K B ) is AF exchange integral constant between the nearest neighbors Mn ions.[13, 18]The Hamiltonian of each Mn ion is in form of Eq. 2. In MC simulations, the truncationradius R cut =5¯ a nn , S Mn =5/2, g Mn =2, and the number of MC steps is 10 at each temperaturepoint. H i = − X R ij ≤ R cut J ( R ij ) S ˆ s i · ˆ s j − g Mn µ B → B · → S i (2)Figure 4 show the typical MC simulation results of the 2D AF cluster, including the ¯ M y -T curves ( ¯ M y denotes the average magnetic moment of all Mn ions along magnetic fielddirection) and the spin configurations of ground state at different temperatures and magneticfields. When magnetic field is 100 Gs, the ¯ M y -T curve emerges both cusp structure (SGtransition) and negative value (paramagnetic-diamagnetic transition below 4 K), moreover,the spin configurations of ground state prove that a quasi-static spin wave really exists and7 y H S M n L - N = J =- k B R cut = a nn G D M SG H a L T H K L B M y T = = H b L SpinWave B M y T = = H c L N = J =- k B R cut = a nn T f = SG H d L T H K L B M y T = = H e L B M y T = = H f L FIG. 4: The typical MC simulation results of a 2D AF cluster with long-range AF interactionsand non-uniform random distribution. At 100 Gs, the ¯ M y -T curve emerges SG transition (about5 K) and GDM (below 4 K) for (a). Meanwhile, the spin configurations of ground state showa quasi-static spin wave which changes with temperature for (b) and (c). However, at 1000 Gs,the ¯ M y -T curve only emerges SG transition and the quasi-static spin wave is faint in the 2D AFcluster, for (d), (e) and (f). gradually disappears as temperature rises, as shown in Figure 4(a), (b) and (c). However,when magnetic field increases to 1000 Gs, the ¯ M y -T curve only emerges cusp structure, andthe quasi-static spin wave is faint or absent which is insufficient to cause paramagnetic-diamagnetic transition at low temperatures, as shown in Figure 4(d), (e) and (f).These MC simulation results of the 2D AF cluster demonstrate that: a quasi-static spinwave really exists and leads to paramagnetic-diamagnetic transition (similar to the GDMin experiments) below the critical temperature, then gradually vanishes as temperature ormagnetic field increases. Meanwhile, these results are basically consistent with the experi-mental results of GDM in Hg − x Mn x Te (NO2 and NO3 samples). Hence, in Hg − x Mn x Te,the model of quasi-static spin waves inducing GDM is reasonable.8n conclusion, the results of DC magnetic measurements prove Hg − x Mn x Te ( x ≥ . c and H c ). The susceptibility of GDM isabout 100-1000 times than that of classic diamagnetic. These novel magnetic phenomenacome from the effects of long-range AF exchange interactions and non-uniform randomdistribution of Mn ions in Hg − x Mn x Te, such as inducing a quasi-static spin wave whichproduces the GDM. In addition, in a 2D AF cluster with long-range AF interactions andnon-uniform random distribution, Monte Carlo simulations confirm that quasi-static spinwaves really exist, which lead to paramagnetic-diamagnetic transition in the SG regime andgradually disappear as temperature or magnetic field increases. Hence, it is possible toemerge long-range magnetic order structure in SG state.The authors thank Wanqi Jie and Tao Wang of Northwestern Polytechnical Univer-sity for the preparation of the HgMnTe monocrystal sample. This work was sponsored bythe STCSM (09JC1415600), the NSF (10927404, 60723001 and 60821092) and the NBRP(2007CB924902) of China. ∗ Electronic mail: [email protected] † Electronic mail: [email protected][1] K. Binder et al. , Rev. Mod. Phys. , 801 (1986).[2] J. S. Gardner et al. , Rev. Mod. Phys. , 53 (2010).[3] D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. , 1792 (1975).[4] R. R. Galazka et al. , Phys. Rev. B , 3344 (1980).[5] Jacek et al. , Diluted Magnetic Semiconductors (Academic Press, Landon, 1988).[6] N. Brandt et al. , Advances in Physics , 193 (1984).[7] V.-C. Lee, Phys. Rev. B , 8849 (1988).[8] S. Nagata et al. , Phys. Rev. B , 3331 (1980).[9] A. Mycielski et al. , Solid State Commun. , 257 (1984).[10] J. R. Anderson et al. , Phys. Rev. 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