Magnetic domains and the walls between are the subject of great interest because of the role they play in determining the electrical properties of ferromagnetic materials and as a means of manipulating electron spin in spintronic devices. However, much less attention has been paid to these effects in antiferromagnets, primarily because there is less awareness of their existence in antiferromagnets, and in addition they are hard to probe since they exhibit no net magnetic moment. In this paper, we discuss the electrical properties of chromium, which is the only elemental antiferromagnet and how they depend on the subtle arrangement of the antiferromagnetically ordered spins. X-ray measurement of the modulation wavevector Q of the incommensurate antiferromagnetic spin-density wave shows thermal hysteresis, with the corresponding wavelength being larger during cooling than during warming. The thermal hysteresis in the Q vector is accompanied with a thermal hysteresis in both the longitudinal and Hall resistivity. During cooling, we measure a larger longitudinal and Hall resistivity compared with when warming, which indicates that a larger wavelength at a given temperature corresponds to a smaller carrier density or equivalently a larger antiferromagnetic ordering parameter compared to a smaller wavelength. This shows that the arrangement of the antiferromagnetic spins directly influences the transport properties. In thin films, the sign of the thermal hysteresis for Q is the same as in thick films, but a distinct aspect is that Q is quantized.
A new approach to store and process information by using the spin in carriers in addition to the charge became prominent after the discovery of the giant magnetoresistance (GMR) effect in ferromagnetic layers separated by a non-ferromagnetic metallic layer . Depending on the relative magnetic orientation of the ferromagnetic layers, the resistance of the multi-layer device switches between low and high resistance, with the low resistance corresponding to the magnetic configuration where the magnetic orientations of the layers are parallel. This is a demonstration of spintronics  (spin-based electronics), where the spin of the carrier (or magnetization) influences the electrical properties of the multi-layer device. A variation of this device is a magnetic tunnel junction, where an insulating tunnelling barrier takes the place of the non-ferromagnetic metallic layer . Except for circumstances where the density of states of the majority spin is a minority spin at the Fermi surface for one of the ferromagnetic electrodes, the resistance is lower when the magnetizations of the two ferromagnetic electrodes are parallel, same as in the GMR device. These magnetic devices that consist of ferromagnetic multi-layers separated by a non-ferromagnetic layer (either metallic or insulating) are called spin valves since by manipulating the spin, the current through the device can be turned on or off. Great advances in the engineering of spin valve devices have led to the commercialization of these devices as sensors for magnetic fields and as memory elements where the memory is stored in the magnetization.
The physical principle underlying these spintronic devices is spin-dependent transport, where the spin of the carrier affects the transport across the device. In a parallel magnetic configuration, the spin of the majority carrier, as well as that of the minority carrier, is the same on both magnetic layers, and leads to a low-resistance state based on the fact that the majority carriers on one electrode easily find an empty state of the same spin on the other electrode. In an antiparallel magnetic configuration, the majority carrier on one electrode is the minority carrier on the other electrode, and therefore has a low density of states on the other electrode, leading to a large scattering rate and high resistance .
The spin valves are artificial devices consisting of multi-layers, and therefore a natural quest is to realize similar effects in a naturally occurring device . A long-standing candidate has been magnetic domains and the walls separating them in ferromagnets . Magnetoresistance across magnetic domains in ferromagnets has shown evidence of resistance owing to the presence of magnetic domain walls, with the largest effect being of the order of 1 per cent [6,7]. Larger magnetoresistance owing to domain walls has been observed in systems where the magnetic domain wall is associated with a grain boundary. Large magnetoresistance owing to spin-polarized tunnelling across grain boundaries in manganite has been observed in polycrystalline samples  as well as across artificial grain boundaries in bicrystal films [9,10]. One important fact is that single crystal manganite samples do not exhibit large low-field magnetoresistance even though they have magnetic domain walls , implying that domain walls alone in the manganites are not sufficient to give rise to spin-dependent transport and that grain boundaries are crucial to display such an effect. This is probably owing to the fact that a sharp boundary that separates one magnetized state from another magnetized state is necessary to display a spin valve behaviour and the grain boundaries play the role of the non-ferromagnetic spacer or tunnelling barrier in the artificial multi-layered devices.
This paper expands on work published previously , where we present electrical effects in an antiferromagnet that we associate with the subtle arrangement of the spins. These results are rather novel since they demonstrate that even in antiferromagnets the magnetic configuration can alter the charge transport. Although the size of the effect is small, it is comparable to the domain wall resistance observed in ferromagnets without grain boundaries. Our system of choice is chromium (Cr) films. Cr is the only elemental antiferromagnet and a classical system for spin-density wave antiferromagnetism. The band structure of Cr (figure 1a) yields Fermi surfaces that include electron and hole octahedra centred around the Γ and H points. The hole octahedra are slightly larger than the electron octahedra, and pairing between the electron and hole with opposite spins is possible by overlapping the two octahedra by a wavevector Q=2π/a(1±δ) that is slightly incommensurate from K/2, with K being a reciprocal lattice vector [13,14]. There is a Fermi surface nesting instability between the two octahedra, which leads to the formation of the antiferromagnetic spin-density wave and opening of a gap at the Fermi level. The carriers that participate in the spin-density wave become localized resulting in a loss of mobile carriers. The spin-density wave induces a charge-density wave, which in turn, owing to the electron–phonon coupling induces a strain wave. The periodicity λ of the spin-density wave is twice that of the charge-density wave and strain wave and is given by a/δ. It depends on temperature, with λ evolving from around 8 nm near the Néel point to about 6 nm at low temperature .
2. Experimental procedure
We were able to observe new electrical effects by growing high-quality single crystal Cr films. The films were grown on MgO substrates, which are electrically inert, and have a small lattice mismatch with Cr and therefore lead to high-quality single crystal films. After growing the films, we patterned Hall bars of 1 mm width using standard optical lithography and wet etching. The voltage leads to measure the longitudinal resistance were either 0.5 or 1 mm apart, and the Hall voltage leads were 1 mm apart.
The electrical measurements were performed using a Quantum Design Physical Property Measurement System (PPMS). Current was applied in square waves of 7 Hz to eliminate thermal voltages. Hall voltage was measured using magnetic fields applied in both perpendicular directions to offset any longitudinal resistance. The temperature-dependent measurements were performed in two ways: by measuring on the run as the system was either cooling down or warming up at a fixed rate, or at discrete temperatures after 1 h wait after the system equilibrated at a given temperature.
The X-ray measurements of the strain wave associated with the spin-density wave were performed at Argonne National Laboratory at beamline 4-ID-D and 33-BM, at Diamond at beamline I16 and at ESRF at beamline ID20.1
Longitudinal and Hall resistivity were measured on four Cr films of thickness 175, 430, 506 and 3500 Å, respectively, under the presence of a magnetic field perpendicular to the plane of the Hall bar (figure 1b). All films show a kink in the longitudinal resistivity and a large loss of carriers at the Néel temperature as a result of the onset of the spin-density wave and opening of an energy gap. The Hall conductance, which can be derived by the measured longitudinal and Hall resistivity, does not show any appreciable change at the transition temperature.
We measured the resistivity on cooling and warming and observe a thermal hysteresis below the Néel temperature for all films with the measured longitudinal and Hall resistivity always being larger on cooling than on warming (figure 2a,b). In the thinnest sample, the hysteresis is prominent even in the raw data as shown in figure 3a,b. The hysteresis curves are independent of the strength of the magnetic field applied and they open and close at the same temperature for both the longitudinal and Hall channels indicating that the underlying mechanism is the same for both. For the 506 Å sample, we observe two hysteresis loops (H1 and H2), with the one occurring at lower temperatures (H2) showing a different amount of hysteresis (figure 2a) depending on the region of the sample (A or B). In addition, the derivative of the resistivity (figure 4a,b) displays kinks below the Néel temperature in two of the films (175 and 506 Å) with the kink occurring at a lower temperature on cooling than on warming, and the location of the kinks coinciding with the position where the hysteresis loop closes (for the 506 Å film, it coincides with the end points of the hysteresis loop H1 at the higher temperature window). In contrast, the Hall conductance shows negligible or much less hysteresis for all samples (figures 2c and 3c). Moreover, the Hall conductance does not show any kink in the derivatives (figure 4c,d) except for 1/σH at the Néel temperature. In figure 5, we plot the ratio of the resistance measured by two different pairs of leads for all samples. It shows that the ratio is temperature dependent for all samples below the Néel temperature.
In order to understand the electrical behaviour, we conducted X-ray scattering experiments on all four films. We searched for the strain wave associated with the spin-density wave around (001), (101) and (112) Bragg peaks by scanning along H, K and L independently. No superstructure was detected in the 430 Å film, which is the most disordered among all four films based on the X-ray data that show the lowest crystallinity, the atomic force microscopy data that show the highest roughness and the transport measurements that show the highest residual resistivity. For all other films, we detected satellite peaks only along the L direction, which is perpendicular to the film plane. We tracked the position of the satellite peaks as a function of temperature both on cooling and warming. There are several notable points. First, the Q vector only points in the direction perpendicular to the film plane . Second, it is hysteretic, with the incommensurability δ being smaller on cooling than on warming . Third, for two of the films (175 and 506 Å), Q is quantized such that only half integer (N+1/2) periods of strain wave are accommodated within the film thickness (in our previous work , we reported it to be full integers, but they are half integers). Fourth, for the 506 Å film, at various temperature windows, there is coexistence of two adjacent quantized Q values.
By combining the spin-density wave information obtained by X-rays, which we indirectly gain by studying the associated strain wave, with the transport data, we can make a direct connection between the antiferromagnetism and the measured resistivity. The largest electrical effect that is a result of the influence of the antiferromagnetism is the kink in the resistivity and loss of carriers at the Néel temperature (figure 1b) owing to the formation of the spin-density wave, a fact that has been established for a long time. This effect is more visible in the derivative of the resistivity and Hall coefficient as seen in figure 4a,b. One natural question is why the Hall conductance does not show as prominent an effect owing to the loss of carriers as the resistivity or Hall coefficient does (compare figure 4c,d with figure 4a,b at the Néel temperature). This can be explained by the expression for the Hall conductance where τ is the relaxation time, f is the Fermi distribution function, v is the electron group velocity and ( and ) are inverse mass coefficients . The Hall conductance is weighted by inverse mass coefficients. Since the mass is large at the flat parts of the Fermi surface, these parts of the Fermi surface do not contribute appreciably to the Hall conductance. At the Néel transition, it is the flat parts of the Fermi surface that undergo a change. Owing to the inverse mass coefficients, the changes in the flat parts of the Fermi surface are not easily detected in the Hall conductance .
For the explanation of the thermal hysteresis in the resistivity, we rule out hysteresis of the order parameter in a first-order phase transition since the sign of the hysteresis—larger Hall resistivity during cooling, which means smaller carrier density and larger order parameter during cooling—is opposite to what would be for a first-order phase transition. Furthermore, we rule out hysteresis owing to the reorientation of Q and associated change of the resistivity owing to the anisotropy in the resistivity, which depends on whether J is parallel or perpendicular to Q . The X-ray measurements clearly indicate that Q is out of the plane throughout the whole temperature range both on cooling and warming. In our prior work , we speculated domain wall scattering as the underlying mechanism for the hysteresis in the 3500 Å film. Our recent X-ray scattering experiments suggest that there is no need to invoke domain wall scattering in order to explain the hysteresis in the electrical measurements. The temperature window of the hysteresis loops in the longitudinal and Hall resistivity coincides with the hysteresis loops for Q in the X-ray measurement of the strain wave in all three samples where we were able to detect a superstructure. This suggests that the thermal hysteresis in resistivity observed below the Néel temperature can be explained by the difference in the incommensurability δ on cooling versus warming. From the fact that in the temperature windows in which there is hysteresis, the smaller δ value on cooling corresponds to a larger longitudinal resistance and larger Hall resistivity (smaller carrier density), we can infer that a spin-density wave with smaller δ value (larger wavelength) has a larger gap and a larger antiferromagnetic ordering parameter compared with a spin-density wave with a larger δ value at the same temperature. It is worthwhile to note that this is not related to the temperature dependence of Q which shows that as the temperature is lowered and the antiferromagnetic order increases, the wavelength of the spin-density wave decreases .
The kinks in the derivative of the resistivity for the 175 and 506 Å samples coincide with temperatures at which the quantization value for the number of strain wave periods switches by one. In the case of the 175 Å sample, the strain wave is quantized such that only two values of δ, corresponding to 5.5 or 4.5 periods accommodated across the film thickness, are observed. This corresponds to 2.75 or 2.25 spin-density wave periods (figure 3d). As we cool down the sample, at the Néel temperature, a quantized strain wave forms corresponding to 4.5 number of periods. As we continue cooling further, the quantization value switches to 5.5 at T∼220 K, which agrees with the temperature at which the kink in the derivative is observed on cooling (223 K). On the warm up, the switching from 5.5 to 4.5 number of periods occurs at a higher temperature T∼250 K, again coinciding with the temperature at which the kink in the derivative occurs (248 K). For the 506 Å sample, as we cycle the temperature, four different quantized values of Q, corresponding to 15.5, 16.5, 17.5 and 18.5 strain wave periods, are observed. The kinks in the resistivity occur when Q switches from 16.5 to 17.5 strain wave periods during cooling, and from 17.5 to 16.5 strain wave periods during warming.
The analysis of the Hall conductance enables us to identify which parts of the Fermi surface contribute to the new electrical effects we discovered. The suppression of thermal hysteresis in the Hall conductance (figures 2c and 3c) suggests that the thermal hysteresis in the resistivity is largely owing to the difference in the flat parts of the Fermi surface on cooling versus warming. The absence of kinks in the derivative of the Hall conductance (figure 4c,d) below the Néel temperature suggests that the kinks in the derivative of the longitudinal and Hall resistivity below the Néel temperature are due to abrupt changes in the flat parts of the Fermi surface when the wavevector of the spin-density wave changes in a discrete manner in the quantized regime.
We can explain the difference in the amount of thermal hysteresis observed in the two regions of the 506 Å sample by taking into account that the temperature window in which this happens (H2 in figure 2) coincides with the temperature window in which there is thermal hysteresis of Q with Q being a single quantized value, corresponding to 17.5 strain wave periods, on cooling but exhibiting two different quantized values, 17.5 and 18.5, on warming. Based on prior arguments, spin-density waves with different Q vectors at the same temperature correspond to different strengths in the order parameter or in other words carrier densities. The coexistence of two different Q values translates to the coexistence of two partially gapped metals with slightly different carrier densities. If the distribution of the two different types of Q domains is statistically different in the two different regions, then the two regions will exhibit a different resistance even after being normalized by the geometrical factors (cross section and length of the resistance strips), resulting in a different amount of thermal hysteresis in the resistivity. The coexistence of two different Q values in the same sample, which is compositionally and structurally homogeneous, is a remarkable example of intrinsic electronic inhomogeneity in a metal.
The electronic inhomogeneity is clearly illustrated by plotting the ratio of the resistance in the two regions on cooling and warming, normalized by the ratio at the highest temperature measured (figure 5b,c). In a normal metal, the ratio of the resistance in two regions of the same material should be a temperature independent constant, which is simply determined by the geometrical dimensions of the regions. In all four samples, we observe a temperature dependence in the ratio when the system is in the antiferromagnetic state. This means that resistivity is not a good physical quantity for representing the resistance in the antiferromagnetic state. One of the most notable examples in which resistivity breaks down is in two-dimensional electron gas systems in narrow constrictions of submicrometre width, in which case the resistance does not scale with the length of the channel, but depends on the number of one-dimensional channels contributing to transport . Another example in which resistivity breaks down is when the dimensions of the system become small enough such that there is statistical fluctuation in the scattering events in one region compared with another region. In metals, this requires the system to be in the micrometre scale since the characteristic length between scattering events is the mean free path, which is of the order of 10 nm at ambient and 100 nm at 2 K for our Cr films. Our observation of resistance fluctuation from region to region when the size of the system is in the millimetre range is due to the distribution of Q domains or Q values and corresponding local variation of the carrier density depending on the wavelength of the antiferromagnetically aligned spins. We speculate that the length scale over which the spin-density wave order parameter varies, i.e. the size of the quantized Q domains in the 506 Å thick film, is substantially larger than the mean free path since we are able to observe resistance fluctuations or mesoscopic effects in resistors whose geometrical dimensions are 1×1 mm (width of Hall bar and separation between the leads). It is known that in bulk single crystal Cr, the size of the Q domains is large, of the order of 100 μm .
Our results clearly demonstrate the direct influence of the spin arrangement on the electrical properties of an antiferromagnet—depending on the wavelength of the spin-density wave, the carrier density and consequently the resistance changes. Furthermore, in high-quality thin films, the spin-density wave is quantized, such that when the quantization value changes, the carrier density changes in a discrete manner such that it is detected as a kink in the derivative of the resistivity. The coexistence of quantized Q domains with different Q values results in the coexistence of two metals with slightly different Fermi surfaces and gives rise to a local variation of the resistance depending on the distribution of Q domains or values.
- This journal is © 2011 The Royal Society