The Influence of Magnetic Domain Walls on Longitudinal and Transverse Magnetoresistance in

The In?uence of Magnetic Domain Walls on Longitudinal and Transverse Magnetoresistance in Tensile Strained (Ga,Mn)As Epilayers

arXiv:0705.3213v1 [cond-mat.mtrl-sci] 22 May 2007

G. Xiang and N. Samarth? Physics Department & Materials Research Institute, Penn State University, University Park PA 16802

We present a theoretical analysis of recent experimental measurements of magnetoresistance in Ga1?x Mnx As epilayers with perpendicular magnetic anisotropy. The model reproduces the ?eldantisymmetric anomalies observed in the longitudinal magnetoresistance in the planar geometry (magnetic ?eld in the epilayer plane and parallel to the current density), as well as the unusual shape of the accompanying transverse magnetoresistance. The magnetoresistance characteristics are attributed to circulating currents created by the presence of magnetic domain walls.
PACS numbers: 75.50.Pp,75.70.Ak,75.50.-d




Contemporary developments in spintronics1 have generated much interest in the interplay between electrical transport and magnetic domain walls (DWs) in ferromagnetic metals2,3,4,5 and in ferromagnetic semiconductors.6,7,8,9 It is of particular relevance in this context to study the relationship between spin transport and DWs in the “canonical” ferromagnetic semiconductor Ga1?x Mnx As10,11,12 because this material is a model system for proof-of-concept semiconductor spintronic devices.13,15 Indeed, recent experimental studies of Ga1?x Mnx As devices have demonstrated that the presence of DWs directly in?uences measurements of the longitudinal magnetoresistance (Rxx (H )) due to contributions from the transverse magnetoresistance (Rxy (H )): in samples with in-plane magnetic anisotropy, this arises because of the giant planar Hall e?ect,6,7 while in samples with perpendicular magnetic anisotropy, the admixture is created by the anomalous Hall e?ect.16 In the latter case, Rxx (H ) shows remarkable ?eld-antisymmetric anomalies in the planar geometry when the external magnetic ?eld (H ) is applied parallel to the current density (j ) and perpendicular to the magnetic easy axis (? z ). This is similar to observations of a ?eld-antisymmetric magnetoresistance in metallic ferromagnetic multilayers with perpendicular anisotropy, but we note that in the metallic case the external ?eld was applied along the easy axis.5 Theoretical modeling has shown how circulating currents in the vicinity of a DW can result in an admixture of Rxy (H ) in the measurement of Rxx (H ) for H applied along the easy axis of a ferromagnetic thin ?lm.5,17 . In this paper, we extend these calculations to a di?erent experimental geometry where H is applied along a hard axis of a ferromagnetic thin ?lm with perpendicular magnetic anisotropy. The magnetization reversal process in this case involves a 3 dimensional process rather than a 1 or 2 dimensional one as in the calculations published earlier. Our theoretical analysis is aimed at explaining the unusual hard axis magnetoresistance observed in tensile-strained Ga1?x Mnx As epilayers with perpendicular magnetic anisotropy.16 When we measure Rxx (H ) and Rxy (H ) in tensile-strained Ga1?x Mnx As in the planar geometry (H ||j ⊥z ?), we observe the following characteristics: 1. The background longitudinal magnetoresistance is symmetric with respect to the direction of the magnetic ?eld H but there are resistance “spikes” with an antisymmetric deviation ?R(H ) = ??R(?H ). These anomalies arise when the magnetization reverses. 2

2. The transverse magnetoresistance shows a hysteresis loop with an unusual shape, with Rxy (H ) = 0 at the ?eld where Rxx (H ) shows a maximum or minimum spike. The model presented in this paper shows that our observations in tensile-strained Ga1?x Mnx As can be explained by the same concepts used to understand the magnetoresistance in unstrained Ga1?x Mnx As with planar anisotropy6,7,17 and in metallic multilayers with perpendicular magnetic anisotropy:5 circulating currents near a DW located between the voltage probes produce Hall e?ect contributions to Rxx . This e?ect manifests itself particularly during magnetization reversal via DW nucleation and propagation. In our model, we treat the device as a rectangular Hall bar of width w and thickness t (Fig. 1). The ?lm is in the xy plane and the length is along the x axis. We begin with the assumption that a DW is positioned at x = 0 in the yz plane, separating the thin ?lm into two domains (i = 1 and i = 2 to the left and right of the DW, respectively) with opposite magnetization. In each domain, the electric ?eld and the current density are related by Ei = ρi ji , where the resistivity tensor is given by: ? ? ρ ?ρH ? ρ1 = ? ρH ρ ? ? ρ ρH ? ρ2 = ? ?ρH ρ



In the above equations, ρ and ρH are the diagonal and o?-diagonal components of the resistivity tensor; note that the latter change sign between the two domains because of the anomalous Hall e?ect. Equations (1) and (2) may then be written as: ? ? ? ? ? ? ??Vi ρ ρH sgn(x) j ? ?x ? = ? ? × ? xi ? ??Vi ?ρH sgn(x) ρ jyi ?y and current density satisfy the following boundary conditions : ?2 V i = 0 , jxi (±∞, y ) = j0 , jyi (±∞, y ) = 0, jyi (0, 0) = 0, jyi (0, w ) = 0. 3 (4) (5) (6) (7) (8)


Assuming no static charges accumulate in the Ga1?x Mnx As sample, the electric potential

Also, the continuities of the electric potential and x component of current at the interface require: jx1 (0, y ) = jx2 (0, y ), V1 (0, y ) = V2 (0, y ). (9) (10)

Using the above boundary conditions, we solve for the electric potential and the current density in the limit β =
ρH ρ

<< 1 to obtain:18,19

V1 (x, y ) = V10 ? j0 ρ(x + βy ) ? ρ
n=1 ∞

An exp(

nπy nπy nπx )[cos( ) + sin( )] w w w nπx nπy nπy )[cos( ) + sin( )] w w w

(11) (12) (13) (14) (15) (16)

V2 (x, y ) = V20 ? j0 ρ(x ? βy ) + ρ

An exp(?

jx1 = j0 + jy1 = ? jx2 jy2 where π w

π w

nAn exp(

nπx nπy ) cos( ) w w

nAn exp(
n=1 ∞

nπx nπy ) sin( ) w w nπx nπy ) cos( ) w w

π = j0 + w π = + w

nAn exp(?

nAn exp(?

nπx nπy ) sin( ) w w

A1 = (j0 w/π )(4β/π )[1 ? 0.205(4β/π )2 + ...], A2 = (j0 w/4π )(4β/π )[(4β/π ) ? 0.412(4β/π )3 + ...], A3 = (j0 w/9π )(4β/π )[1 + 0.297(4β/π )2 + ...], 4 A4 = (j0 w/16π )(4β/π )[ (4β/π ) ? 0.397(4β/π )3 + ...], 3 2 V20 ? V10 = ?j0 wβρ{1 + (4/π )(4β/π ) × [1.052 ? 0.181(4β/π )2 + ...]}.

(17) (18) (19) (20) (21)

We note that the transverse ?eld Ey due to the Hall e?ect changes sign from ?∞ to +∞. By symmetry, Ey vanishes in the vicinity of x = 0 where the domain wall is located. The Lorentz force is then not balanced due to the lack of an electric force eEy , and the carriers are de?ected towards one side of the sample, causing a nonuniform circulating current around the DW at x = 0. 4



Using the above model, we quantitatively calculate the Hall voltage at x when the domain wall is located at x = 0. VH (x) = Vi (x, 0) ? Vi (x, w ). To ?rst order in β , the Hall voltage is 8 VH (x) = (ρH j0 w )sgn(x)(1 ? 2 π e?πn|x|/w ). 2 n n=odd



More generally, when the domain wall is located at x = xDW , 8 VH (x) = (ρH j0 w )sgn(x ? xDW )(1 ? 2 π e?πn|x?xDW |/w ). 2 n n=odd


We now use eq. 24 to calculate Rxy (H ) in the presence of an in-plane external magnetic ?eld. Although H is nominally in the xy plane during the experiment, in practice, there is always a slight misalignment towards z ? characterized by an angle δ 1? between H and j .16 We divide our discussion of Rxy (H ) into four di?erent regimes (see Fig. 2): 1. In regime I, the sample is in a single domain state while we sweep the external magnetic ?eld from -2 T to HI = ?5400 Oe. The in-plane ?eld is strong enough that the magnetization is completely aligned in the xy plane and hence Rxy = 0. 2. In regime II, the sample is still in a single domain state. As the in-plane magnetic ?eld is further reduced (|H | < |HI |) the magnetization of the sample starts to rotate toward the perpendicular direction, with the symmetry being broken by the slight misalignment. We assume the magnetization rotates coherently as a sine function of the external ?eld, until the magnetization is totally aligned along +? z . During this process, the angle α between the magnetization and the xy plane is: α = |H | π ), (1 ? 2 HI (25)

and the Hall resistance measured at x = 0 is given by: Rxy (H ) = RH 0 sin α, where RH 0 is the Hall resistance at zero ?eld. 5 (26)

3. In region III, the external ?eld changes sign and is swept from 0 to HI . The z component of the external ?eld is now opposite to the magnetization of the sample, initiating magnetization reversal through the nucleation and propagation of DWs. For simplicity, we assume that a single DW starts from one end of the device and moves to the other end. Given the length of the actual Hall bar L = 1500 ?m, the domain wall is located at xDW =
L . 2

From the experimental data, the ?eld at which Rxy (HC ) = 0
L 2

marks the point at which xDW = 0. We assume that the position of the DW varies linearly with H ,5,17 so that xDW = the external in-plane ?eld. Then, the Hall resistance measured by the Hall probe at x = 0 in region III is π 8 |H | Rxy (H ) = RH 0 sin (1 ? )sgn(?xDW )(1 ? 2 2 HI π e?πn|xDW |/w ). 2 n n=odd


H ? HC . HC

Further, as shown above in eqs. (25)

and (26), Rxy (H ) also changes because the out-of-plane magnetization rotates with


4. In region IV, the sample is in a single-domain state again. The external ?eld (> HI ) forces the magnetization to be fully aligned in the xy plane and the Hall resistance stays at zero. We show a representative ?t in ?gure 2. An identical calculation can be applied to the process when the magnetic ?eld sweeps from positive to negative (not shown). The calculated transverse MR is in good agreement with the experimental results, indicating that the model used here is appropriate.



We now calculate Rxx (H ), noting that the value of Rxx depends on the relative locations of the domain wall and the electrodes. For two electrodes placed at points x = l/2 (l = 450 ?m) along the lower edge (y = 0), there exist three possible cases of xDW . To ?rst order in β , Rxx is given by:


1. xDW < ?l/2, Rxx
l l V 2 (? 2 , 0) ? V2 ( 2 , 0) = j0 wt

= RS + RS ( ( ρH w 4 )I = ρ l π2

)I exp(? nπ w
l 2

(28) + xDW ) ? exp(? nπ w n2
l 2

? xDW )



where RS = ρl/(wt), t is the thickness of the sample, and w is the width of the Hall bar. 2. ?l/2 < xDW < l/2 Rxx =
l l V 1 (? 2 , 0) ? V2 ( 2 , 0) j0 wt

= RS ? RS [( ( 3. xDW > l/2 Rxx )II ρH w 4 = ρ l π2

)II ? 1] exp(? nπ w
l 2

(30) + xDW ) + exp(? nπ w n2
l 2

? xDW )



l l V 1 (? 2 , 0) ? V1 ( 2 , 0) = j0 wt

= RS ? RS (

l 2

(32) ? xDW ) ? exp(? nπ w n2
l 2

( Note that


exp(? nπ ρH w 4 w = ρ l π 2 n=odd

+ xDW )


ρH w ρH wt = ρ l t ρl = Rxy (H )/RS


Again, recall that Rxy (H ) varies with the out-of-plane rotation of the magnetization, as discussed in the last section. |H | π ), Rxy (H ) = RH 0 sin (1 ? 2 HI Overall, ?Rxx = (Rxx ? RS )/(RS ) is given by ?R = RH 0 π |H | 4 l sin (1 ? )[ 2 ( ) ? θ( ? |xDW |)] RS 2 HI π 2 ∞ nπ l l exp(? w 2 + xDW ) ? sgn(|xDW | ? 2 ) exp(? nπ w = 2 n n=odd 7 (35)

l 2

? xDW )



where xDW =

L 2


H ? HC . HC

The calculated ?Rxx is shown as the solid line in the ?gure 3 and is qualitatively in agreement with the experimental results. We speculate that the magnitude of the calculated anomalous spike is higher than that of the measured one because the measured magnetoresistance is sensitive to the actual structure of the DW. In our highly idealized model, the DW is assumed to be a perfect plane in the yz plane, perpendicular to the long edge of the Hall bar. In reality, the DW structure is likely to be far more complicated, as suggested by recent magneto-optical imaging of the easy axis magnetization reversal process of tensile-strained Ga1?x Mnx As samples.20 The reasonable agreement between the model and experiment is hence quite surprising and better than might be anticipated. Finally, it is important to compare these results with the antisymmetric anomalies reported in the metallic multilayer samples with perpendicular magnetic anisotropy. In that case, the antisymmetric magnetoresistance s observed for magnetic ?elds applied along the easy axis (perpendicular to the sample plane). However, in the Ga1?x Mnx As samples studied here, we do not observe any such antisymmetric magnetoresistance in the perpendicular geometry.16 Noting that the coercive ?eld for easy axis magnetization reversal in tensile-strained Ga1?x Mnx As is very small (? 20 Oe), we speculate that DW nucleation and propagation occurs very rapidly during easy axis magnetization reversal. The absence of the MR anomalies in the perpendicular geometry can then be attributed to the lack of experimental resolution in current experiments. In contrast, in the planar geometry, the e?ective coercive ?eld for magnetization reversal is much larger (? 2200 Oe) so that magnetization reversal occurs adiabatically with a very slow nucleation and propagation of DWs across the sample. Finally, we note again that magneto-optical images of the easy axis magnetization reversal process show a very complicated domain nucleation and propagation that probably statistically averages out the contributions to Rxx from circulating currents.



In summary, we have shown that – in the planar geometry – the key features of the unusual longitudinal and transverse magnetoresistance in tensile-strained Ga1?x Mnx As can be readily explained by extending an earlier model applied to easy axis magnetoresistance in ferromagnets. Since the current model addresses the hard axis magnetoresistance in samples 8

with perpendicular anisotropy, the interplay between the magnetization reversal process and magnetoresistance is more complicated than in earlier studies. Our model reveals that there are two di?erent magnetic states involved in the sample during the magnetic ?eld sweep: one is a single-domain state, when the in-plane magnetic ?eld is strong enough to align the magnetization in the xy plane, or when the magnitude of the magnetic ?eld decreases from maximum to zero and the magnetization of the sample spontaneously rotates from the xy plane to the z axis. The other state is a two-domain state, ideally separated by a single DW. When the magnetic ?eld changes sign, the z component of the in-plane ?eld, due to the unintentional misalignment, initiates DW nucleation and propagation in the sample. The observed ?eld-antisymmetric anomalies and unusual Hall loops then arise from AHE contributions when a DW is located in between the voltage probes. This research has been supported by grant numbers ONR N0014-05-1-0107 and NSF DMR-0305238. References

? 1 2 3 4 5 6

Electronic address: nsamarth@psu.edu S. Wolf et al., Science 294, 1488 (2001). A. D. Kent, J. Yu, U. Rudiger, and S. S. P. Parkin, J. Phys. Condens. Matter 13, R461 (2001). T. Ono et al., Science 284, 468 (1999). D. A. Allwood et al., Science 296, 2003 (2002). X. M. Cheng et al., Phys. Rev. Lett. 94, 017203 (2005). H. X. Tang, R. K. Kawakami, D. D. Awschalom, and M. L. Roukes, Phys. Rev. Lett. 90, 107201 (2003).

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H. X. Tang et al., Nature 431, 52 (2004). A. W. Holleitner et al., Appl. Phys. Lett. 85, 5622 (2004). M. Yamanouchi, D. Chiba, F. Matsukura and H. Ohno, Nature 428, 539 (2004). H. Ohno in Semiconductor Spintronics and Quantum Computation, D. D. Awschalom, D. Loss, and N. Samarth (Eds.), (Springer-Verlag, Berlin, 2002).


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Figure Captions Fig. 1. Schematic of a thin ?lm with one 180o domain wall at x = 0. The ?lm is assumed to be in?nitely long. The width and the thickness of the ?lm are w and t, respectively. Fig. 2. Measured (solid line) and calculated (open circles) transverse magnetoresistance in a tensile-strained Ga1?x Mnx As epilayer at T = 80K. The cartoons show the magnetization con?guration with respect to the Hall bar at di?erent ?elds. The circle locates the ?eld at which the Hall voltage is zero, indicating that the DW is located between midway between a pair of voltage probes. Fig. 3. Measured (solid line) and calculated (open circles) longitudinal magnetoresistance in a tensile-strained Ga1?x Mnx As epilayer at T = 80K. The cartoons show the magnetization con?guration with respect to the Hall bar at di?erent ?elds.





w t Domain 1 z DW y x Domain 2




M xxxxx


80 K
uuu uuu xxx xxx

Rxy (Rbe) (:)


f M

e d M

a b c

-50 -10000



H (Oe)


uuu uuu

xxx xxx



80 K

'Rxx ('Rac) (:)


xxxxx xxxxx

f M a b c e d M








H (Oe)