- Email: [email protected]
Constrained phase coexistence in thin MBE-grown MnAs films on GaAs
Materials Science and Engineering B91– 92 (2002) 433– 436 www.elsevier.com/locate/mseb
Constrained phase coexistence in thin MBE-grown MnAs films on GaAs Bernd Jenichen *, Vladimir M. Kaganer, Frank Schippan, Wolfgang Braun, Lutz Da¨weritz, Klaus H. Ploog Paul-Drude-Institut fu¨r Festko¨rperelektronik, Haus6ogteiplatz 5 -7, D-10117 Berlin, Germany
Abstract We present experimental evidence for the equilibrium coexistence between crystalline phases in thin MnAs layers grown epitaxially on GaAs. The phases, which can coexist in the bulk system only at one temperature point, are simultaneously present in the heterostructures over a temperature interval of more than 20 °C, which varies with the thickness of the MnAs layer. This phase coexistence is explained by the constraint on the lateral expansion of epitaxial layers which gives rise to strain in the layer. For the thicker MnAs layer a hysteresis of the temperature dependence of the phase composition is observed. The curvature of the samples is strongly anisotropic, with the larger curvature along the c-axis of the MnAs layer, and originates from thermal contraction of the film on cooling from growth temperature. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Phase transition; Magnetic material; Epitaxy; X-ray PACS numbers: 68.35.Rh; 64.70.Kb; 61.50.Ks; 05.70.Np
1. Introduction The molecular beam epitaxy (MBE) growth of MnAs on GaAs attracts considerable interest as a way to integrate materials with ferromagnetic and semiconducting properties in epitaxially grown heterostructures [1 –5] with the aim to obtain spin injection [6]. Bulk MnAs crystals are ferromagnetic at room temperature. Slightly above room temperature, at about 40 °C, they experience a first-order transition from the ferromagnetic to the paramagnetic phase. This transition is accompanied by a large (1%) change of the lattice parameters. Recently we have found [7] that the transition in epitaxial MnAs layers on GaAs(001) proceeds in a way qualitatively different from the bulk phase transition, with a wide temperature range where the two phases coexist in the layer. The bulk structure of MnAs is well known [8–11]. Below 40 °C, it forms the ferromagnetic hexagonal phase aMnAs. At approximately 40 °C, a first-order phase transition to the paramagnetic or* Corresponding author. Tel.: + 49-30-203-77324; fax: + 49-30203-77201. E-mail address: [email protected] (B. Jenichen).
thorhombic phase bMnAs occurs. This transition is rather unusual (ferromagnetic transitions are commonly continuous at the Curie temperature). Qualitatively, the transition is explained as the result of a sufficiently strong dependence of the exchange interaction on the distance between magnetic atoms [12]. This results in a giant magnetoelastic coupling [13] and gives rise to a lattice parameter jump at the transition as large as 1%. In the present paper, we report x-ray measurements of MnAs layers on GaAs(001) in the temperature interval 5–50 °C. We find that the two phases of MnAs coexist in a broad temperature interval and that the temperature dependence of the phase composition shows a hysteresis. We measured also the sample curvature to characterize the strain in the films. During epitaxy of MnAs on GaAs(001) the (1( 100) side facet of the hexagonal hMnAs prism is attached to the GaAs(001) surface with its c-axis parallel to GaAs[1( 10]. The mismatch between the MnAs and GaAs lattices in this direction amounts to 33%. Transmission electron microscopy studies [2] show that every sixth GaAs{220} plane coincides with every fourth MnAs{0002} plane, which reduces the actual mismatch to 5%. This mismatch, as well as the mismatch along the perpendicular direction (7.7%), is released by regu-
0921-5107/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 1 ) 0 0 9 9 2 - 8
434
B. Jenichen et al. / Materials Science and Engineering B91–92 (2002) 433–436
lar arrays of misfit dislocations. The unique orientation of the epitaxial MnAs layers on GaAs(001) surface, with (1( 100) MnAs (001) GaAs and [0001] MnAs [11( 0] GaAs, is only achieved for growth on As-stable GaAs. It was checked in the present study by in situ reflection high-energy electron diffraction and subsequent ex-situ grazing incidence X-ray diffraction. At the aMnAs – bMnAs transition, the unit cell shrinks in the hexagonal plane, while the height of the prism (perpendicular to the plane of the figure) does not change.
2. Experiment The MnAs layers were grown by solid source MBE on 100 nm thick GaAs buffer layers at 250 °C with a growth rate of 19 nm h − 1 [2]. The thicknesses of the MnAs films are 60 nm (sample 1) and 100 nm (sample 2). X-ray diffractometric measurements were performed
in a high-resolution X-ray diffractometer with a temperature controlled sample stage. The sample temperature was controlled by resistive heating together with cooling by liquid nitrogen with an estimated systematic uncertainty in temperature determination of at most 2 °C. Temperature-dependent double crystal X-ray measurements were performed using a symmetrically cut four-reflection Du Mond–Bartels type Ge 220 monochromator placed at a distance of 30 cm from the sample. The angular acceptance of the detector was 0.1°. The curvature of sample 2 (due to the remaining misfit between the MnAs film and the GaAs substrate) was measured in a double crystal topographic camera equipped with a plane Si 440 collimator crystal [14] by observing the displacement of the diffraction spot (GaAs 135 or 115) over the sample surface with the change of the angle of incidence. A 135 reflection of GaAs was used to minimize the dispersion of the double crystal arrangement. The scattering plane made an angle 74° with respect to c-axis of the MnAs film. The diffraction peak intensities I( ) of the layer were fitted to a sum of Lorentzians with an adjustable exponent p, An {1+[( − n )/D ]2}p n = 1,2
I( )= %
Fig. 1. Diffraction curve ( –2q scan) of sample 1 near the GaAs(002) reflection measured at a temperature of 43.4 °C with Cu Ka1 radiation. The aMnAs(1( 100) and the bMnAs(020) reflections are resolved.
(1)
where is the glancing angle of incidence on the sample surface and n is the position of the nth peak. We obtained the good fits presented below by using a fixed value of p= 3 and taking equal widths D for the two peaks. Then, the ratio of the peak intensities An of the two peaks observed at the same temperature was equal to the ratio of the integrated intensities of the peaks.
3. Results and discussion
Fig. 2. Diffraction curve ( –2q scan) of sample 1 near the aMnAs(1( 100) and the bMnAs(020) reflections measured at different temperatures using Cu Ka1 radiation. The ratio of the integrated intensities of the peaks is changing with the sample temperature.
Fig. 1 presents the −2q X-ray diffraction scan of sample 1 at 43.4 °C. 2q denotes the detector angle with respect to the incident beam. The peak of the GaAs substrate and the peaks of the aMnAs and bMnAs phases can be distinguished clearly. When the temperature is changed, the intensities of the two MnAs layer peaks change as shown in Fig. 2. The aMnAs peak shifts to larger incidence angles on heating, which is a result of a strong thermal contration of the aMnAs phase [8–11], opposite to the usual thermal expansion. The measurements were performed in several thermal cycles of stepwise heating and cooling between 31 and 52 °C with a measurement at each temperature. The ratio of the integrated intensities from sample 1 does not show a hysteresis and does not change from one cycle to the other, which indicates an equilibrium coexistence of the two phases in this sample.
B. Jenichen et al. / Materials Science and Engineering B91–92 (2002) 433–436
435
Table 1 presents the results of the curvature measurements of sample 2 at 30 °C. The curvature radius is maximum along the direction GaAs[11( 0] (the c-axis of the MnAs film) and minimum in the perpendicular direction (the hexagonal plane of the film). The curvature radii for intermediate directions follow the Euler theorem, 1/R = sin2h/Ra + cos2h/Rc, where Ra and Rc are the curvature radii in the hexagonal plane and in the c-direction of MnAs, respectively, and h is the angle between the given direction in the surface plane and the c-axis. The measured curvature radii R are related to the lateral strain m of the film in the corresponding directions by [15,16] 1 6tf = m R t 2s Fig. 3. Temperature dependence of the fraction of aMnAs calculated from the integrated intensities of the MnAs X-ray reflections for (a) sample 1 and (b) sample 2. Triangles pointing upwards (downwards) correspond to heating (cooling) of the sample. For sample 2 a hysteresis of the phase composition is observed. Table 1 Radius of curvature R and lateral misfit [by Eq. (2)] of the MnAs layer of sample 2 at 30 °C along different directions Reflection 11( 5 315 31( 5 115
Angle to c-axis (°) 0 26 74 90
R (m) 6.78 7.77 19.4 32.82
Strain m (%) 2.7 2.3 0.94 0.55
The angle is with respect to the c-axis of MnAs (parallel to GaAs [11( 0]).
The ratio of the integrated intensities of the aMnAs and bMnAs peaks directly gives the ratio of the volume fractions of the phases in the layer since the structure factors of both reflections are almost equal. The fraction of the aMnAs phase in sample 1 is plotted as a function of temperature in Fig. 3(a). In contrast to the bulk phase transition, only a small fraction of the layer has the structure of the low-temperature aMnAs phase just below the transition. The fraction of this phase increases almost linearly with decreasing temperature in an interval of about 10 °C. Sample 2 [Fig. 3(b)] reveals a wider temperature range of phase coexistence, which amounts to 20 °C, and a hysteresis of about 5 °C. The difference in temperature behavior of the two samples remains unclear for us. We note that the hysteresis in Fig. 3(b) cannot be explained as a usual temperature hysteresis at a first order transition caused by the absence of nucleation sites for the new phase. In the present case, upon heating and cooling both phases are present at any temperature in the phase coexistence range in comparable, albeit different, amounts.
(2)
where tf and ts are the film and substrate thicknesses (tf ts). These are tf = 100 nm and ts = 330 mm for sample 2. The values of the misfit in the hexagonal plane and in the c-direction are to be compared with the variations of the lattice parameters of MnAs in the temperature interval from 250 °C (the growth temperature) to room temperature [8], which are 0.6 and 1.5%, respectively. We conclude that, at the growth temperature, the misfit in the hexagonal plane is completely relieved by a network of misfit dislocations while the residual misfit in c-direction is 1%. The temperature dependence of the curvature in the phase coexistence range was measured in the (135) reflection which minimizes the dispersion broadening (chromatic aberration) with the use of the Si(440) collimator crystal. In this way the curvature in the direction making an angle of 74° to the c-axis was measured. We found a non-reversible temperature behavior, which is not understood at present. This behavior is shown in Fig. 4(a). In the first route of cooling from 60 to 0 °C (arrow 1), the curvature radius increases from 9 to 13 m. During the subsequent heating (arrow 2), the curvature radius does not return to the initial value but becomes 7.2 m. In the second cooling route (arrow 3), the curvature radius increases again to 13 m, and after the second heating it returns to the initial high-temperature value of 9 m. Further cycling (not shown in the figure) gives intermediate values between 7 and 9 m. The accuracy of the curvature radius measurements was estimated to be better than 0.5 m, i.e. considerably smaller than the observed variations. Fig. 4(b) presents the temperature dependence of the lateral strain calculated from the measured curvature radii with the use of Eq. (2). The misfit obtained from the curvature measurements is compared with the one calculated (with the use of the Euler theorem) as m= psin2h+ mT, where p is the phase transformation strain in the hexagonal plane, mT is the thermal expansion contribution, which we took equal in the hexagonal
436
B. Jenichen et al. / Materials Science and Engineering B91–92 (2002) 433–436
plane and along the c-axis (dmT/dT =3.5 × 10 − 5) and h = 74° is the angle between the c-axis and the direction along which the curvature is measured. We took into account the mean phase transformation strain in the hexagonal plane, p =p0x, where p0 =0.012 is the phase transformation strain between aMnAs and bMnAs and x is the fraction of aMnAs which was determined directly from the data of Fig. 3. The calculated strain [full symbols and the line in Fig. 4(b)] is in a reasonable agreement with the results of the curvature measurements. The temperature hysteresis in Fig. 3(b) is also visible in the calculated misfit. Any equilibrium first-order phase transition proceeds as a discontinuous motion of the system from one minimum of its free energy to another, which becomes deeper at the transition temperature. At a first-order phase transition, the system can remain in the higher local minimum of the free energy because of the absence of nucleation centers. In particular, it was argued [12] that the magnetoelastic effect in MnAs suppresses this nucleation, which gives rise to a large temperature hysteresis of the magnetization at the phase transition in bulk material. However, as soon as the new phase is nucleated, the whole system transforms to the new phase. The magnetization in bulk MnAs changes abruptly at the phase transition, albeit with hysteresis [12,13,17]. The phase coexistence
which we observe in the layer is hence a consequence of the epitaxy.
4. Summary We have observed an equilibrium phase coexistence between the hexagonal aMnAs and orthorhombic bMnAs phases of MnAs/GaAs(001) thin heteroepitaxial layers. The fraction of the low-temperature phase aMnAs linearly increases upon cooling below the bulk phase transition temperature in an interval of 10–20 °C. The temperature range of phase coexistence and the thermal hysteresis are different for two samples investigated, which may be caused by the difference in thickness as well as quality of the samples. The presence of the phase coexistence in a large temperature interval is explained by the constraint on the changes of the lateral dimensions of the film at the phase transition imposed by the epitaxy.
Acknowledgements We gratefully acknowledge technical support by J. Richter (Humboldt Universita¨ t) and H.v. Kiedrowski, and stimulating discussions with P. Santos.
References
Fig. 4. (a) Temperature dependence of the curvature radius of sample 2 along the direction making an angle of 74° with the c-axis of MnAs. Different symbols and the arrows 1 –4 refer to data obtained by subsequent cooling and heating of the sample. (b) The strain obtained from the measured curvature radii (empty symbols) and calculated using the measured temperature dependence of the fraction of the aMnAs phase x (full symbols). Solid lines are guides for the eye only.
[1] M. Tanaka, J.P. Harbison, M.C. Park, Y.S. Park, T. Shin, G.M. Rothberg, J. Appl. Phys. 76 (1994) 6278. [2] F. Schippan, A. Trampert, L. Da¨ weritz, K.H. Ploog, J. Vac. Sci. Technol. 17 (1999) 1716. [3] F. Schippan, G. Behme, L. Da¨ weritz, K.H. Ploog, B. Dennis, K.-U. Neumann, K.R.A. Ziebeck, J. Appl. Phys. 88 (2000) 2766. [4] H. Ohno, Science 281 (1998) 951. [5] Y. Ohno, D.K. Young, B. Beschoten, F. Matsukura, H. Ohno, D.D. Awschalom, Nature 402 (1999) 790. [6] P.R. Hammar, B.R. Bennett, M.J. Yang, M. Johnson, Phys. Rev. Lett. 83 (1999) 203. [7] V.M. Kaganer, B. Jenichen, F. Schippan, W. Braun, L. Da¨ weritz, K.H. Ploog, Phys. Rev. Lett. 85 (2000) 341. [8] B.T.M. Willis, H.P. Rooksby, Proc. Phys. Soc. (Lond.) B 67 (1954) 290. [9] R.H. Wilson, J.S. Kasper, Acta Crystallogr. 17 (1964) 95. [10] (a) G.A. Govor, Fiz. Tverd. Tela 23 (1981) 1444; (b) G.A. Govor, Sov. Phys. Solid State 23 (1981) 841. [11] G.A. Govor, K. Ba¨ rner, J.-W. Schu¨ nemann, Phys. Stat. Sol. (a) 113 (1989) 403. [12] C.P. Bean, D.S. Rodbell, Phys. Rev. 126 (1962) 104. [13] V.A. Chernenko, L. Wee, P.G. McCormick, R. Street, J. Appl. Phys. 85 (1999) 7833. [14] B. Jenichen, R. Ko¨ hler, W. Mo¨ hling, J. Phys. E 21 (1988) 1062. [15] G.G. Stoney, Proc. R. Soc. A 32 (1909) 172. [16] S.N.G. Chu, A.T. Macrander, K.E. Strege, W.D. Johnston, J. Appl. Phys. 57 (1985) 249. [17] A. Zieba, K. Seite, A. Kjekshus, A.F. Andersen, Acta Chem. Scand. A 32 (1978) 173.
Constrained phase coexistence in thin MBE-grown MnAs films on GaAs Bernd Jenichen *, Vladimir M. Kaganer, Frank Schippan, Wolfgang Braun, Lutz Da¨weritz, Klaus H. Ploog Paul-Drude-Institut fu¨r Festko¨rperelektronik, Haus6ogteiplatz 5 -7, D-10117 Berlin, Germany
Abstract We present experimental evidence for the equilibrium coexistence between crystalline phases in thin MnAs layers grown epitaxially on GaAs. The phases, which can coexist in the bulk system only at one temperature point, are simultaneously present in the heterostructures over a temperature interval of more than 20 °C, which varies with the thickness of the MnAs layer. This phase coexistence is explained by the constraint on the lateral expansion of epitaxial layers which gives rise to strain in the layer. For the thicker MnAs layer a hysteresis of the temperature dependence of the phase composition is observed. The curvature of the samples is strongly anisotropic, with the larger curvature along the c-axis of the MnAs layer, and originates from thermal contraction of the film on cooling from growth temperature. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Phase transition; Magnetic material; Epitaxy; X-ray PACS numbers: 68.35.Rh; 64.70.Kb; 61.50.Ks; 05.70.Np
1. Introduction The molecular beam epitaxy (MBE) growth of MnAs on GaAs attracts considerable interest as a way to integrate materials with ferromagnetic and semiconducting properties in epitaxially grown heterostructures [1 –5] with the aim to obtain spin injection [6]. Bulk MnAs crystals are ferromagnetic at room temperature. Slightly above room temperature, at about 40 °C, they experience a first-order transition from the ferromagnetic to the paramagnetic phase. This transition is accompanied by a large (1%) change of the lattice parameters. Recently we have found [7] that the transition in epitaxial MnAs layers on GaAs(001) proceeds in a way qualitatively different from the bulk phase transition, with a wide temperature range where the two phases coexist in the layer. The bulk structure of MnAs is well known [8–11]. Below 40 °C, it forms the ferromagnetic hexagonal phase aMnAs. At approximately 40 °C, a first-order phase transition to the paramagnetic or* Corresponding author. Tel.: + 49-30-203-77324; fax: + 49-30203-77201. E-mail address: [email protected] (B. Jenichen).
thorhombic phase bMnAs occurs. This transition is rather unusual (ferromagnetic transitions are commonly continuous at the Curie temperature). Qualitatively, the transition is explained as the result of a sufficiently strong dependence of the exchange interaction on the distance between magnetic atoms [12]. This results in a giant magnetoelastic coupling [13] and gives rise to a lattice parameter jump at the transition as large as 1%. In the present paper, we report x-ray measurements of MnAs layers on GaAs(001) in the temperature interval 5–50 °C. We find that the two phases of MnAs coexist in a broad temperature interval and that the temperature dependence of the phase composition shows a hysteresis. We measured also the sample curvature to characterize the strain in the films. During epitaxy of MnAs on GaAs(001) the (1( 100) side facet of the hexagonal hMnAs prism is attached to the GaAs(001) surface with its c-axis parallel to GaAs[1( 10]. The mismatch between the MnAs and GaAs lattices in this direction amounts to 33%. Transmission electron microscopy studies [2] show that every sixth GaAs{220} plane coincides with every fourth MnAs{0002} plane, which reduces the actual mismatch to 5%. This mismatch, as well as the mismatch along the perpendicular direction (7.7%), is released by regu-
0921-5107/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 1 ) 0 0 9 9 2 - 8
434
B. Jenichen et al. / Materials Science and Engineering B91–92 (2002) 433–436
lar arrays of misfit dislocations. The unique orientation of the epitaxial MnAs layers on GaAs(001) surface, with (1( 100) MnAs (001) GaAs and [0001] MnAs [11( 0] GaAs, is only achieved for growth on As-stable GaAs. It was checked in the present study by in situ reflection high-energy electron diffraction and subsequent ex-situ grazing incidence X-ray diffraction. At the aMnAs – bMnAs transition, the unit cell shrinks in the hexagonal plane, while the height of the prism (perpendicular to the plane of the figure) does not change.
2. Experiment The MnAs layers were grown by solid source MBE on 100 nm thick GaAs buffer layers at 250 °C with a growth rate of 19 nm h − 1 [2]. The thicknesses of the MnAs films are 60 nm (sample 1) and 100 nm (sample 2). X-ray diffractometric measurements were performed
in a high-resolution X-ray diffractometer with a temperature controlled sample stage. The sample temperature was controlled by resistive heating together with cooling by liquid nitrogen with an estimated systematic uncertainty in temperature determination of at most 2 °C. Temperature-dependent double crystal X-ray measurements were performed using a symmetrically cut four-reflection Du Mond–Bartels type Ge 220 monochromator placed at a distance of 30 cm from the sample. The angular acceptance of the detector was 0.1°. The curvature of sample 2 (due to the remaining misfit between the MnAs film and the GaAs substrate) was measured in a double crystal topographic camera equipped with a plane Si 440 collimator crystal [14] by observing the displacement of the diffraction spot (GaAs 135 or 115) over the sample surface with the change of the angle of incidence. A 135 reflection of GaAs was used to minimize the dispersion of the double crystal arrangement. The scattering plane made an angle 74° with respect to c-axis of the MnAs film. The diffraction peak intensities I( ) of the layer were fitted to a sum of Lorentzians with an adjustable exponent p, An {1+[( − n )/D ]2}p n = 1,2
I( )= %
Fig. 1. Diffraction curve ( –2q scan) of sample 1 near the GaAs(002) reflection measured at a temperature of 43.4 °C with Cu Ka1 radiation. The aMnAs(1( 100) and the bMnAs(020) reflections are resolved.
(1)
where is the glancing angle of incidence on the sample surface and n is the position of the nth peak. We obtained the good fits presented below by using a fixed value of p= 3 and taking equal widths D for the two peaks. Then, the ratio of the peak intensities An of the two peaks observed at the same temperature was equal to the ratio of the integrated intensities of the peaks.
3. Results and discussion
Fig. 2. Diffraction curve ( –2q scan) of sample 1 near the aMnAs(1( 100) and the bMnAs(020) reflections measured at different temperatures using Cu Ka1 radiation. The ratio of the integrated intensities of the peaks is changing with the sample temperature.
Fig. 1 presents the −2q X-ray diffraction scan of sample 1 at 43.4 °C. 2q denotes the detector angle with respect to the incident beam. The peak of the GaAs substrate and the peaks of the aMnAs and bMnAs phases can be distinguished clearly. When the temperature is changed, the intensities of the two MnAs layer peaks change as shown in Fig. 2. The aMnAs peak shifts to larger incidence angles on heating, which is a result of a strong thermal contration of the aMnAs phase [8–11], opposite to the usual thermal expansion. The measurements were performed in several thermal cycles of stepwise heating and cooling between 31 and 52 °C with a measurement at each temperature. The ratio of the integrated intensities from sample 1 does not show a hysteresis and does not change from one cycle to the other, which indicates an equilibrium coexistence of the two phases in this sample.
B. Jenichen et al. / Materials Science and Engineering B91–92 (2002) 433–436
435
Table 1 presents the results of the curvature measurements of sample 2 at 30 °C. The curvature radius is maximum along the direction GaAs[11( 0] (the c-axis of the MnAs film) and minimum in the perpendicular direction (the hexagonal plane of the film). The curvature radii for intermediate directions follow the Euler theorem, 1/R = sin2h/Ra + cos2h/Rc, where Ra and Rc are the curvature radii in the hexagonal plane and in the c-direction of MnAs, respectively, and h is the angle between the given direction in the surface plane and the c-axis. The measured curvature radii R are related to the lateral strain m of the film in the corresponding directions by [15,16] 1 6tf = m R t 2s Fig. 3. Temperature dependence of the fraction of aMnAs calculated from the integrated intensities of the MnAs X-ray reflections for (a) sample 1 and (b) sample 2. Triangles pointing upwards (downwards) correspond to heating (cooling) of the sample. For sample 2 a hysteresis of the phase composition is observed. Table 1 Radius of curvature R and lateral misfit [by Eq. (2)] of the MnAs layer of sample 2 at 30 °C along different directions Reflection 11( 5 315 31( 5 115
Angle to c-axis (°) 0 26 74 90
R (m) 6.78 7.77 19.4 32.82
Strain m (%) 2.7 2.3 0.94 0.55
The angle is with respect to the c-axis of MnAs (parallel to GaAs [11( 0]).
The ratio of the integrated intensities of the aMnAs and bMnAs peaks directly gives the ratio of the volume fractions of the phases in the layer since the structure factors of both reflections are almost equal. The fraction of the aMnAs phase in sample 1 is plotted as a function of temperature in Fig. 3(a). In contrast to the bulk phase transition, only a small fraction of the layer has the structure of the low-temperature aMnAs phase just below the transition. The fraction of this phase increases almost linearly with decreasing temperature in an interval of about 10 °C. Sample 2 [Fig. 3(b)] reveals a wider temperature range of phase coexistence, which amounts to 20 °C, and a hysteresis of about 5 °C. The difference in temperature behavior of the two samples remains unclear for us. We note that the hysteresis in Fig. 3(b) cannot be explained as a usual temperature hysteresis at a first order transition caused by the absence of nucleation sites for the new phase. In the present case, upon heating and cooling both phases are present at any temperature in the phase coexistence range in comparable, albeit different, amounts.
(2)
where tf and ts are the film and substrate thicknesses (tf ts). These are tf = 100 nm and ts = 330 mm for sample 2. The values of the misfit in the hexagonal plane and in the c-direction are to be compared with the variations of the lattice parameters of MnAs in the temperature interval from 250 °C (the growth temperature) to room temperature [8], which are 0.6 and 1.5%, respectively. We conclude that, at the growth temperature, the misfit in the hexagonal plane is completely relieved by a network of misfit dislocations while the residual misfit in c-direction is 1%. The temperature dependence of the curvature in the phase coexistence range was measured in the (135) reflection which minimizes the dispersion broadening (chromatic aberration) with the use of the Si(440) collimator crystal. In this way the curvature in the direction making an angle of 74° to the c-axis was measured. We found a non-reversible temperature behavior, which is not understood at present. This behavior is shown in Fig. 4(a). In the first route of cooling from 60 to 0 °C (arrow 1), the curvature radius increases from 9 to 13 m. During the subsequent heating (arrow 2), the curvature radius does not return to the initial value but becomes 7.2 m. In the second cooling route (arrow 3), the curvature radius increases again to 13 m, and after the second heating it returns to the initial high-temperature value of 9 m. Further cycling (not shown in the figure) gives intermediate values between 7 and 9 m. The accuracy of the curvature radius measurements was estimated to be better than 0.5 m, i.e. considerably smaller than the observed variations. Fig. 4(b) presents the temperature dependence of the lateral strain calculated from the measured curvature radii with the use of Eq. (2). The misfit obtained from the curvature measurements is compared with the one calculated (with the use of the Euler theorem) as m= psin2h+ mT, where p is the phase transformation strain in the hexagonal plane, mT is the thermal expansion contribution, which we took equal in the hexagonal
436
B. Jenichen et al. / Materials Science and Engineering B91–92 (2002) 433–436
plane and along the c-axis (dmT/dT =3.5 × 10 − 5) and h = 74° is the angle between the c-axis and the direction along which the curvature is measured. We took into account the mean phase transformation strain in the hexagonal plane, p =p0x, where p0 =0.012 is the phase transformation strain between aMnAs and bMnAs and x is the fraction of aMnAs which was determined directly from the data of Fig. 3. The calculated strain [full symbols and the line in Fig. 4(b)] is in a reasonable agreement with the results of the curvature measurements. The temperature hysteresis in Fig. 3(b) is also visible in the calculated misfit. Any equilibrium first-order phase transition proceeds as a discontinuous motion of the system from one minimum of its free energy to another, which becomes deeper at the transition temperature. At a first-order phase transition, the system can remain in the higher local minimum of the free energy because of the absence of nucleation centers. In particular, it was argued [12] that the magnetoelastic effect in MnAs suppresses this nucleation, which gives rise to a large temperature hysteresis of the magnetization at the phase transition in bulk material. However, as soon as the new phase is nucleated, the whole system transforms to the new phase. The magnetization in bulk MnAs changes abruptly at the phase transition, albeit with hysteresis [12,13,17]. The phase coexistence
which we observe in the layer is hence a consequence of the epitaxy.
4. Summary We have observed an equilibrium phase coexistence between the hexagonal aMnAs and orthorhombic bMnAs phases of MnAs/GaAs(001) thin heteroepitaxial layers. The fraction of the low-temperature phase aMnAs linearly increases upon cooling below the bulk phase transition temperature in an interval of 10–20 °C. The temperature range of phase coexistence and the thermal hysteresis are different for two samples investigated, which may be caused by the difference in thickness as well as quality of the samples. The presence of the phase coexistence in a large temperature interval is explained by the constraint on the changes of the lateral dimensions of the film at the phase transition imposed by the epitaxy.
Acknowledgements We gratefully acknowledge technical support by J. Richter (Humboldt Universita¨ t) and H.v. Kiedrowski, and stimulating discussions with P. Santos.
References
Fig. 4. (a) Temperature dependence of the curvature radius of sample 2 along the direction making an angle of 74° with the c-axis of MnAs. Different symbols and the arrows 1 –4 refer to data obtained by subsequent cooling and heating of the sample. (b) The strain obtained from the measured curvature radii (empty symbols) and calculated using the measured temperature dependence of the fraction of the aMnAs phase x (full symbols). Solid lines are guides for the eye only.
[1] M. Tanaka, J.P. Harbison, M.C. Park, Y.S. Park, T. Shin, G.M. Rothberg, J. Appl. Phys. 76 (1994) 6278. [2] F. Schippan, A. Trampert, L. Da¨ weritz, K.H. Ploog, J. Vac. Sci. Technol. 17 (1999) 1716. [3] F. Schippan, G. Behme, L. Da¨ weritz, K.H. Ploog, B. Dennis, K.-U. Neumann, K.R.A. Ziebeck, J. Appl. Phys. 88 (2000) 2766. [4] H. Ohno, Science 281 (1998) 951. [5] Y. Ohno, D.K. Young, B. Beschoten, F. Matsukura, H. Ohno, D.D. Awschalom, Nature 402 (1999) 790. [6] P.R. Hammar, B.R. Bennett, M.J. Yang, M. Johnson, Phys. Rev. Lett. 83 (1999) 203. [7] V.M. Kaganer, B. Jenichen, F. Schippan, W. Braun, L. Da¨ weritz, K.H. Ploog, Phys. Rev. Lett. 85 (2000) 341. [8] B.T.M. Willis, H.P. Rooksby, Proc. Phys. Soc. (Lond.) B 67 (1954) 290. [9] R.H. Wilson, J.S. Kasper, Acta Crystallogr. 17 (1964) 95. [10] (a) G.A. Govor, Fiz. Tverd. Tela 23 (1981) 1444; (b) G.A. Govor, Sov. Phys. Solid State 23 (1981) 841. [11] G.A. Govor, K. Ba¨ rner, J.-W. Schu¨ nemann, Phys. Stat. Sol. (a) 113 (1989) 403. [12] C.P. Bean, D.S. Rodbell, Phys. Rev. 126 (1962) 104. [13] V.A. Chernenko, L. Wee, P.G. McCormick, R. Street, J. Appl. Phys. 85 (1999) 7833. [14] B. Jenichen, R. Ko¨ hler, W. Mo¨ hling, J. Phys. E 21 (1988) 1062. [15] G.G. Stoney, Proc. R. Soc. A 32 (1909) 172. [16] S.N.G. Chu, A.T. Macrander, K.E. Strege, W.D. Johnston, J. Appl. Phys. 57 (1985) 249. [17] A. Zieba, K. Seite, A. Kjekshus, A.F. Andersen, Acta Chem. Scand. A 32 (1978) 173.