- Email: [email protected]
Phase separation in diluted magnetic semiconductor quantum wells
Physica E 12 (2002) 388 – 390
www.elsevier.com/locate/physe
Phase separation in diluted magnetic semiconductor quantum wells L. Brey ∗ , F. Guinea Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid, Spain
Abstract The phase diagram of diluted magnetic semiconductor quantum wells is studied. We *nd that the interaction between the carriers in the hole gas can lead to *rst-order ferromagnetic transitions. These transitions can induce phase separation in double quantum wells systems. We make some experimental predictions for observing these *rst-order phase transitions. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 75.50.Pp; 73.61.Ey; 75.70.Cn Keywords: Magnetic semiconductors; Quantum wells
Recent advances in the MBE growing techniques have made possible the fabrication of Mn-based diluted magnetic semiconductors (DMS) with a rather high ferromagnetic–paramagnetic critical temperature Tc [1]. In semiconductors, it is possible to modulate spatially the density of carriers by changing the doping pro*les but in DMS it is also suitable to vary the magnetic order of the carriers by changing the magnetic ion densities. The combination of these two possibilities opens a rich *eld of applications for these materials. The high critical temperature DMSs have a high concentration, c, of Mn2+ ions, randomly located. The itinerant carriers in the Ga1−x Mnx As systems are holes and their density, c∗ , is much smaller than the magnetic ion density. In a doped semiconductor, the spin S = 52 Mn2+ ions feel a long-range ferromagnetic interaction created by the coupling mediated by the itinerant spin polarized carriers [2–5]. ∗
Corresponding author. Fax: +34-91-372-0623. E-mail address: [email protected] (L. Brey).
In this work, we investigate the nature of the phase diagram of diluted magnetic semiconductor quantum wells, with emphasis on the existence of abrupt transitions for experimentally accessible parameters [6]. We analyze quantum wells made of GaMnAs and with thickness w. The system is described by the following Hamiltonian: H = Hh + J
I; i
SI · si (ri − RI );
(1)
where Hh is the part of the Hamiltonian which describe the itinerant holes. It is the sum of the kinetic energy of the holes and the hole–hole interaction energy. The in-plane motion of the holes is approximated by a single parabolic band of eEective mass m∗ . The interaction between the electrical carriers is described with the local spin density approximation (LSDA) [7]. The last term is the antiferromagnetic exchange interaction between the spin of the Mn2+ ions located at RI and the spins , ˜si , of the itinerant carriers. The interaction between ions mediated by the conduction holes is of long range. Thus, we will assume that the thermal distribution of the orientation of the Mn spins is that
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 3 1 7 - 4
L. Brey, F. Guinea / Physica E 12 (2002) 388 – 390
20
Paramagnetic 15
T(K)
induced by an eEective *eld, due to the holes, which should be calculated self-consistently. The aim of this work is to study the magnetic phase diagram of DMS quantum wells with a 2D density of holes n2D = c∗ w. For doing that we write the free energy F as a function of the carrier spin polarization, = n↑ − n↓ =n2D , which is the order parameter. Here, n is the 2D density of carriers with spin . The critical temperatures for the ferromagnetic to paramagnetic transition in DMS is typically smaller than 100 K, and for these temperatures we can consider that the electron gas is degenerate. Hence, the only temperature dependence in F is due to thermal Huctuations of the Mn spins. Treating the holes in the LSDA the free energy per unit area takes the form ˜2 F = Fions + ∗ n22D (1 + 2 ) + Exc (n2D ; ): (2) m 2 Here, Exc is the hole exchange correlation energy and Fions is the contribution of ion spins to the free energy: sinh [h(S + 1=2)=T ] Fions = −Tcw log ; (3) sinh (h=2T ) J n2D : (4) with h = 2 w In obtaining Eq. (3) we have assumed that the hole wave function in the z-direction has the form w−1=2 . The phase diagram with parameters J =0:15 eV nm3 , ion concentration c = 1 nm−3 and w = 10 nm, is shown in Fig. 1. We include a single hole band of eEective mass m|| = 0:11me , and a dielectric constant 0 = 12:2 [5]. We use the expression given by Vosko et al. [7] for Exc . The phase diagram is obtained by minimizing the free energy equation (2) with respect to , for different values of T . A *nite value of indicates a ferromagnetic state whereas = 0 corresponds to the paramagnetic phase. An abrupt=continuous change, as a function of T , in the value of is the mark of a *rst=second-order transition. In the high density region, the dashed line which represents the second-order phase transition agrees with that obtained from the divergences of the magnetic susceptibility [5]. The main novelty of our calculation is the identi*cation of a *rst-order transition to a fully polarized state at intermediate densities. This transition takes place at higher T than that at which the magnetic susceptibility of the system diverges. The
389
10
5
0
Ferromagnetic
0.2
0.4
0.6
0.8
kF (nm-1)
Fig. 1. Phase diagram of a DMS quantum well, using the parameters described in the text. Full and broken lines represent, respectively, *rst- and second-order phase transitions between ferromagnetic (F) and paramagnetic (P) phases. Upon lowering the temperature, we obtain a continuous transition between the P and the F phases, followed by an abrupt transition between two F phases with diEerent values of . As commented in the text, this last transition is probably spurious.
existence of a *rst-order transition between the paramagnetic and the ferromagnetic phases implies that, if the chemical potential is kept *xed, the carrier density will change abruptly. Conversely, if the average hole density is *xed, a region where inhomogeneous solutions are stable will appear near the transition, leading to phase separation. At higher densities, when a continuous transition between = 0 and = 0 occurs, we also *nd a *rst-order phase transition between the partially polarized system, ¡ 1, and the fully polarized phase, = 1. We believe that this is a spurious transition related with the LSDA [6]. The existence of discontinuous transitions leads to the possibility of phase separation. The Maxwell construction gives the region where two coexisting phases (paramagnetic and ferromagnetic) of diEerent densities are energetically favored. From the dependence on densities of the ground state free energy we *nd a region of phase separation near the line of *rst-order phase transitions. As the two phases have diEerent hole densities there are, associated with the phase separation, electrostatic eEects which need to be included. These eEects tend to reduce the size of the phases, and made diJcult the experimental observation of the phase separation. We propose that phase separation
390
L. Brey, F. Guinea / Physica E 12 (2002) 388 – 390
can occur in a double quantum well system made with DMS. Suppose two wells, each of them with a nominal 2D density of holes n. In the phase separation state, the right well is ferromagnetic and has a density n(1 + ) and the left well is paramagnetic and has a density n(1 − ). The system is unstable against phase separation when the following condition is satis*ed: 2F[n] ¿ (1 + )F[n(1 + )] + (1 − )F[n(1 − )] + EH (n; );
(5)
where F(n0 ) is the minimum free energy at density n0 , and EH (n0 ; ) is the capacitive energy due to the imbalance of the charge density in the bilayer, w 2 2e2 n2 d− : EH (n; ) = (6) 4 3 Here d is the distance between the center of the wells. The existence of a discontinuity in the chemical potential in the *rst-order phase transitions, made it possible that at a given T Eq. (5) can be satis*ed in a range of densities near the ferromagnetic–paramagnetic critical density nc . We have checked numerically that this range of densities is very narrow and the charge transfer small, in such a way that inequality (5) can be written as 0 ¿ nc ( + − − ) + nc 2 ( + + − ) + EH (nc ; ); (7)
and this can signi*cantly change the transport properties of the double quantum well system. In a bilayer system, the *rst-order transitions also can be induced by an applied electric *eld. The *eld compensates the diEerence in the chemical potential of the two layers, which leads to a charge transfer. By suitably tuning the parameters, the density in one of the layers will reach the value at which the *rst-order transition discussed above takes place. At this point, there will be an abrupt change in the charge transfer [6], which can be measured with standard capacitive techniques. In conclusion, we have analyzed the possible discontinuous transitions in 2D diluted magnetic semiconductors, where a single subband is occupied. We *nd that the interaction between carriers can lead to *rst-order transitions in quantum wells. At the transition, the holes become fully polarized. This transition can be induced by a change in the density of the hole gas, the temperature, and, in multilayer systems, an applied electric *eld. At these transitions, the minority spin band becomes empty, and the density of states at the Fermi level is reduced by one-half. In double layer systems, signi*cant electrostatic barriers can appear near the transition. This change can alter signi*cantly the transport properties. Thus, it can be important in the operation of devices made with these materials.
+(−)
being the chemical potential at nc , in the ferromagnetic (paramagnetic) phase. This equation can be satis*ed provided the phase transition is discontinuous. The value of can be obtained by minimizing the free energy, + − − =− : 2( + + − ) + (2e2 =)(nc =2)(d − (w=3)) (8) Therefore, the diEerence in chemical potential of these two phases induces a transfer of charge between the layers. In equilibrium, the charge transfer induces an electrostatic potential which compensates the discontinuity in the chemical potential. For reasonable values of barriers and well thickness we *nd that the charge transfer is small, ∼ 0:008. However, the electrostatic shift is of the order of meV
Acknowledgements We acknowledge the *nancial support from grants PB96-0875 and PB96-0085 (MEC, Spain). References [1] H. Ohno, J. Magn. Magn. Mater. 200 (1999) 110; H. Ohno, Science 281 (1998) 951. [2] T. Dietl et al., Phys. Rev. B 45 (1997) R3347. [3] H. Takahashi, Phys. Rev. B 56 (1997) 7839. [4] T. Jungwirth et al., Phys. Rev. B 59 (1999) 9818. [5] B. Lee et al., Phys. Rev. B 61 (2000) 15606. [6] L. Brey, F. Guinea, Phys. Rev. Lett. 85 (2000) 2384. [7] S.H. Vosko et al., Can. J. Phys. 58 (1980) 1200.
www.elsevier.com/locate/physe
Phase separation in diluted magnetic semiconductor quantum wells L. Brey ∗ , F. Guinea Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid, Spain
Abstract The phase diagram of diluted magnetic semiconductor quantum wells is studied. We *nd that the interaction between the carriers in the hole gas can lead to *rst-order ferromagnetic transitions. These transitions can induce phase separation in double quantum wells systems. We make some experimental predictions for observing these *rst-order phase transitions. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 75.50.Pp; 73.61.Ey; 75.70.Cn Keywords: Magnetic semiconductors; Quantum wells
Recent advances in the MBE growing techniques have made possible the fabrication of Mn-based diluted magnetic semiconductors (DMS) with a rather high ferromagnetic–paramagnetic critical temperature Tc [1]. In semiconductors, it is possible to modulate spatially the density of carriers by changing the doping pro*les but in DMS it is also suitable to vary the magnetic order of the carriers by changing the magnetic ion densities. The combination of these two possibilities opens a rich *eld of applications for these materials. The high critical temperature DMSs have a high concentration, c, of Mn2+ ions, randomly located. The itinerant carriers in the Ga1−x Mnx As systems are holes and their density, c∗ , is much smaller than the magnetic ion density. In a doped semiconductor, the spin S = 52 Mn2+ ions feel a long-range ferromagnetic interaction created by the coupling mediated by the itinerant spin polarized carriers [2–5]. ∗
Corresponding author. Fax: +34-91-372-0623. E-mail address: [email protected] (L. Brey).
In this work, we investigate the nature of the phase diagram of diluted magnetic semiconductor quantum wells, with emphasis on the existence of abrupt transitions for experimentally accessible parameters [6]. We analyze quantum wells made of GaMnAs and with thickness w. The system is described by the following Hamiltonian: H = Hh + J
I; i
SI · si (ri − RI );
(1)
where Hh is the part of the Hamiltonian which describe the itinerant holes. It is the sum of the kinetic energy of the holes and the hole–hole interaction energy. The in-plane motion of the holes is approximated by a single parabolic band of eEective mass m∗ . The interaction between the electrical carriers is described with the local spin density approximation (LSDA) [7]. The last term is the antiferromagnetic exchange interaction between the spin of the Mn2+ ions located at RI and the spins , ˜si , of the itinerant carriers. The interaction between ions mediated by the conduction holes is of long range. Thus, we will assume that the thermal distribution of the orientation of the Mn spins is that
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 3 1 7 - 4
L. Brey, F. Guinea / Physica E 12 (2002) 388 – 390
20
Paramagnetic 15
T(K)
induced by an eEective *eld, due to the holes, which should be calculated self-consistently. The aim of this work is to study the magnetic phase diagram of DMS quantum wells with a 2D density of holes n2D = c∗ w. For doing that we write the free energy F as a function of the carrier spin polarization, = n↑ − n↓ =n2D , which is the order parameter. Here, n is the 2D density of carriers with spin . The critical temperatures for the ferromagnetic to paramagnetic transition in DMS is typically smaller than 100 K, and for these temperatures we can consider that the electron gas is degenerate. Hence, the only temperature dependence in F is due to thermal Huctuations of the Mn spins. Treating the holes in the LSDA the free energy per unit area takes the form ˜2 F = Fions + ∗ n22D (1 + 2 ) + Exc (n2D ; ): (2) m 2 Here, Exc is the hole exchange correlation energy and Fions is the contribution of ion spins to the free energy: sinh [h(S + 1=2)=T ] Fions = −Tcw log ; (3) sinh (h=2T ) J n2D : (4) with h = 2 w In obtaining Eq. (3) we have assumed that the hole wave function in the z-direction has the form w−1=2 . The phase diagram with parameters J =0:15 eV nm3 , ion concentration c = 1 nm−3 and w = 10 nm, is shown in Fig. 1. We include a single hole band of eEective mass m|| = 0:11me , and a dielectric constant 0 = 12:2 [5]. We use the expression given by Vosko et al. [7] for Exc . The phase diagram is obtained by minimizing the free energy equation (2) with respect to , for different values of T . A *nite value of indicates a ferromagnetic state whereas = 0 corresponds to the paramagnetic phase. An abrupt=continuous change, as a function of T , in the value of is the mark of a *rst=second-order transition. In the high density region, the dashed line which represents the second-order phase transition agrees with that obtained from the divergences of the magnetic susceptibility [5]. The main novelty of our calculation is the identi*cation of a *rst-order transition to a fully polarized state at intermediate densities. This transition takes place at higher T than that at which the magnetic susceptibility of the system diverges. The
389
10
5
0
Ferromagnetic
0.2
0.4
0.6
0.8
kF (nm-1)
Fig. 1. Phase diagram of a DMS quantum well, using the parameters described in the text. Full and broken lines represent, respectively, *rst- and second-order phase transitions between ferromagnetic (F) and paramagnetic (P) phases. Upon lowering the temperature, we obtain a continuous transition between the P and the F phases, followed by an abrupt transition between two F phases with diEerent values of . As commented in the text, this last transition is probably spurious.
existence of a *rst-order transition between the paramagnetic and the ferromagnetic phases implies that, if the chemical potential is kept *xed, the carrier density will change abruptly. Conversely, if the average hole density is *xed, a region where inhomogeneous solutions are stable will appear near the transition, leading to phase separation. At higher densities, when a continuous transition between = 0 and = 0 occurs, we also *nd a *rst-order phase transition between the partially polarized system, ¡ 1, and the fully polarized phase, = 1. We believe that this is a spurious transition related with the LSDA [6]. The existence of discontinuous transitions leads to the possibility of phase separation. The Maxwell construction gives the region where two coexisting phases (paramagnetic and ferromagnetic) of diEerent densities are energetically favored. From the dependence on densities of the ground state free energy we *nd a region of phase separation near the line of *rst-order phase transitions. As the two phases have diEerent hole densities there are, associated with the phase separation, electrostatic eEects which need to be included. These eEects tend to reduce the size of the phases, and made diJcult the experimental observation of the phase separation. We propose that phase separation
390
L. Brey, F. Guinea / Physica E 12 (2002) 388 – 390
can occur in a double quantum well system made with DMS. Suppose two wells, each of them with a nominal 2D density of holes n. In the phase separation state, the right well is ferromagnetic and has a density n(1 + ) and the left well is paramagnetic and has a density n(1 − ). The system is unstable against phase separation when the following condition is satis*ed: 2F[n] ¿ (1 + )F[n(1 + )] + (1 − )F[n(1 − )] + EH (n; );
(5)
where F(n0 ) is the minimum free energy at density n0 , and EH (n0 ; ) is the capacitive energy due to the imbalance of the charge density in the bilayer, w 2 2e2 n2 d− : EH (n; ) = (6) 4 3 Here d is the distance between the center of the wells. The existence of a discontinuity in the chemical potential in the *rst-order phase transitions, made it possible that at a given T Eq. (5) can be satis*ed in a range of densities near the ferromagnetic–paramagnetic critical density nc . We have checked numerically that this range of densities is very narrow and the charge transfer small, in such a way that inequality (5) can be written as 0 ¿ nc ( + − − ) + nc 2 ( + + − ) + EH (nc ; ); (7)
and this can signi*cantly change the transport properties of the double quantum well system. In a bilayer system, the *rst-order transitions also can be induced by an applied electric *eld. The *eld compensates the diEerence in the chemical potential of the two layers, which leads to a charge transfer. By suitably tuning the parameters, the density in one of the layers will reach the value at which the *rst-order transition discussed above takes place. At this point, there will be an abrupt change in the charge transfer [6], which can be measured with standard capacitive techniques. In conclusion, we have analyzed the possible discontinuous transitions in 2D diluted magnetic semiconductors, where a single subband is occupied. We *nd that the interaction between carriers can lead to *rst-order transitions in quantum wells. At the transition, the holes become fully polarized. This transition can be induced by a change in the density of the hole gas, the temperature, and, in multilayer systems, an applied electric *eld. At these transitions, the minority spin band becomes empty, and the density of states at the Fermi level is reduced by one-half. In double layer systems, signi*cant electrostatic barriers can appear near the transition. This change can alter signi*cantly the transport properties. Thus, it can be important in the operation of devices made with these materials.
+(−)
being the chemical potential at nc , in the ferromagnetic (paramagnetic) phase. This equation can be satis*ed provided the phase transition is discontinuous. The value of can be obtained by minimizing the free energy, + − − =− : 2( + + − ) + (2e2 =)(nc =2)(d − (w=3)) (8) Therefore, the diEerence in chemical potential of these two phases induces a transfer of charge between the layers. In equilibrium, the charge transfer induces an electrostatic potential which compensates the discontinuity in the chemical potential. For reasonable values of barriers and well thickness we *nd that the charge transfer is small, ∼ 0:008. However, the electrostatic shift is of the order of meV
Acknowledgements We acknowledge the *nancial support from grants PB96-0875 and PB96-0085 (MEC, Spain). References [1] H. Ohno, J. Magn. Magn. Mater. 200 (1999) 110; H. Ohno, Science 281 (1998) 951. [2] T. Dietl et al., Phys. Rev. B 45 (1997) R3347. [3] H. Takahashi, Phys. Rev. B 56 (1997) 7839. [4] T. Jungwirth et al., Phys. Rev. B 59 (1999) 9818. [5] B. Lee et al., Phys. Rev. B 61 (2000) 15606. [6] L. Brey, F. Guinea, Phys. Rev. Lett. 85 (2000) 2384. [7] S.H. Vosko et al., Can. J. Phys. 58 (1980) 1200.