1. Title-affiliation Spin-polarization and magnetization of conduction-band dilute-magnetic-semiconductor quantum wells with non-step-like density of states Constantinos Simserides 1,2 1 Leibniz Institute for Neurobiology, Special Lab for Non-Invasive Brain Imaging, Magdeburg, Germany 2 University of Athens, Physics Department, Solid State Section, Athens, Greece

2. Keywords – Things to remember DOS = density of states DMS = dilute magnetic semiconductor conduction band,narrow to wide,DMSQWs B in-plane magnetic field e.g. n-doped DMS ZnSe / Zn 1-x-y Cd x Mn y Se / ZnSe QWs

3. outline considerable fluctuation of M How does in-plane B modify DOS ? - DOS diverges significantly from ideal step-like 2DEG form severe changes to physical properties e.g. Shannon entropy, S spin-subband populations, spin polarization internal energy, Ufree energy, F in-plane magnetization, M (if vigorous competition between spatial and magnetic confinement)

4. DOS in simple structures B applied parallel to quasi 2DEG ● the DOS deviates from the famous step-like (B → 0) form. not only the general shape of the DOS varies, but this effect is also quantitative. ● The eigenvalue equation has to be solved for each i and k x [1]. ● E i (k x ) must be determined self-consistently for quantum wells which are not ideally narrow [1]. ● The van Hove singularities, are not - in general - simple saddle points [1]. ● The singularities are e.g. crucial for the interpretation of magnetoresistance measurements [1,2]. interplay between spatial and magnetic confinement

5. DETAILS… DOS in simple structures Limit B → 0, E i (k x ) = E i + ħ 2 k x 2 /(2m * ). DOS recovers simple famous 2DEG form Limit simple saddle point, E i (k x ) = E i – ħ 2 k x 2 /(2n * ), (n * > 0). DOS diverges logarithmically -ln|ε-E i |n(ε)n(ε)

6. Basic Theory and Equations Comparison with characteristic systems knowing DOS  we can calculate various electronic properties…

7. DOS in DMS structures enhanced energy splitting between spin-up and spin-down states (all possible degrees of freedom become evident ) for any type of interplay between spatial and magnetic confinement i.e. for narrow as well as for wide QWs i, k x, σ

8. Enhanced electron spin-splitting, U oσ proportional to the cyclotron gap spin-spin exchange interaction between s- or p- conduction band electrons and d- electrons of Μn +2 cations Low temperatures. spin-splitting maximum, ~ 1/3 of conduction band offset Higher temperatures. spin-splitting decreases enhanced contribution of spin-up electrons Feedback mechanism due to n down (r) - n up (r).

9. Equations

10. Results and discussion (a) Low temperatures, N = constant, T = constant

11. L = 10 nm (spatial confinement dominates) ~ parabolic spin subbands increase B  more flat dispersion  few % DOS increase A single behavior of Internal Energy Free Energy Entropy

12. L = 30 nm (drastic dispersion modification) Spin-subband dispersion and DOS

13. L = 30 nm Spin-subband Populations Internal energy Free Energy Entropy + Depopulation of higher spin-subband

14. L = 60 nm (~ spin-down bilayer system) Spin-subband dispersion and DOS

15. L = 60 nm Spin-subband Populations Internal Energy Free Energy Entropy + Depopulation of higher spin-subband

16. Magnetization considerable fluctuation of M (if vigorous competition between spatial and magnetic confinement)

17. Magnetization considerable fluctuation of M (if vigorous competition between spatial and magnetic confinement) Magnetization fluctuation: 5 A/m (as adding 10 17 cm -3 Mn).

18. Results and discussion (b) Higher temperatures, N = constant

19. Spin-subband populations – Depopulation ● exploit the depopulation of the higher subbands to eliminate spin-up electrons ● choose the parameters so that only spin-down electrons survive or Subband populations, L = 30 nmSubband populations, L = 60 nm

20. Spin Polarization ● spin-polarization 10 nm N s = N s,up + N s,down (free carrier 2D concentration)

21. Synopsis - Conclusion - Results for different degrees of magnetic and spatial confinement. Spin-subband Populations, Spin Polarization Internal energy Free energy Entropy Magnetization - How much the classical staircase 2DEG DOS must be modified, under in-plane B. - Valuable system for conduction-band spintronics. - The DOS modification causes considerable effects on the system’s physical properties. We predict a significant fluctuation of the M when the dispersion is severely modified by the parallel magnetic field.

22. Bibliography 1 [1] C. D. Simserides, J. Phys.: Condens. Matter 11 (1999) 5131. [2] O. N. Makarovskii, L. Smr\u{c}ka, P. Va\u{s}ek, T. Jungwirth, M. Cukr and L. Jansen, Phys. Rev. B 62 (2000) 10908. [3] H. Ohno, J. Magn. Magn. Mater. 200 (1999) 110 ; ibid. 242-245 (2002) 105. [4] S. P. Hong, K. S. Yi, J. J. Quinn, Phys. Rev. B {\bf 61} (2000) 13745. [5] B. Lee, T. Jungwirth, A. H. MacDonald, Phys. Rev. B {\bf 61} (2000) 15606. [6] H. J. Kim and K. S. Yi, Phys. Rev. B {\bf 65} (2002) 193310. [7] C. Simserides, to be published in Physica E. [8] H. Venghaus, Phys. Rev. B {\bf 19} (1979) 3071 ; S. Adachi and T. Taguchi, Phys. Rev. B 43 (1991) 9569. [9] C. E. Shannon, Bell Syst. Tech. J. {\bf 27} (1948) 379. [10] $N = \Gamma \sum_{i,\sigma} \int_{-\infty}^{+\infty} \! dk_x I$, $S = -k_B \Gamma \sum_{i,\sigma} \int_{-\infty}^{+\infty} \! dk_x K$, $U = \Gamma \sum_{i,\sigma} \int_{-\infty}^{+\infty} \! dk_x [E_{i,\sigma}(k_x) I + J]$. $\Gamma = \frac {A \sqrt{2m^*}}{4 \pi^2 \hbar}$. $I = \int_{0}^{+\infty} \! \frac {da}{\sqrt{a}} \Pi$, $J = \int_{0}^{+\infty} \! da \sqrt{a} \Pi$, $K = \int_{0}^{+\infty} \! \frac {da}{\sqrt{a}} \Pi ln\Pi$, $\Pi = (1+exp(\frac{a+E_{i,\sigma}(k_x)-\mu}{k_B T}))^{-1}$. [11]M. S. Salib, G. Kioseoglou, H. C. Chang, H. Luo, A. Petrou, M. Dobrowolska, J. K. Furdyna, A. Twardowski, Phys. Rev. B {\bf 57} (1998) 6278. [12] W. Heimbrodt, L. Gridneva, M. Happ, N. Hoffmann, M. Rabe, and F. Henneberger, Phys. Rev. B {\bf 58} (1998) 1162. [13] M. Syed, G. L. Yang, J. K. Furdyna, M. Dobrowolska, S. Lee, and L. R. Ram-Mohan,Phys. Rev. B {\bf 66} (2002) 075213. [14] S. Lee, M. Dobrowolska, J. K. Furdyna, and L. R. Ram-Mohan, Phys. Rev. B {\bf 61} (2000) 2120.

23. Bibliography 2 [1] H. Ohno, J. Magn. Magn. Mater. (2004) in press ; J. Crystal Growth 251, 285 (2003). [2] M. Syed, G. L. Yang, J. K. Furdyna, et al, Phys. Rev. B 66, 075213 (2002). [3] S. Lee, M. Dobrowolska, J. K. Furdyna, and L. R. Ram-Mohan, Phys. Rev. B 61, 2120 (2000). [4] C. Simserides, J. Comput. Electron. 2, 459 (2003); Phys. Rev. B 69, 113302 (2004). [5] S. P. Hong, K. S. Yi, J. J. Quinn, Phys. Rev. B 61, 13745 (2000). [6] H. J. Kim and K. S. Yi, Phys. Rev. B 65, 193310 (2002). [7] C. Simserides, Physica E 21, 956 (2004). [8] H.Venghaus, Phys. Rev. B 19, 3071 (1979). [9] H. W. Hölscher, A. Nöthe and Ch. Uihlein, Phys. Rev. B 31, 2379 (1985). [10] B. Lee, T. Jungwirth, A. H. MacDonald, Phys. Rev. B 61, 15606 (2000). [11] L. Brey and F. Guinea, Phys. Rev. Lett. 85, 2384 (2000). [12] For holes, the value J pd = 0.15 eV nm 3, is commonly used [10,11]. ZnSe has a sphalerite type structure and the lattice constant is 0.567 nm. Hence, -J sp-d ~ 12 10 -3 eV nm 3.

24. Acknowledgments Many thanks to Prof. G. P. Triberis Prof. J. J. Quinn Prof. Kyung-Soo Yi

25. End

26. Landau Levels DOS

27. Spin Polarization, to be continued … ● spin-polarization 30 nm ● spin-polarization 60 nm ● comparison 10 nm, 30 nm, 60 nm We keep N = constant !

28. Spin Polarization – Non homogeneous spin-splitting ● non-homogeneous spin-splitting, 30 nm

29. Spin Polarization – Non homogeneous spin-splitting ● non-homogeneous spin-splitting, 60 nm

30. DETAILS… quasi 2DEG DOS modification under in-plane B Sometimes this step-like DOS may become a stereotype, although even e.g. in the excellent old review [AFS] the authors pointed out that “for more complex energy spectra” - than the simple parabola – “the density of states must generally be found numerically”. At that time most of the calculations referred to parabolic bands and the in-plane magnetic field was treated as a perturbation which gave a few percent correction in the effective mass. This is one of the two asymptotic limits of the present case. The need to understand and calculate self-consistently the dispersion of a quasi 2DEG in the general case of interplay between spatial and magnetic localization, when the system is subjected to an in-plane magnetic field, can be justified in [SIMS] and [MAKAR] and in the references therein. Nice calculations of the DOS under in-plane magnetic field can be found in Lyo’s paper [LYO], in a tight-binding approach for narrow double quantum wells. The crucial features of the present DOS, i.e. the van Hove singularities, are not - in general - simple saddle points [1] because the E i (k x ), as we approach the critical points, are not of the form -ak x 2, a > 0. Simple analytical models are insufficient to explain e.g. the magnetoresistance and have to be replaced by self-consistent calculations in the case of wider quantum wells [SIMS,MAKAR]. Limit B → 0, E i (k x ) = E i + ħ 2 k x 2 /(2m * ). DOS recovers simple famous 2DEG form Limit simple saddle point, Ei(kx) = Ei – ħ 2 k x 2 /(2n * ), (n * > 0). DOS diverges logarithmically -ln|ε-E i | n(ε)n(ε)

31. Enhanced electron spin-splitting DETAILS

32. Detailed Equations

33. Spin Polarization N s = N s,up + N s,down (free carrier 2D concentration) For conduction band electrons

34. Spin-subband populations – Depopulation ● exploit the depopulation of the higher subbands to eliminate spin-up electrons ● choose the parameters so that only spin-down electrons survive or