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### Title-affiliation

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

Spin-polarization and magnetization of

conduction-band

dilute-magnetic-semiconductor quantum wells

with

non-step-likedensity of states

e.g. n-doped DMS ZnSe / Zn1-x-yCdxMnySe / ZnSe QWs

B in-plane

magnetic field

Keywords – Things to remember- DOS = density of states

- DMS = dilute magnetic semiconductor

conduction band, narrow to wide, DMS QWs

outline

- How does in-plane B modify DOS ?

- DOS diverges significantly from ideal step-like 2DEG form

- considerable fluctuation of M

- severe changes to physical properties e.g.

- spin-subband populations, spin polarization

- internal energy, U free energy, F

- Shannon entropy, S

- in-plane magnetization, M

(if vigorous competition between spatial and magnetic confinement)

B applied parallel to quasi 2DEG

DOS in simple structuresnot only the generalshape of the DOS varies,

but this effect is also quantitative.

● the DOS deviates from the

famous step-like (B→0) form.

- interplay between spatial and magnetic confinement

● Ei(kx) must be determined self-consistently for quantum wells which are not ideally narrow [1].

● The eigenvalue equation has to be solved for each i and kx [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].

n(ε)

-ln|ε-Ei|

DETAILS… DOS in simple structuresLimitB → 0,

Ei(kx) = Ei+ ħ2kx2/(2m*).

DOS recovers simple famous 2DEG form

Limit simple saddle point,

Ei(kx) = Ei – ħ2kx2/(2n*), (n* > 0).

DOS diverges logarithmically

Basic Theory and Equations

Comparison with characteristic systems

knowing DOS we can calculate various electronic properties…

DOS in DMS structures

enhanced energy splitting

between spin-up and spin-down states

(all possible degrees of freedombecomeevident)

i, kx, σ

for any typeofinterplay

between spatial and magnetic confinement

i.e.

for narrow as well as for wide QWs

proportional to the cyclotron gap

spin-spin exchange interaction

between

s- or p- conduction band electrons

and

d- electrons of Μn+2 cations

Enhanced electron spin-splitting, UoσHigher temperatures.

spin-splitting decreases

enhanced contribution of spin-up electrons

Feedback mechanism due tondown(r) - nup(r).

Low temperatures.

spin-splitting maximum,

~ 1/3 of conduction band offset

Results and discussion

(a) Low temperatures, N = constant, T = constant

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

L = 30 nm

Spin-subband Populations

Internal energy

Free Energy

Entropy

+ Depopulation

of higher spin-subband

L = 60 nm

Spin-subband Populations

Internal Energy

Free Energy

Entropy

+ Depopulation

of higher spin-subband

Magnetization

considerable fluctuation of M

(if vigorous competition between

spatial and magnetic confinement)

Magnetization

considerable fluctuation of M

(if vigorous competition between

spatial and magnetic confinement)

Magnetization fluctuation:

5 A/m

(as adding 1017 cm-3 Mn).

Results and discussion

(b) Higher temperatures, N = constant

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 nm

Subband populations, L = 60 nm

Synopsis - Conclusion

- Results for different degrees of magnetic and spatial confinement.

- Valuable system for conduction-band spintronics.

- How much the classical staircase 2DEG DOS must be modified, under in-plane B.

- The DOS modification causes considerable effects on the system’s physical properties.

Spin-subband Populations,

Spin Polarization

Internal energy

Free energy

Entropy

Magnetization

We predict a significant fluctuation of the M

when the dispersion is severely modified

by the parallel magnetic field.

Bibliography 1

[1] C. D. Simserides, J. Phys.: Condens. Matter11(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.

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 Jpd = 0.15 eV nm3, is commonly used [10,11].

ZnSe has a sphalerite type structure and the lattice constant is 0.567 nm.

Hence, -Jsp-d ~ 12 10-3 eV nm3.

Spin Polarization, to be continued …

● spin-polarization 30 nm

● comparison 10 nm, 30 nm, 60 nm

We keep N = constant !

● spin-polarization 60 nm

Spin Polarization – Non homogeneous spin-splitting

● non-homogeneous spin-splitting, 30 nm

Spin Polarization – Non homogeneous spin-splitting

● non-homogeneous spin-splitting, 60 nm

-ln|ε-Ei|

n(ε)

DETAILS… quasi 2DEG DOS modification under in-plane BLimitB → 0,

Ei(kx) = Ei+ ħ2kx2/(2m*).

DOS recovers simple famous 2DEG form

- 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 Ei(kx), as we approach the critical points, are not of the form -akx2, 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 simple saddle point,

Ei(kx) = Ei – ħ2kx2/(2n*), (n* > 0).

DOS diverges logarithmically

Spin-subband populations– Depopulation

● choose the parameters so that

only spin-down electrons survive

or

● exploit the depopulation

of the higher subbands

to eliminate spin-up electrons

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