Title-affiliation

Spin-polarization and magnetization of Spin-polarization and magnetization of conduction-band conduction-band

dilute-magnetic-semiconductor quantum dilute-magnetic-semiconductor quantum wells wells withwith

non-step-likenon-step-like density of statesdensity of statesConstantinos Simserides1,2

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

Keywords – Things to remember• DOS = density of states

• DMS = dilute magnetic semiconductor

conduction band, narrow to wide, DMS QWs

B in-plane

magnetic field

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

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, U free energy, F

• in-plane magnetization, M

(if vigorous competition between spatial and magnetic confinement)

DOS in simple structuresB applied parallel to quasi

2DEG

● the DOS deviates from the famous step-like (B→0) form.

i xi

xix

kE

kEdk

mAn

)(

))((

2

2)(

2

*

i i )E(

Am)(n

*

2

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 kx [1].

● Ei(kx) 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

DETAILS… DOS in simple

structures

i iE

Amn )()( 2

*

Limit B → 0,

Ei(kx) = Ei + ħ2kx2/(2m*).

DOS recovers simple famous 2DEG form

Limit simple saddle point,Ei(kx) = Ei – ħ2kx

2/(2n*), (n* > 0).

DOS diverges logarithmically

-ln|ε-Ei| n(ε)

Basic Theory and EquationsBasic 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

,i x,i

x,ix

*

)k(E

))k(E(dk

mA)(n

222

2

(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, kx, σ

Enhanced electron spin-splitting, Uoσ

)(2 0

**

Sdspce

o SBJyNm

mgU

Tk

nnSJSBg

B

updowndspBMn 2

)()( rr

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 ndown(r) - nup(r).

Equations

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(drastic dispersion modification)

Spin-subband dispersion

and

DOS

L = 30 nm

Spin-subband Populations

Internal energy

Free Energy

Entropy

+ Depopulation of higher spin-subband

L = 60 nm(~ spin-down bilayer system)

Spin-subband dispersion

and

DOS

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

Spin Polarization

● spin-polarization 10 nm

s

upsdowns

N

NN ,,

Ns = Ns,up + Ns,down

(free carrier 2D concentration)

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.

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.

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.

Acknowledgments

Many thanks to

Prof. G. P. Triberis

Prof. J. J. Quinn

Prof. Kyung-Soo Yi

End

Landau Levels DOS

Spin Polarization, to be continued …

● spin-polarization 30 nm

● spin-polarization 60 nm

s

upsdowns

N

NN ,,

● comparison 10 nm, 30 nm, 60 nmWe keep N = constant !

Spin Polarization – Non homogeneous spin-splitting

● non-homogeneous spin-splitting, 30 nm

Spin Polarization – Non homogeneous spin-splitting

● non-homogeneous spin-splitting, 60 nm

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 Ei(kx), as we approach the critical points, are not of the form -akx

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].

i iE

Amn )()( 2

*

Limit B → 0,

Ei(kx) = Ei + ħ2kx2/(2m*).

DOS recovers simple famous 2DEG form

Limit simple saddle point,Ei(kx) = Ei – ħ2kx

2/(2n*), (n* > 0). DOS diverges

logarithmically

-ln|ε-Ei| n(ε)

Enhanced electron spin-splitting DETAILS

Detailed Equations

Spin Polarization

Ns = Ns,up + Ns,down (free carrier 2D concentration)

For conduction band electrons

s

up,sdown,s

N

NN

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