Transcript simserides 17 nov 2004 talk MMN2004

Title-affiliation
Spin-polarization and magnetization of
conduction-band
dilute-magnetic-semiconductor quantum wells
with
non-step-like density of states
Constantinos 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
e.g. n-doped DMS
conduction band,
ZnSe / Zn1-x-yCdxMnySe / ZnSe
narrow to wide,
B in-plane
magnetic field
DMS
QWs
QWs
outline
• How does in-plane B modify DOS ?
- DOS diverges significantly from ideal step-like 2DEG form
• 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
• considerable fluctuation of M
(if vigorous competition between spatial and magnetic confinement)
DOS in
simple structures
B applied parallel to quasi 2DEG
n(  ) 
m* A
 2

i
not only the general shape of the DOS varies,
 (   Ei )
but this effect is also quantitative.
● the DOS deviates from the
famous step-like (B→0) form.
• interplay between spatial and magnetic confinement
n( ) 
A 2m
2
2 
*

i


dk x
(  Ei ( k x ))
  Ei ( k x )
● 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].
DETAILS…
DOS in
simple structures
Limit B → 0,
Ei(kx) = Ei +
ħ2kx2/(2m*).
Limit simple saddle point,
Ei(kx) = Ei – ħ2kx2/(2n*), (n* > 0).
*
n ( ) 
m A

2
 (  E )
i
i
DOS recovers simple famous 2DEG form
n(ε)
 -ln|ε-Ei|
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 freedom become evident)
n(  ) 
*
A 2m
2  2
2


 
i,


dk x
 (   Ei , ( k x ))
  Ei , ( k x )
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
U o
g * m*

c  yN 0 J spd SBS ( )    
2me
proportional to the cyclotron gap
spin-spin exchange interaction
between
s- or p- conduction band electrons
and
d- electrons of Μn+2 cations

g Mn  B SB  J sp d S
ndown (r )  nup (r )
2
k BT
Higher temperatures.
Low temperatures.
spin-splitting decreases
enhanced contribution of spin-up electrons
spin-splitting maximum,
~ 1/3 of conduction band offset
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
● choose the parameters so that
only spin-down electrons survive
Subband populations, L = 30 nm
or
● exploit the depopulation
of the higher subbands
to eliminate spin-up electrons
Subband populations, L = 60 nm
Spin Polarization
 
N s ,down  N s ,up
Ns
Ns = Ns,up + Ns,down
● spin-polarization 10 nm
(free carrier 2D concentration)
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. 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.
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
N s ,down  N s ,up
Ns
● comparison 10 nm, 30 nm, 60 nm
We 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
Limit B → 0,
Ei(kx) = Ei + ħ2kx2/(2m*).
*
n ( ) 
m A

2
 (  E )
i
i
DOS recovers simple famous 2DEG form
Limit simple saddle point,
Ei(kx) = Ei – ħ2kx2/(2n*), (n* > 0).
•
n(ε)  -ln|ε-Ei|
DOS diverges logarithmically
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].
Enhanced electron spin-splitting DETAILS
Detailed Equations
Spin Polarization
For conduction band electrons
 
Ns = Ns,up + Ns,down
N s ,down  N s ,up
Ns
(free carrier 2D concentration)
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