Antônio J. Roque da Silva (IFUSP), Adalberto Fazzio (IFUSP), Raimundo R. dos Santos (UFRJ), and Luiz Eduardo Oliveira

(UNICAMP)

Parametrization of Mn-Mn interactions in

Ga1-xMnxAs semiconductors

Workshop de Nanomagnetismo – 24 e 25/6/2004

Rede Virtual de Nanociência e Nanotecnologia do Estado do Rio de Janeiro/FAPERJ

Financiamento: •CNPq •FAPERJ•FAPESP •Inst Milênio de Nanociências, •Rede Nacional de Materiais Nanoestruturados

Motivation

Combination of semiconductor technology with magnetism should give rise to new devices: Spin-polarized electronic transport

long spin-coherence times (~ 100 ns) have been observed in semiconductors

manipulation of quantum states at a nanoscopic level

Magnetic semiconductors

• Early 60’s: EuO and CdCr2S4 very hard to grow

• Mid-80’s: Diluted Magnetic Semiconductors II-VI (e.g., CdTe and ZnS) II Mn difficult to dope direct Mn-Mn AFM exchange interaction

PM, AFM, or SG (spin glass) behavior

• 90’s: Low T MBE (In,Mn)AsUniform (Ga,Mn)As films on GaAs substrates: FM; heterostructures- Possibility of useful devices

Ga: [Ar] 3d10 4s2 4p1

Mn: [Ar] 3d5 4s2

Mn atoms: provide both magnetic moments and holes hole-mediated ferromagnetism

Ga

As

Ga1-xMnxAs

Resistance measurements on samples with different Mn concentrations:

Metal R as T Insulator R as T

Reentrant MIT

[Ohno, JMMM 200, 110(1999)]

Ga1-xMnxAs

0.00 0.02 0.04 0.06 0.08 0.1020

40

60

80

100

120

circles: Matsukura et al, PRB57, R2037 (1998) squares: Edmonds et al, APL 81, 3010 (2002)up triangles: Seong et al, PRB 66, 033202 (2002) and Potashnik et al, APL 79, 1495 (2001)diamonds: van Esch et al, PRB 56, 13103 (1997)down triangles: Asklund et al, PRB 66, 115319 (2002)

open star = Yu et al, PRB 65, 201303 (2002); APL81, 844 (2002)full star = Moriya-Munekata, JAP 93, 4603 (2003)full squares = Potashnik et al, PRB66, 012408 (2002)

Tc(

K)

Mn composition (x)

Reproducibility?

0.00 0.02 0.04 0.06 0.08 0.100

2

4

6

8

open star = Yu et al, PRB 65, 201303 (2002); APL81, 844 (2002)full star = Moriya-Munekata, JAP 93, 4603 (2003)

circle: Ohno et al, JMMM 200, 110 (1999)squares: Edmonds et al, APL 81, 3010 (2002)up triangles: Seong et al, PRB 66, 033202 (2002) Potashnik et al, APL 79, 1495 (2001)

ho

le c

on

cen

trat

ion

(10

20 c

m-3)

Mn composition x

Hole concentration vs Mn concentration

1 hole/Mn atom

A simple mean field treatment† yields

1h/MnNotice maximum of p(x) within the M phase correlate with MIT

Early predictions

[Matsukura et al., PRB 57, R2037 (1999)]

log!

†[RRdS, LE Oliveira, and J d’Albuquerque e Castro, JPCM (2002)]

First principles calculations should shed light into these issues

Experimental data very sensitive to growth conditions

what are the dominant mechanisms behind the origin of ferromagnetism in DMS?

• how delocalized are the holes (are effective mass theories meaningful)?

• what is the effective Mn-Mn interaction? RKKY?

• what is the role of disorder?

Method

Ab initio total energy calculations – DFT -VASP

Ultra-soft pseudopotential

Supercell calculations – 128/250 atoms (fcc)• Spin polarized• GGA (Perdew, Burke, Ernzerhof) for exchge-correl’n• Plane waves basis set – (cutoff of 230eV, k = L)

• Final forces smaller than 0.02 eV/Å

MnAs

Ga

Single Mn atom

20.3 Å

Isosurfaces for the net local magnetization MnGa )()()( rrrm

Ground state: quite localized hole interacting antiferromagnetically with S=5/2 of Mn(d 5 )

Green=0.004e/A3

Blue= -0.004e/A3

We now consider two Mn atoms per unit cell

Assume all possible non-equivalent positions

For a given relative position, we consider FM and AFM relative Mn orientations, and work out the energy difference

Fit this energy difference to a Heisenberg interaction:

21MnMn SS JH

thus estimates for J (r1– r2)

Ferromagnetic

Mn Mn

As As

Mn-Mn 1st NN

Antiferromagnetic

Ferromagnetic

Mn MnMn Mn

As AsAs As

Mn-Mn 1st NN

Antiferromagnetic

Ferromagnetic

Mn MnMn Mn

As AsAs As

As

As

Mn

Mn

Mn-Mn 1st NN

Mn-Mn 2nd NN

Antiferromagnetic

Ferromagnetic

Mn MnMn Mn

Mn

Mn

As AsAs As

As

As

Mn

MnAs

As

Mn-Mn 1st NN

Mn-Mn 2nd NN

Antiferromagnetic

Again, note quite localized character of the holes

12<110>

6<100>

24<211>

12<110>

24<310>

8<111>

The ferro-antiferro total energy differences yield...

12

6

24

12

248

the effective coupling between Mn spins (JMn-

MnSMn·SMn)

Therefore:

• impurity levels are localized

effective-mass picture for holes may be quite inadequate

• Mn-Mn interaction mediated by AFM coupling Mn-hole

• J Mn-Mn always ferromagnetic non-RKKY

• estimates for anisotropy and direction dependences for effective J Mn-Mn

Our current agenda:

1) Effects of disorder?

2) Effects of concentration?

Preliminary results

Strategy (in principle): Randomly place Mn atoms in the Ga sublattice and use a look up table for J’s

Ga

MnJ1

J4

J2

We start with 4 Mn in our 128 atoms supercell:

Roadmap1) Randomly place 4 Mn atoms in the Ga sublattice

2) Calculate, using same ab initio scheme, the total energies for:

a) (Mn1,Mn2,Mn3,Mn4)=(up,up,up,up) – Ferro

b) (Mn1,Mn2,Mn3,Mn4)=(down,up,up,up) – Flip Mn1

c) (Mn1,Mn2,Mn3,Mn4)=(up,down,up,up) – Flip Mn2

d) Etc.

3) Calculate energy differences E(Flip-Mn1)-Ferro, etc.

4) Write up same energy differences using an effective Heisenberg Hamiltonian, and extract effective Jn

5) Compare with previous results with only two Mn

4 Mn in 128 cell: - disorder inside unit cell - images are taken care of (unwanted

order!) - Mn concentration – 0.0625 (6.25 %)

- Different from 1 Mn in 32 atoms unit cell or

2 Mn in 64 atoms unit cellGa

Mn

J1

Ji

4 Mn in 128 atoms unit cell

Ab initio results

Ferro = -553.737 eV

Flip 1 = -553.547 eV

Flip 2 = -553.586 eV

Flip 3 = -553.302 eV

Flip 4 = -553.508 eV

Ferro - lowest energy configuration

1-Ferro = 0.190 eV

2-Ferro = 0.151 eV

3-Ferro = 0.435 eV

4-Ferro = 0.229 eV

4 Mn in 128 atoms unit cellHeisenberg Hamiltonian results

Ferro =

Flip 1 = Flip 2 =

Flip 3 =

Flip 4 =

1-Ferro =

2-Ferro =

3-Ferro =

4-Ferro =

For the particular realization, the Hamiltonian is

434232

413121

134

315

2

2

MnMnMnMnMnMn

MnMnMnMnMnMn

SSJSSJSSJ

SSJSSJSSJH

4

25

ji MnMn SS

5431 22224

25JJJJ

54 224

25JJ

5431 22224

25JJJJ

5431 22224

25JJJJ

543 2224

25JJJ

531 4224

25JJJ

543 4424

25JJJ

41 444

25JJ

31 424

25JJ

2 Mn in 128 atoms unit cellClassical x Quantum Heisenberg Hamiltonian results

4

25

ji MnMn SS

ji SSJH ˆˆ

Classical Quantum

J1 -23.2 -19.3

J2 -10.4 -8.7

J3 -13.6 -11.3

J4 -5.6 -4.7

J5 -2.6 -2.2

J6 -4.4 -3.7

ji SSJH

J (meV)

Same trend,

Classical or

Quantum

2 Mn x 4 Mn in 128 atoms unit cellClassical Heisenberg Hamiltonian

4

25

ji MnMn SS

2Mn 4Mn

J1 -23.2 -12.6

J2 -10.4 -

J3 -13.6 -2.8

J4 -5.6 -4.8

J5 -2.6 0.1

J6 -4.4 -

ji SSJH

J (meV)

2 Mn x 4 Mn in 128 atoms unit cellClassical Heisenberg Hamiltonian

4

25

ji MnMn SS

2Mn (4Mn)1 (4Mn)2

J1 -23.2 -12.6 -13.0

J2 -10.4 - -4.7

J3 -13.6 -2.8 -6.0

J4 -5.6 -4.8 -

J5 -2.6 0.1 -1.3

J6 -4.4 - -

ji SSJH

J (meV)

2 Mn x 3Mn x 4 Mn in 128 atoms unit cellClassical Heisenberg Hamiltonian

4

25

ji MnMn SS

2Mn 3Mn (4Mn)1 (4Mn)2

J1 -23.2 -19.2 -12.6 -13.0

J2 -10.4 - - -4.7

J3 -13.6 -8.4 -2.8 -6.0

J4 -5.6 - -4.8 -

J5 -2.6 -1.0 0.1 -1.3

J6 -4.4 - - -

ji SSJH

J (meV)

2Mn: x = 0.03125

3Mn: x = 0.046875

4Mn: x = 0.0625

2 Mn x 3Mn x 4 Mn in 128 atoms unit cell

Large reduction in the values of some of the J’s

– Possible reasons:

-Effective Heisenberg Hamiltonian may not be appropriate to describe “magnetic” excitations

-Effective Hamiltonian ok to describe low-energy magnetic excitations, but our spin flip excitations may have too high an energy (non-collinear spin ab initio calculations?)

-Disorder and/or concentration may have an important effect in the effective J couplings

Next steps (1):• Perform more calculations with random structures – obtain a distribution for effective J’s

• Perform similar calculations for different Mn concentrations

• Non-collinear spin calculations

•If we conclude that we have a physically correct description through effective J’s + classical Heisenberg Hamiltonian, perform calculations for T > 0 (Monte Carlo)Next steps (2):

• Study (ab initio) how defects (e.g., interstitial Mn) change this picture by placing them in the, for example, 4 Mn in 128 atoms supercell – local disorder + defects

Conclusions:

• Effective mass descriptions (and improvements thereof) not reliable

• Effective Mn-Mn interactions not RKKY

• Disorder strongly influences effective Mn-Mn interactions; simple model?

• Heterostructures: -doping, Be co-doping

Mn-hole exchange coupling

Jhd = 0.083 eV; 250 atoms, x = 0.008

Jhd = 0.11 eV; 128 atoms, x = 0.0156

• We have performed total energy calculations based on the density-functional theory (DFT) within the generalized-gradient approximation (GCA) for the exchange-correlation potential.

• The electron-ion interactions are described using ultra-soft pseudopotentials and plane wave expansion up to 200 eV as implemented in the VASP code.

• We used a 128-atom and 250-atom fcc supercell and the L-point for the Brillouin sampling. The positions of all atoms in the supercell were relaxed until all the force components were smaller than 0.05 eV/Å.

)()( rr

Isosurfaces for the difference between

calculated for the MnGa ground state and the GaAs host

m(r) = (r)-(r)

m(r) = +0.5 e-/Å3

Sub-Si n=p

n.5p.oo5

As

As

As

As Ga

a1

t2

As

As

As

As

a1

t2

As

As

As

As Mn

a1

t2

t2

e

•F. Matsukura, H. Ohno, A. Shen, and Y. Sugawara, Phys. Rev. B 57, R2037 (1998)

MBE at low growth T (200 - 300 OC) on GaAs (001) substrates

x = 0.015 – 0.071

200 nm thick Ga1-xMnxAs samples

•A. van Esch et al, PRB 56, 13103 (1997)

Ga1-xMnx As layers grown on GaAs (100) substrates

GaAs grown by MBE at low temperatures (200 – 300 OC)

samples of 3 m thick with Mn concentrations up to 9%

•K. W. Edmonds et al, APL 81, 3010 (2002)

metallic behavior for 0.015 x 0.08

Ga1-xMnxAs layers grown on semi-insulating GaAs (001) substrates by low-temperature (180 – 300 OC) MBE using As2

samples: 45 nm thick

•S. J. Potashnik et al, APL 79, 1495 (2001)

temperature during growth: 250 OC

Ga1-xMnxAs layers: thicknesses in range 110 – 140 nm

•M. J. Seong et al, PRB 66, 033202 (2002)

samples grown as in Potashnik et al: 250 OC and 120 nm

used a Raman-scattering intensity analysis of the coupled plasmon-LO phonon mode

and the unscreened LO phonon.

•H. Asklund et al, PRB 66, 115319 (2002)

angle-resolved photoemission; 1% - 6%

growth temperature of LT-GaAs and GaMnAs was typically 220 0C

Mn concentrations accurate within 0.5 %

NOTE THAT

•T. Hayashi et al, APL 78, 1691 (2001)

“a 10 oC difference in the substrate temperature during growth can lead to a

considerable difference in the transport properties as well as in magnetism even

though there is no difference in the growth mode as observed by electron diffraction

2 Mn atoms as nearest-neighbors (Ga sub-lattice)

Antiferromagnetic couplingm(r) = +0.004 e-/Å3

m(r) = -0.004 e-/Å3

• VERY DILUTED DOPING LIMIT: Mn FORMS ACCEPTOR LEVEL 110 meV ABOVE VALENCE BAND

• ANGLE-RESOLVED PHOTOEMISSION SPECTROSCOPY OBSERVES IMPURITY BAND NEAR EF.

• INFRARED MEASUREMENTS OF THE ABSORPTION COEFFICIENT ALSO REVEAL A STRONG RESONANCE NEAR THE ENERGY OF THE Mn ACCEPTOR IN GaAs.

• E. J. Singley, R. Kawakami, D. D. Awschalom, and D. N. Basov, PRL 89, 097203 (02)

conductivity data: estimate the effective mass to be 0.7 mo < m* < 15 mo for the x = 0.052 sample, and larger at all other dopings, which suggest that the carriers do not simply reside in the unaltered GaAs valence band

favor a picture of the electronic structure involving

impurity states at EF rather than of holes doped into

an unaltered GaAs valence band

work obtained by using “complete” Kohn-Luttinger

formalism (magnetic anisotropy, strain, etc):

• M. Abolfath, T. Jungwirth, J. Brum, and A. H. MacDonald, PRB 63, 054418 (2001).

• T. Dietl, H. Ohno, and F. Matsukura, PRB 63, 195205 (2001).

Isosurfaces for the net local magnetization: two MnGa defects

In (a) and (b) the two Mn are nearest neighbors with their S=5/2 spins alligned parallel and antiparallel, respectively

Mn Mn

As As

Mn-Mn 1st nn

Mn Mn

As As

)()()( rrrm

Green=0.004e/A

Blue= -0.004e/A