Shallow centers in diluted magnetic semiconductors

0038-1098/85 $3.00 + .OO Pergamon Press Ltd.

Solid State Communications, Vo1.53,No.12. pp.1097-1101, 1985. Printed in Great Britain.

Shallow

Centers

in Diluted

Magnetic

Semiconductors

J. K. Furdyna Physics Department,

Purdue

University,

West Lafayette,

IN

47907

U.S.A.

The effect of spin-spin exchange interaction between shallow centers in diluted magnetic semiconductors and localized magnetic moments is reviewed and illustrated by three distinct examples: (1) spin-flip transitions observed on donors in Cd,_,Mn,Se, illustrating the role of exchange as a “magnetic field intensifier;” (2) the bound magnetic polaron, illustrating the effect of exchange on shallow centers in DMS in the absence of an external magnetic field; and (3) the giant magnetoresistance observed in p-type Hgl_,Mn,Te, which arises as a result of exchange-induced “swelling” of the acceptor wave functions with increasing magnetic field.

I. Introduction

exchange can be qualitatively expressed through effective g-factor, which can be written as ’

semiconductors (DMS)-magnetic Diluted semimagnetic also referred to as frequently semiconductors--are mixed semico+n+ducting crystals in ) are incorporated which magnetic ions (usually Mn in substitutional positions of the host (usually a II-VI) crystal lattice. Hg,_,Mn,Te and Cd,_,Mn,Se are two examples of such compound semiconductors. These alloys are interesting for three reasons’. First, the semiconducting properties of these materials--such as the energy gap and the effective mass--can be “tuned” in a controlled fashion by varying the composition. Second, DMS--being disordered magnetic alloys--are of interest for their magnetic properties, displaying, e.g., a low-temperature spin glass transition ‘, formnt,ion of antiferromagnetic clusters3, and magnon excitat.ions ‘. Third, the presence of magnetic ions in the DMS lattice leads to spin-spin exchange interaction between the localized magnetic moments and the band electrons’. This in turn leads to new and exciting effects in electrical and optical properties of DMS, which have been largely responsible for the amount of attention devoted to these materials in recent years. In this paper we describe the striking consequences of exchange interaction on shallow donor and acceptor levels. We shall illustrate this with three in which exchange interaction manifests examples, itself in very distinct ways: the electric-dipole-induced spin resonance of donor electrons in Cd,_,Mn,Se, the bound magnetic polaron in Cd,_,Mn,Se and Cd _,Mn,Te, and the giant negative magnetoresitance in kg,_,Mn,Te. II.

One exchange ment of spin-down applied. parameters electrons,

Ekctric

D$o~~f;eResonance IX X

in

of the most interesting manifestations of interaction in DMS is a dramatic enhancethe energy difference between spin-up and states when an external magnetic field is In the case of the parabolic conduction band in Cdl_,Mn,Se--which also describe donor of interest in this paper-the effect of

get7 =

kc’+

N,CY

=g*_-

an

0M

fleB gM&B ’ where g’ is the g-factor determined purely by the band parameters, a is the s-d exchange integral, N, is the concentration of Mn ions, is the z-average of the Mn spin, M is the magnetization, j+, is the Bohr magneton, gM* is the g-factor of Mn ions, an-d B is the magnetic field (assumed parallel to i). Here two features are of importance: the effective g-factor is temperature-dependent because of the presence of M, which obeys the Curie-Weiss law; and the exchange contribution to gefl is extremely large (typically in excess of 100 for 10% Mn at low temperatures). In wide-gap DMS, where g’ is of the order of 2, this contribution is overwhelming, and it is for that reason that the effect of Mn in DMS is often viewed as a “magnetic field intensifier,” since it allows the observation of phenomena that in non-magnetic semiconductors would require megagauss fields. This behavior is graphically illustrated by the exchange-enhanced spin splitting of the 1s ground state of donor electrons in Cd,_,Mn,Se, observed by electric-dipole-induced spin resonance (EDSR) at fartransitions infrared (FIR) frequencies ‘. Direct between two spin states in the Is level of a hydrogenic donor in a semiconductor are forbidden by electric dipole seIection rules. However, in pyroelectric crystals (uniaxial crystals without inversion symmetry, such as CdS, CdSe, or Cd,_,Mn,Se) in the presence of spin-orbit interaction, electron states are described by a Hamiltonian which contains a term’

(2) where S is the unit vector along the c axis, T and z are the spin and momentum operators of the electron, respectively, and X is a constant. This term, regarded as a perturbation, mixes states of different orbital angular momentum with respect to the i axis, but. conserves total (orbital plus spin) angular momentum about that axis. Thus the perturbed conduction band 1097

I098

SHALLOW CENTERS IN DILUTED ,MAGNETIC SEMICONDUCTORS

states (which also describe shallow donors) contain a mixture of "up" and "down" spin, which in turn permits the electric dipole spin resonance transition either for the conduction or for the donor-bound electrons. The resulting absorption then provides a direct means of determining X. Using first-order perturbation theory, with H' of Eq. (2) as the perturbing Hamiltonian. we can calculate the ~ound state donor wave functions. It is convenient to perform the calculation for two distinct geometries ~ i 6 and ~ ~. where ~ is the dc magnetic field. For example, for ~ i c (takino ~H i, cU x), perturbation theory gives the first-order wave functions of ls,[ and ls,1 states approximately in the form s i!

•

'

I

'

Vol. I

'

I

i

53,

No.

12

[

Faraday qllBIIc X=118.8/~m

z 0

~

15K

~r

z<[ F-

iXm" 1

Xm"

AE

(3) II

iXm" ] [ls

Xm'

>

AE

z[lsT>,

I

0

(4) w h e r e / l s f > and Ils~> designate unperturbed states, ~x~s is the spin-flip energy t~

= g,,~UB B ,

(5)

Fig. 1.

m" is the effective mass, and AE is the excitation energy between the Is ground state and the excited donor states of higher angular momentum. Equations (3) and (4) are approximate in that AE has been treated as a constant, approximately equal to the energy of the Is --* 2p transition. This is a reasonable approximation because all excited states lie rather close to each other compared to the Is level. The corresponding EDSR matrix element obtained with Eqs. (3) and (4) has the form

X

~

I

2 4 6 MAGNETIC FIELO (T)

'

,

I

i

I

8

FIR magnetotransmission in Cd0.0Mn0.1Se in the Faraday configuration at several temperatures. EDSR is clearly seen as the prominent absorption dip, superposed on the slowly varying magnetic-field- dependent background. As the temperature increases above 10K, a new line appears on the lowfield side of the original resonance and grows in intensity, possibly due to the thermally ionized conduction electrons (after Ref. 6).

AEr~

<~PI,tJ z J !hi, 1 > - 2R" ( A E ) 2 - ( ~ s ) 2 '

(6)

where R" is the effective Rydberg. For thisjz;eometry (~i6), the transition is only allowed for ~]1 ~ (the ordinary Voigt configuration). Figure l shows typical FIR transmission data, observed using the 118.8 /lm wavelength of an optically pumped FIR laser at several temperatures ¢. Note that the position of the resonance is strongly temperature-dependent. This illustrates precisely the influence of the magnetization in the expression for ge~ (Eq. l). As M decreases with increasing temperatures, increasingly larger fields are required to achieve the same spin splitting (corresponding to the liS.8 pm laser line). Note also that the absorption line splits into a doublet as the temperature increases. The reason for this is not presently clear, but a possibility has been suggested 6 that the onset of a second line can be due to free electron spin-flip contribution, as these become ionized to the conduction band. Equation (6) also shows that the EDSR intensity must vary strongly with the photoi~ energy (note the denominator of the matrix element). The physical reason for this is that, since the transition is made possible by the admixture from 2p,i (and higher spin-down levels) into the ls, r level, and since these levels approach each other as the spin splitting increases, the higher the spin splitting the greater will be the mL'(ing and the stronger the ls, i --. Is,[ transition. This is illustrated in Fig. 2, which shows the dramatic increase of the EDSR intensity with the FIR photon frequency.

HI.

Bound

Magnetic

Polarons

in DMS

The exchange phenomena affecting band electrons and impurity states in DMS can be traced to the existence of finite magnetization M due to the localized magnetic moments in the lattice. This occurs, of course, in the presence of a magnetic field, which produces a magnetization on a macroscopic scale, and most exchange phenomena involving extended states (free electrons, excitons, etc.) are therefore observed in a finite field B. However, in the case of localized states--such as donors or acceptors--finite magnetization on the scale of the localized state can exist even when B =0, due to two causes: local fluctuations of magnetization, and electron-spin-induced local magnetization. Consider an electron bound in a donor level. The hydrogen-like orbit of the electron encloses a large number of Mn + + ions (ca. 400). There is a finite probability that at any given time the net local magnetization of the enclosed magnetic ions is nonzero simply due to thermodynamic considerations ~. In addition, the donor electron itself will interact with the nearby moments through spin-spin exchange, which then polarizes the Mn "v+ spins enclosed in the Bohr orbit, creating a finite magnetization on the local scale. The fluctuation mechanism dominates at higher temperatures, whereas at low temperatures the local magnetization of the Mn + + ions is primarily induced by the bound electron. Whatever the cause, there exists a local magnetization at the donor (or acc.eptor) site, which in turn influences the energy of the impurity state. Specifically, in this situation it would

Vol. 53, No.

12

SHALLOW CENTERS

IN DILUTED MAGNETIC

/

(CRA)

'E

/o

v

IO 2

z o

I-CL cr o u-) o3 < c3 LLI IO l

w I.-z

,00

~

t

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t

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!

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5

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IV. Giant Negative Magnetoreslstance in Hgi_.MnxTe

'

I0

Another unique manifestation of the effect of exchange interaction on shallow centers is the observation at low temperatures of gigantic negative ma¢netoresistance in open gap p-type Hgl_x.MnxTe l&r~. This example demonstrates how dramatically the exchange-induced phenomena can affect ordinary electrical properties of a semiconductor. Figure 4 shows the dependence of transverse resistivity for a Hgt_xMnxTe sample (x-~0.11) on magnetic field 14, observed at 1.4 K. The sample contains ca. 10 l° cm -3 acceptors. As shown in the figure, the resistance is seen to drop precipitously as a function of field (by seven orders of magnitude in 7

PHOTON ENERGY (meV) Fig. 2.

Dependence of the integrated absorption ~ r of EDSR on the photon energy. White and black dots show experimental data obtained on Cd0.QMn0ASe in the cyclotron-resonanceactive polarization at 4.7 and 9.8K, respectively. The solid curve is the best theoretical fit, obtained using E.q. (6} with AE = 15.5 eV and k : 6.05 X l 0 - 1 0 e V - c m , and with a donor concentration of 2.0 x l0 Is em -3 determined by Hall measurements (see Ref. 6).

Cdl. x Mn~Se

•

A

1099

require energy to flip the spin of the donor electron even when there is no external magnetic field. This is the bound magnetic polaron (BMP}. BMP has been observed in various situations in DMS, the most direct observation being in Raman scattering in Cdl_xMnxSe and Cdt_xMnxTe containing donors s - n . Figure 3 shows Raman scattering data on a sample of Cdt_xMnxSe (x =0.10} at several temperatures, manifesting a Raman shift of between 5 and 10 cm -l even when B =0. The dots in the Figure are experimental datal2, and the solid curves are calculated using the theory of Dietl and Spalek s. Note that in this temperature regime both the fluctuation and the self-induced local magnetization contributions are important. As aa external magnetic field is applied, the BMP spin-flip phenomenon gradually transforms info ordinary spin-flip of donors (with the exception that the spin splitting between the ls,~ and ls,I states is tremendously enhanced by the presence of M, much as in the case of EDSR, described above) and that the splitting is not linear with B owing to the Brillouinfunction character of the macroscopic magnetization g' ~0.

IO 3

CdogMno tSe

SEMICONDUCTORS

X = 010

~3 >-

T=5OK

T=ISK

1

!,

H ,o

li !I

T=IOK

T=2OK H=O

H:o

-7

;/

i

\

z~e

/

7.

/ IO

20

/ tO

20

RAMAN

Fig. 3.

.

0 SHIFT

bO

20

Cm-l )

Raman spectra for Cdl_x.MnxSe ( x - 0 . 1 0 ) . showing bound magnetic polaron spin-flip scattering at several temperatures (Ref. 12). The dots are experimental; the solid curves are calculated using the Diet[-Spalek theory, taking account of fluctuation and selfinduced-magnetization contributions at different temperatures.

SHALLOW CENTERS IN DILUTED ~AGNETIC SEMICONDUCTORS

llO0 lOs

I

I

I

i

I

I

H g l _ x Mn x Te

IOz

X = O.JJ

E i

iOs

>-

~_ Io5 tCO JO4 W W

~n

iO3

I,I

z

< or" t.-

I0 z

IO°0

I

I

I

I

I

I

I

2

3

4

5

6

MAGNETIC

Fig. 4.

7

FIELO ( Teslo )

Giant negative magnetoresistance observed in p-type Hgl_xMn_Te 3(x~0.11, acceptor concentration -~101¢cm - , T = I . 4 K; after Ref. 14). The effect is ascribed to the spatial growth of the acceptor wave functions with increasing magnetic field•

Tesla!). This effect, intimately related to the behavior of acceptors in this material, can be understood as follows, at least for fairly large energy gaps (corresponding in Hgl_xMnxTe to, say, x > 0.10). In the presence of the magnetic field, the four-fold degenerate r s valence band (which also determines the behavior of shallow acceptor levels} is Zeeman-split into four subbands 15, each of these being further split into a ladder of Landau levels. For Hgl_xMn~Te with x > 0.1, the exchange interaction can make the Zeeman splitting much larger than the Landau splitting. The effect of exchange also makes each of the bands anisotropic in the coordinates defined by the applied magnetic field, with the components of the hole masses transverse and longitudinal to the field corresponding to particular combinations of the light and heavy hole masses• Specifically, the mass ellipsoid of the uppermost Zeeman-split band is cigar-shaped, oriented along ]~, with the effective mass transverse to the applied field close to that of the light-hole mass, and the longitudinal mass approximately that of the heavy hole mass Is. This sub-band is closest to the acccptor level. When the acceptor binding energy is less than the Zeeman splitting (which can be easily satisfied when exchange interaction is present), the acceptor wave function will then be predominantly described by the parameters of this closest-lying valence sub-band. Now the Bohr radius of a hydrogen-like state is inversely proportional to the mass. As the situation

Vol.

53,

No.

12

just described becomes established, the transverse size of the Bohr radius (which at zero field is determined by the heavy hole mass) gradually goes over to being described by the light hole mass and thus increases compared to the zero field situation 16,17. This very unusual feature of a growing acceptor wave function with increasing magnetic field has two immediate consequences: first,the ionization energy of the acceptor decreases with field, leading to a "boiloff" of free carriers (opposite of freeze-out) with increasing magnetic field; second, the increase in the spatial extent of the acceptor wave function leads to increasing impurity conduction (and possibly even a magnetically-induced Mott transition). Both effects are observed in the form of a dramatic negative magnetoresistance 13,14,shown in Fig. 4. At low temperatures, where conduction processes are ascribed to impurity conduction, magnetoresistance manifests the anisotropy expected from the disc-like spatial form of the acceptor wave function 13 It should be emphasized that this effect cannot be understood simply as a "magnetic field intensification," as was the case in Sec I--i.e., it could not be observed in a non-magnetic semiconductor even if megaguass fields were available. What makes this possible is the fact that the exchange interaction affects preferentially the spin states--it intensifies spin-splitting, while having relatively little effect on the orbital (Landau) states--and it is thi preferential treatment of the spin states that underlies the phenomenon of giant negative magnetoresistance in Hgl-xMnxTe.

V.

Concluding

Comments

In conclusion, it is important to emphasize the distinctly different physical role played by the exchange interaction in the three examples cited above. In the case of the EDSR, this interaction plays predominantly the role of a magnetic-field intensifier: except for the temperature dependence and saturation effects, the same behavior would be observable in a non-magnetic semiconductor--but at several million gauss. In the case of the BMP, no magnetic field is present to be intensified. The spin splitting is exchange-induced by the finite local magnetization, which exists on the scale of the donor or acceptor due to fluctuations and/or as a self-induced effect. The case of the giant negative magnetoresistance illustrates a behavior unique to DMS (which could not be observed in a non-magnetic semiconductor), relying on the fact that the exchange interaction acts preferentially on the spin states, enhancing the spin splitting dramatically, while leaving the Landau levels practically unaffected.

Acknowledgment The author wishes to thank Profs. J. Mycielski and A.K. Ramdas for enlightening discussions on various aspects of shallow centers in DMS. He also gratefully acknowledges the support of the National Science Foundation (Grant DMR-80-2024g and DMR82-18783) and of the Office of Naval Research (Contract N00014-82-K0563).

Vol. 53, No. 12

SHALLOWCEN~ERS IN DILUTED.MAGNETICSEMICON~UCTORS

[I01

References

[1] [2] [3] [4] [5]

[6]

For a recent review, see, e.g., J.K. Furdyna, J. Appl. Phys. 53, 7&37 (1982), and references cited therein. R.R. Galazka, S. Nagata and P.H. Keesom, Phys. Rev. B22, 3344 (1980). G. Dolling, T.M. Holden, V.F. Sears, J.K. Furdyna, and W. Giriat, J. Appl. Phys. 53, 7644 (1982). A.K. Ramd~, J. Appl. Phys. 53, 7649 (1982}. For a lucid presentation see, e.g., J. Mycielski, Recent Developments in Condensed Matter Physics, Vol. 1, edited by J.T. Devreese (Plenum Press, New York, 1981}, p. 725; and R.R. Galazka and J. Kossut, in Narrow Gap Semiconductors: Physics and Applications, Lecture Notes in Physics, No. 133 (Springer, Berlin, Ig80}, p. 245. M. Dobrowolska, H.D, Drew, J.K. Furdyna, T. Ichiguchi, A. Witowski, and P.A. Wolff, Phys. Rev. Lett. 49, 845 (1982}; Phys. Rev. B29, 6652

(1984).

[7] [8] [9]

R.C. Casella, Phys. Rev. Lett. 5,371 (1960). T. Dietl and J. Spalek, Phys. Rev. Lett. 48, 355 (1982); Phys. Rev. Bo~., 1548 (1983). D. Heiman, P.A. Wolff and G. Warnock, Phys. Rev. B27, 4848 {1983); P.A. Wolff and J. Warnook, J. Appl. Phys. ,5,5, 2300 (1984).

[101 M. Nawrocki, R. Planel, G. Fishman and R.R. Galazka, Phys. Rev. Lett. 46, 735 (1981). [11] D.L. Peterson, A. Petrou, M. Dutta, A.K. Ramdas, and S. Rodriguez, Solid State Commun. ~3 667 (1982). [12] D.L. Peterson, D.U. Bartholomew, U. Debska, A.K. Ramdas, and S. Rodriguez (to be published). [13] A. Mycielski and J. Mycielski, Prec. 15th Int. Conf. on the Physics of Semiconductors, Kyoto, 1980, J. Phys. Soc. Japan 4£9 (1980}, Suppl. A., p. 807. [141 T. Wojtowicz and A. Mycielski, Proc. 16th Int. Conf. on the Physics of Semiconductors, Montpellier, 1982, Physica L [ 7 _ ~ , 476 ( 1983}. [15] J.A. Gaj, J. Ginter and R.R. Galazka, Phys. Status Solidi (b) 89, 655 (t978). [16] J. Mycielski, Prec. Conf. on Applications of High Magnetic Fields in the Physics of Semiconductors, Grenoble, 1982, edited by G. Landwehr (Springer, Berlin, 1983), p. 431. [17] T.R. Gawron and J. Trylski, Prec. 4th Int. Conf. on the Physics of Narrow Gap Semiconductors, Linz, 1981, Lecture Notes in Physics No. 152 (Springer, Berlin, 1982), p. 312.