- Email: [email protected]
Zeeman studies of CdTe-Cd1 − xMnxTe multiquantum wells
Solid.State Electronics Vol. 37. Nos 4-6. pp. 1129-1132. 1994 Copyright ~ 1994 ELsevier Science Ltd 0038-1101(93)E0027-X Printed in Great Britain. All rights reserved 0038-1101/94 $6.00 + 0.00
Verga.on
ZEEMAN STUDIES OF CdTe-Cdl_xMnxTe MULTIQUANTUM WELLS S. JACKSON, S. R. BARDORF,T. STIRNER,W. E. HAGSTON,P. HARRISONand J. E. NICHOLLS Department of Applied Physics, University of Hull, Hull HU6 7RX, England Abstract--Various novel features of the magnetic field splitting associated with the photoluminescence excitation spectra (PLE) and the photoluminescence spectra (PL) for CdTe/Cd t _ ~Mn,Te quantum well structures are described. The unusual Zeeman splitting pattern of the heavy-hole barrier exciton state, together with that of the light-hole well exciton state, is shown to be consistent with the magnetic response of the monolayers adjacent to an interface being different from that of the bulk. In particular it is shown that for the carrier in the conduction band, the magnitude of the exchange integral with the magnetic ions in the first two monolayers is of approximately the same magnitude but of opposite sign to that occurring in the bulk.
i. INTRODUCTION
Dilute magnetic semiconductors (DMS) such as the Cdl_/MnxTe system described here, are characterized by large magneto-optical effects resulting from the carrier-magnetic ion exchange interaction[l]. In the following we describe the manner in which these effects manifest themselves in the photoluminescence (PL) and photoluminescence excitation (PLE) spectra, and we show how such data can be used to obtain information about the structures, such as the extent of the spatial distribution of the donors throughout the system and the influence of interfaces on the carrier-magnetic ion exchange interaction. 2. EXPERIMENT AND THEORY
Since the effects to be described are characteristic of many different quantum well structures we present detailed results for only one sample. The latter was formed from 15 wells of CdTe of width 72 A sandwiched between Cd0.gz5Mn0.075Te barriers of width 156 A. The sample was grown with a VG Semicon V80H molecular beam epitaxy (MBE) system on InSb (001) substrate. Layer thicknesses were determined from a calibration of the molecular flux rates and checked by double crystal X-ray diffraction. A typical PLE spectrum obtained at 1.7 K is shown in Fig. 1. The high quality of the material is evidenced in the spectrum by the presence of both the heavyand light-hole IS and 2S states and the width of the lines (~< 2 meV). For the purpose of the present paper the main focus of interest is the Zeeman splitting (in the Faraday configuration) of the heavy- and lighthole 1S states, the Zeeman splitting of the heavy-hole barrier states and the magnetic field dependence of the PL emission intensities. With regard to the latter, the PL spectra consist of 4 emission bands as shown
in Fig. 2. Two of these are "free" excitons at different intra-well widths together with their associated donor bound excitons ~ 3 meV lower in energy. The relative increase in the higher energy component of each pair (i.e. free or bound) at elevated temperature is consistent with the assignment to intra-well width (as opposed to inter-well width) fluctuations. This follows from the fact that if the levels arose from intra-well width fluctuations, the relative increase in the upper energy component at elevated temperature is simply a consequence of the Boltzmann distribution. However, if the levels were due to inter-well width fluctuations one would not expect any appreciable change in the relative intensities of the two lines, since the tunnelling rates through such wide barriers would not be significant at these low temperatures. Furthermore, if tunnelling did occur, it would be expected to populate the lower energy level at the expense of the higher energy level, which is contrary to what is observed. The energy separation between the free and bound excitons is consistent with the assignment of the latter as an exciton bound to a neutral donor. This assignment is based on envelope calculations we have carried out which show that the binding energy of an electron to a donor varies from ,,-20 meV when the donor is in the centre of the well to ~ 5 meV when the donor is in the centre of the barrier. It is well known from calculations in bulk material, that the binding energy of an exciton to a neutral donor is a small fraction ~ of the binding energy of the donor electron itseltI2]. Use of Hayne's rule[2] suggests ~ >~0.1 for our system. From these considerations it follows that excitons bound to neutral donors will have binding energies varying from ~>2 meV in the centre of the well to >0.5 meV in the centre of the barrier. It is clear that the zero field peak separation ~<3 meV between the "free exciton" and the "bound exciton peak" is consistent with these arguments and would
1129
1130
S. JACKSONet B
HH 1S
1~.1
/ f//
~
II
l
"~
A
fl
1760
Barrier HH I
~62o.~
,~d
al.
"
HH 2S
I
,7097
I
n=2HH
~
I
le95.8
\
j,
~
"k~
e~
o'~"
oI
I
F 1720
/e
.
m
•
I.H 2S
1618
1740
1717 ExcRetion energy (meV)
Barrier HH
,,,
17oo
..i
1680
.\
1 o~
1660 0
' 1
' 2
4 . 3
.
. 4
. 5
6
7
Magnetic field (Tesla)
Fig. 1. A typical PLE spectrum showing the barrier transition and the IS and 2S light- and heavy-hole states.
Fig. 3. Barrier heavy hole (HH) transition as a function of magnetic field in Faraday configuration.
imply that the donors are distributed throughout the well and barrier region. Independent confirmation of the latter conclusion is provided by the magnetic field dependence of the intensity ratio of the free to bound exciton emission, where it is found that, in the Faraday configuration with increasing magnetic field the bound exciton emission intensity decreases whilst the free exciton intensity increases. The explanation of this is based on the following considerations. When a magnetic field is applied, the effective barrier height for the spin-up electron increases whilst that for the spin down electron decreases. Consequently the one electron energy level for the spin up electron increases relative to the bottom of the (non-magnetic) well, whilst that for the spin down electron decreases. Similar considerations apply to the electron bound to the donor atom. However, when an exciton is bound
to a neutral donor, the two electrons (one from the exciton and the other from the donor) pair their spins which means, as a first approximation, that there is negligible change in the energy of such spin-singlet electron pairs, and only the hole (from the exciton) responds to the applied magnetic field. Consequently a stage will be reached where, depending on the initial binding energy of the exciton to the donor (which in turn depends on the spatial position of the donor in the quantum well system), the energy of a free exciton plus neutral donor in the magnetic field, will be lower than that of the neutral donor-bound exciton complex. In other words the applied magnetic field will destabilize the neutral donor-bound exciton complex[3], with the donors in the centre of the barriers (which have the smallest binding energy) being destabilized at the lowest fields, followed successively by excitons bound to neutral donors with spatial positions varying from the centre of the barrier towards the centre of the well at progressively higher magnetic fields. Accompanying the destabilization of the weakest bound exciton complexes, the emission peak separation between the free and bound exciton complexes would be expected to increase, which is also in agreement with observations as can be seen in Fig. 2. The PLE spectra contain two features which are anomalous compared with what would be expected on the basis of observations on the bulk material. However since both these anomalies can be accounted for in terms of the same physical mechanisms, we will deal with them together. Consider first the PLE spectrum associated with excitonic transitions in the barrier. For purposes of clarity, the splitting of the heavy-hole component only is shown, in Fig. 3. The overall magnitude of the splitting is in accord with what we would expect from the bulk measurements. However closer examination of the spectrum in Fig. 3 reveals a clear asymmetry in the + and a - splittings of the heavy-hole exciton components, with the difference between the splittings being ~ 5 meV at a field of 7 T. Such an asymmetry is not present in the Zeeman splitting of dilute magnetic bulk material. If the mean of the ~r + and
1620
>
1618
b
~0 000 000 0
vE
0
1616 OJ r" bJ
• • •
o o •
•
•
1614
HHX 0
~ ~
0 O
0 O
0 o
0
.~_
E
1612
--J D..
1610 DBX 1608 M a g n e t i c Field (Tesla)
Fig. 2. PL emission energy as a function of magnetic field in Faraday configuration.
Zeeman studies of CdTe~d~ _,Mn,Te multiquantum wells a - splittings is fitted to a modified Briilouin function, one can find good overall agreement provided the value of N0~t, which is a measure of the exchange interaction of the electrons in the conduction band with the magnetic ions, is reduced by 40 meV from a value of 220 meV characteristic of the bulk, to a value of 180 meV. To appreciate the significance of this result we note that the magnetic response of the first few monolayers next to an interface will be different from the bulk[4]. Since a barrier has two interfaces, if N is the number of monolayers over which the magnetic response is different, and we denote the corresponding average value of N0~ over these interface regions, by N0a we have, for a barrier of width 50 monolayers, the following result:
2N Noct + (50 - 2N)Noa 50
~"
:1 !
•
x f *
1131
-
+
_
.-::::- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
o
Q -4-~ -5
_
lI
i
0
1
I
I
2 5 magnetic
i
•
•
I
i
4 5 6 field B (T)
•
•
I
i
7
8
Fig. 5. Zeeman splitting of the light-hole exciton transition together with theoretical fits.
= 180,
from which: 1000 No~ -No~x = - - - N
where
This shows that if N ~ 4 then No~ is of opposite sign to N0~. Detailed studies we have carried out on surface roughness effects[5] show that in our samples N ~ 2. Utilization of this result in the above equation shows that: No • ~ - 280 meV, i.e. the value of the exchange interaction term for the first few monolayers is of opposite sign, but approximately the same order of magnitude, as the exchange interaction appropriate to the bulk. This feature is of direct relevance to understanding the apparent anomaly of why the light-hole exciton splittings are comparable with those of the heavyhole excitons in the wells, as shown in Fig. 4. To see why this is anomalous we note that, in a magnetic barrier, the ratio R of the light-hole splitting to the heavy-hole splitting is given by: R
-
~
B -3A _ - 3(B + A )'
1633
.--.----.++ 1628
I,g ..i
1623
HH
1618 1613
,
0
1
2
,
L
i
3 4 6 M a g n e t i c field ( T e s l a )
7
Fig. 4. Light-hole (LH) and heavy-hole (HH) transition energies as a function of magnetic field in Faraday configuration.
B
Nofl
A
N0a
exchange constant for hole exchange constant for electron"
In bulk material B/A = 4 thus giving a value of R=I/15. Correspondingly this ratio should be reflected in the splittings of the excitons in the well region, which is why the light-hole exciton splitting is expected to be much smaller than that of the heavyhole exciton. This is contrary to what is observed. If, however, we replace A by - A the ratio R becomes 7/9 and the two splittings will become comparable, as observed. Since for wide wells the carrier wavefunction penetrates only a short distance into the barrier and thus, in effect, sees the "larger value" of the ratio R, we have a simple explanation of the anomalously large splittings of the light-hole states. A more detailed calculation substantiates the basic validity of this idea. The dashed lines in Fig. 5 show the theoretical splittings of the light-hole state assuming the barrier responds to the magnetic field as in the bulk material. However the much closer fit shown by the full lines is given by introducing "interface potentials". For the curve shown, it was assumed that in the first two monolayers the value of N0~t had the same magnitude as the bulk but was of opposite sign, whereas the sign of Nofl was kept the same as in the bulk but its magnitude was double. Other choices of parameters could of course be made, but the values quoted are taken as being representative of those that yield reasonable agreement with experiment. In summary we see that (due to the regions adjacent to the interface) there will be a difference (A V) between the potential "appropriate" to the "finite" barrier and the potential appropriate to the bulk (i.e. "infinitely" wide barrier). Furthermore the difference (A V) will have a magnetic field dependence. We will now show how this concept enables us to account for the asymmetry in the Zeeman splitting. Consider first
1132
S. JACKSONet al.
the heavy-hole exciton states in the barrier. Strictly speaking we should incorporate the term A V in the usual 8-band k . p perturbation scheme appropriate to the valence and conduction band region[I]. (Note the effects of strain are readily incorporated in this formalism, but in the present system they play a relatively minor role, in that in the well region strain lifts the degeneracy of the light- and heavy-hole states, however the energy changes are small compared with the effects of carrier confinement and the overall energy band offset.) However it is clear, in qualitative terms, what the effects of A V will be. From the viewpoint of perturbation theory A V will produce, in first order, a change in the one-electron energies, whilst in second order it will simply repel adjacent energy levels. This means for example, that the energy level separation between the lowest conduction and (heavy-hole) valence band will become magnetic field dependent--i.e, there will be, in effect, a field dependent bandgap renormalization. This readily explains the asymmetry in the Zeeman splitting between the ~r ÷ and ~r- states, with a total asymmetry of 5 meV for example, corresponding to a bandgap renormalization of 2.5 meV. (Such a renorrealization increases the apparent Zeeman splitting of the a + state by 2.5 meV, whilst decreasing that of the a - state by 2.5 meV.)
3, CONCLUSION The fact that the magnetic response of the first few monolayers in a magnetic barrier could differ from the rest of the barrier region has been noted by other
workers[4]. Similarly a reduction in the average value of N0~ for narrow barriers has been seen in Faraday rotation experiments[6]. However the present results are the first, to our knowledge, to present evidence for the quite dramatic changes in the exchange parameters in the monolayer regions next to an interface--namely a reversal in the sign of N0~ and a doubling in the value of Nofl. This has been made possible because of the high quality of the structures studied, when coupled with the degree of wavefunction localization in these structures. Acknowledgements--The authors wish to thank the Science and Engineering Research Council (U.K.) for supporting this work. One author (T. S.) would like to thank the University of Hull for the award of a Brynmor Jones Scholarship. REFERENCES
1. Semiconductors and Semimetals (R. K. Willardson and A. C. Beer, Treatise Editors; J. K. Furdyna and J. Kossut, Volume Editors), Vol. 25. Academic Press, Boston (1988). 2. Physics and Chemistry o f H - V l Compounds (Edited by J. S. Prener and N. Aven). North Holland, Amsterdam (1969). 3. T. J. Gregory, J. E. Nicholls, J. J. Davies, J. O. Williams and N. Maung, Semicond. Sci. Teehnol. 3, 1193 (1988). 4. D. R. Yakovlev. Festk6rperprobleme, Adt~ances in Solid State Physics, Vol. 32 (Edited by U. Rossler), p. 251. Vieweg, Braunschweig/Wiesbaden (1992). 5. P. Harrison, T. Stirner, S. R. Bardorf, W. E. Hagston, S. Jackson, K. A. Dhese, J. H. C. Hogg, V. Hewer, J. E. Nicholls and M. O'Neill, Superlatt. Microstruet. 13, 431 (1993). 6. M. Kohl, M. R. Freeman, J. M. Hong and D. D. Awschalom, Phys. Rev. B 43, 2431 (1991).
Verga.on
ZEEMAN STUDIES OF CdTe-Cdl_xMnxTe MULTIQUANTUM WELLS S. JACKSON, S. R. BARDORF,T. STIRNER,W. E. HAGSTON,P. HARRISONand J. E. NICHOLLS Department of Applied Physics, University of Hull, Hull HU6 7RX, England Abstract--Various novel features of the magnetic field splitting associated with the photoluminescence excitation spectra (PLE) and the photoluminescence spectra (PL) for CdTe/Cd t _ ~Mn,Te quantum well structures are described. The unusual Zeeman splitting pattern of the heavy-hole barrier exciton state, together with that of the light-hole well exciton state, is shown to be consistent with the magnetic response of the monolayers adjacent to an interface being different from that of the bulk. In particular it is shown that for the carrier in the conduction band, the magnitude of the exchange integral with the magnetic ions in the first two monolayers is of approximately the same magnitude but of opposite sign to that occurring in the bulk.
i. INTRODUCTION
Dilute magnetic semiconductors (DMS) such as the Cdl_/MnxTe system described here, are characterized by large magneto-optical effects resulting from the carrier-magnetic ion exchange interaction[l]. In the following we describe the manner in which these effects manifest themselves in the photoluminescence (PL) and photoluminescence excitation (PLE) spectra, and we show how such data can be used to obtain information about the structures, such as the extent of the spatial distribution of the donors throughout the system and the influence of interfaces on the carrier-magnetic ion exchange interaction. 2. EXPERIMENT AND THEORY
Since the effects to be described are characteristic of many different quantum well structures we present detailed results for only one sample. The latter was formed from 15 wells of CdTe of width 72 A sandwiched between Cd0.gz5Mn0.075Te barriers of width 156 A. The sample was grown with a VG Semicon V80H molecular beam epitaxy (MBE) system on InSb (001) substrate. Layer thicknesses were determined from a calibration of the molecular flux rates and checked by double crystal X-ray diffraction. A typical PLE spectrum obtained at 1.7 K is shown in Fig. 1. The high quality of the material is evidenced in the spectrum by the presence of both the heavyand light-hole IS and 2S states and the width of the lines (~< 2 meV). For the purpose of the present paper the main focus of interest is the Zeeman splitting (in the Faraday configuration) of the heavy- and lighthole 1S states, the Zeeman splitting of the heavy-hole barrier states and the magnetic field dependence of the PL emission intensities. With regard to the latter, the PL spectra consist of 4 emission bands as shown
in Fig. 2. Two of these are "free" excitons at different intra-well widths together with their associated donor bound excitons ~ 3 meV lower in energy. The relative increase in the higher energy component of each pair (i.e. free or bound) at elevated temperature is consistent with the assignment to intra-well width (as opposed to inter-well width) fluctuations. This follows from the fact that if the levels arose from intra-well width fluctuations, the relative increase in the upper energy component at elevated temperature is simply a consequence of the Boltzmann distribution. However, if the levels were due to inter-well width fluctuations one would not expect any appreciable change in the relative intensities of the two lines, since the tunnelling rates through such wide barriers would not be significant at these low temperatures. Furthermore, if tunnelling did occur, it would be expected to populate the lower energy level at the expense of the higher energy level, which is contrary to what is observed. The energy separation between the free and bound excitons is consistent with the assignment of the latter as an exciton bound to a neutral donor. This assignment is based on envelope calculations we have carried out which show that the binding energy of an electron to a donor varies from ,,-20 meV when the donor is in the centre of the well to ~ 5 meV when the donor is in the centre of the barrier. It is well known from calculations in bulk material, that the binding energy of an exciton to a neutral donor is a small fraction ~ of the binding energy of the donor electron itseltI2]. Use of Hayne's rule[2] suggests ~ >~0.1 for our system. From these considerations it follows that excitons bound to neutral donors will have binding energies varying from ~>2 meV in the centre of the well to >0.5 meV in the centre of the barrier. It is clear that the zero field peak separation ~<3 meV between the "free exciton" and the "bound exciton peak" is consistent with these arguments and would
1129
1130
S. JACKSONet B
HH 1S
1~.1
/ f//
~
II
l
"~
A
fl
1760
Barrier HH I
~62o.~
,~d
al.
"
HH 2S
I
,7097
I
n=2HH
~
I
le95.8
\
j,
~
"k~
e~
o'~"
oI
I
F 1720
/e
.
m
•
I.H 2S
1618
1740
1717 ExcRetion energy (meV)
Barrier HH
,,,
17oo
..i
1680
.\
1 o~
1660 0
' 1
' 2
4 . 3
.
. 4
. 5
6
7
Magnetic field (Tesla)
Fig. 1. A typical PLE spectrum showing the barrier transition and the IS and 2S light- and heavy-hole states.
Fig. 3. Barrier heavy hole (HH) transition as a function of magnetic field in Faraday configuration.
imply that the donors are distributed throughout the well and barrier region. Independent confirmation of the latter conclusion is provided by the magnetic field dependence of the intensity ratio of the free to bound exciton emission, where it is found that, in the Faraday configuration with increasing magnetic field the bound exciton emission intensity decreases whilst the free exciton intensity increases. The explanation of this is based on the following considerations. When a magnetic field is applied, the effective barrier height for the spin-up electron increases whilst that for the spin down electron decreases. Consequently the one electron energy level for the spin up electron increases relative to the bottom of the (non-magnetic) well, whilst that for the spin down electron decreases. Similar considerations apply to the electron bound to the donor atom. However, when an exciton is bound
to a neutral donor, the two electrons (one from the exciton and the other from the donor) pair their spins which means, as a first approximation, that there is negligible change in the energy of such spin-singlet electron pairs, and only the hole (from the exciton) responds to the applied magnetic field. Consequently a stage will be reached where, depending on the initial binding energy of the exciton to the donor (which in turn depends on the spatial position of the donor in the quantum well system), the energy of a free exciton plus neutral donor in the magnetic field, will be lower than that of the neutral donor-bound exciton complex. In other words the applied magnetic field will destabilize the neutral donor-bound exciton complex[3], with the donors in the centre of the barriers (which have the smallest binding energy) being destabilized at the lowest fields, followed successively by excitons bound to neutral donors with spatial positions varying from the centre of the barrier towards the centre of the well at progressively higher magnetic fields. Accompanying the destabilization of the weakest bound exciton complexes, the emission peak separation between the free and bound exciton complexes would be expected to increase, which is also in agreement with observations as can be seen in Fig. 2. The PLE spectra contain two features which are anomalous compared with what would be expected on the basis of observations on the bulk material. However since both these anomalies can be accounted for in terms of the same physical mechanisms, we will deal with them together. Consider first the PLE spectrum associated with excitonic transitions in the barrier. For purposes of clarity, the splitting of the heavy-hole component only is shown, in Fig. 3. The overall magnitude of the splitting is in accord with what we would expect from the bulk measurements. However closer examination of the spectrum in Fig. 3 reveals a clear asymmetry in the + and a - splittings of the heavy-hole exciton components, with the difference between the splittings being ~ 5 meV at a field of 7 T. Such an asymmetry is not present in the Zeeman splitting of dilute magnetic bulk material. If the mean of the ~r + and
1620
>
1618
b
~0 000 000 0
vE
0
1616 OJ r" bJ
• • •
o o •
•
•
1614
HHX 0
~ ~
0 O
0 O
0 o
0
.~_
E
1612
--J D..
1610 DBX 1608 M a g n e t i c Field (Tesla)
Fig. 2. PL emission energy as a function of magnetic field in Faraday configuration.
Zeeman studies of CdTe~d~ _,Mn,Te multiquantum wells a - splittings is fitted to a modified Briilouin function, one can find good overall agreement provided the value of N0~t, which is a measure of the exchange interaction of the electrons in the conduction band with the magnetic ions, is reduced by 40 meV from a value of 220 meV characteristic of the bulk, to a value of 180 meV. To appreciate the significance of this result we note that the magnetic response of the first few monolayers next to an interface will be different from the bulk[4]. Since a barrier has two interfaces, if N is the number of monolayers over which the magnetic response is different, and we denote the corresponding average value of N0~ over these interface regions, by N0a we have, for a barrier of width 50 monolayers, the following result:
2N Noct + (50 - 2N)Noa 50
~"
:1 !
•
x f *
1131
-
+
_
.-::::- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
o
Q -4-~ -5
_
lI
i
0
1
I
I
2 5 magnetic
i
•
•
I
i
4 5 6 field B (T)
•
•
I
i
7
8
Fig. 5. Zeeman splitting of the light-hole exciton transition together with theoretical fits.
= 180,
from which: 1000 No~ -No~x = - - - N
where
This shows that if N ~ 4 then No~ is of opposite sign to N0~. Detailed studies we have carried out on surface roughness effects[5] show that in our samples N ~ 2. Utilization of this result in the above equation shows that: No • ~ - 280 meV, i.e. the value of the exchange interaction term for the first few monolayers is of opposite sign, but approximately the same order of magnitude, as the exchange interaction appropriate to the bulk. This feature is of direct relevance to understanding the apparent anomaly of why the light-hole exciton splittings are comparable with those of the heavyhole excitons in the wells, as shown in Fig. 4. To see why this is anomalous we note that, in a magnetic barrier, the ratio R of the light-hole splitting to the heavy-hole splitting is given by: R
-
~
B -3A _ - 3(B + A )'
1633
.--.----.++ 1628
I,g ..i
1623
HH
1618 1613
,
0
1
2
,
L
i
3 4 6 M a g n e t i c field ( T e s l a )
7
Fig. 4. Light-hole (LH) and heavy-hole (HH) transition energies as a function of magnetic field in Faraday configuration.
B
Nofl
A
N0a
exchange constant for hole exchange constant for electron"
In bulk material B/A = 4 thus giving a value of R=I/15. Correspondingly this ratio should be reflected in the splittings of the excitons in the well region, which is why the light-hole exciton splitting is expected to be much smaller than that of the heavyhole exciton. This is contrary to what is observed. If, however, we replace A by - A the ratio R becomes 7/9 and the two splittings will become comparable, as observed. Since for wide wells the carrier wavefunction penetrates only a short distance into the barrier and thus, in effect, sees the "larger value" of the ratio R, we have a simple explanation of the anomalously large splittings of the light-hole states. A more detailed calculation substantiates the basic validity of this idea. The dashed lines in Fig. 5 show the theoretical splittings of the light-hole state assuming the barrier responds to the magnetic field as in the bulk material. However the much closer fit shown by the full lines is given by introducing "interface potentials". For the curve shown, it was assumed that in the first two monolayers the value of N0~t had the same magnitude as the bulk but was of opposite sign, whereas the sign of Nofl was kept the same as in the bulk but its magnitude was double. Other choices of parameters could of course be made, but the values quoted are taken as being representative of those that yield reasonable agreement with experiment. In summary we see that (due to the regions adjacent to the interface) there will be a difference (A V) between the potential "appropriate" to the "finite" barrier and the potential appropriate to the bulk (i.e. "infinitely" wide barrier). Furthermore the difference (A V) will have a magnetic field dependence. We will now show how this concept enables us to account for the asymmetry in the Zeeman splitting. Consider first
1132
S. JACKSONet al.
the heavy-hole exciton states in the barrier. Strictly speaking we should incorporate the term A V in the usual 8-band k . p perturbation scheme appropriate to the valence and conduction band region[I]. (Note the effects of strain are readily incorporated in this formalism, but in the present system they play a relatively minor role, in that in the well region strain lifts the degeneracy of the light- and heavy-hole states, however the energy changes are small compared with the effects of carrier confinement and the overall energy band offset.) However it is clear, in qualitative terms, what the effects of A V will be. From the viewpoint of perturbation theory A V will produce, in first order, a change in the one-electron energies, whilst in second order it will simply repel adjacent energy levels. This means for example, that the energy level separation between the lowest conduction and (heavy-hole) valence band will become magnetic field dependent--i.e, there will be, in effect, a field dependent bandgap renormalization. This readily explains the asymmetry in the Zeeman splitting between the ~r ÷ and ~r- states, with a total asymmetry of 5 meV for example, corresponding to a bandgap renormalization of 2.5 meV. (Such a renorrealization increases the apparent Zeeman splitting of the a + state by 2.5 meV, whilst decreasing that of the a - state by 2.5 meV.)
3, CONCLUSION The fact that the magnetic response of the first few monolayers in a magnetic barrier could differ from the rest of the barrier region has been noted by other
workers[4]. Similarly a reduction in the average value of N0~ for narrow barriers has been seen in Faraday rotation experiments[6]. However the present results are the first, to our knowledge, to present evidence for the quite dramatic changes in the exchange parameters in the monolayer regions next to an interface--namely a reversal in the sign of N0~ and a doubling in the value of Nofl. This has been made possible because of the high quality of the structures studied, when coupled with the degree of wavefunction localization in these structures. Acknowledgements--The authors wish to thank the Science and Engineering Research Council (U.K.) for supporting this work. One author (T. S.) would like to thank the University of Hull for the award of a Brynmor Jones Scholarship. REFERENCES
1. Semiconductors and Semimetals (R. K. Willardson and A. C. Beer, Treatise Editors; J. K. Furdyna and J. Kossut, Volume Editors), Vol. 25. Academic Press, Boston (1988). 2. Physics and Chemistry o f H - V l Compounds (Edited by J. S. Prener and N. Aven). North Holland, Amsterdam (1969). 3. T. J. Gregory, J. E. Nicholls, J. J. Davies, J. O. Williams and N. Maung, Semicond. Sci. Teehnol. 3, 1193 (1988). 4. D. R. Yakovlev. Festk6rperprobleme, Adt~ances in Solid State Physics, Vol. 32 (Edited by U. Rossler), p. 251. Vieweg, Braunschweig/Wiesbaden (1992). 5. P. Harrison, T. Stirner, S. R. Bardorf, W. E. Hagston, S. Jackson, K. A. Dhese, J. H. C. Hogg, V. Hewer, J. E. Nicholls and M. O'Neill, Superlatt. Microstruet. 13, 431 (1993). 6. M. Kohl, M. R. Freeman, J. M. Hong and D. D. Awschalom, Phys. Rev. B 43, 2431 (1991).