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Energies of electrons bound to nitrogen pairs in GaP
Solid State Communications, Vol. 36, pp. 563 —566 Pergamon Press Ltd. 1980. Printed in Great Britain. ENERGIES OF ELECTRONS BOUND TO NITROGEN PAIRS IN GaP F. Thuselt and K. Kreher Sektion Physik der Karl-Marx.Universität, Leipzig, DDR Arbeitsgemeinschaft A111Bv-I-Ialbleiter, DDR.70 10 Leipzig, Linnéstral~e5, DDR and H..J. Wunsche Sektion Physik der Humboldt-Universität zu Berlin, Bereich Theoretische Haibleiterphysik, DDR-1086 Berlin, Postfach 1297, DDR (Received 27 June 19801w A.R. Miedenia)
The binding energies of electrons bound to nitrogen pairs in GaP are evaluated by use of the effective mass theory with a suitable model potential. The variational procedure yields (I) satisfactory agreement with previous experimental values for nitrogen pairs and (ii) no bound state for the isolated nitrogen atom. Our results suggest the electron to be bound in a potential with strain.likc contributions which approaches earlier ideas of Allen. NITROGEN represents an isoelectronic impurity in GaP which has been studied intensively because of the fundamental scientific and practical aspects of its optical properties. With nitrogen doping the lumin. csccnce spectra in Gal’ crystals are dramatically altered both at low [1—3Jand high [4—7] excitation intensities. Sharp absorption and emission zero phonon lines give evidence for bound exciton states produced by the impurity potential. Excitons can also be bound by pairs of nitrogen labelled NN, (1 = 1, 2, ) where i indicates the separation of atoms in the phosphorus sublattice. These NN1 lines form a series raising in energy with increasing pair distance. This series converges to the A/B.line of the isolated nitrogen exciton [I, 8]. Due to precise excitation.spectroscopic measurements of Cohen and Sturge [8] also the existence of excited exciton states for near-neighbour NN-pairs is proved. These states reveal a hydrogenic series of acceptor-like transitions with a continuum resulting from bound NN-electron—free hole states. In contrast to the case of hydrogen-like impurities, the experimental features of both exciton and electron nitrogen states are beyond any successful explanation. As a reason for this failure the lack of knowledge of the real impurity potential within the host may be respon. sible. A generally accepted fact is its short-ranging caused either by: (i) differences in the electron affinities (electronegativities) between impurity and host atom [9, 10] ; . .
(ii) strain fields around the impurity [II, 121; or (iii) both (13, 141. Obviously, the explanation of the binding energies of the NN electron states in terms of screened pseudopotential differences is a hard task. Although we know results which are in broad agreement with experiment [101 the calculated values scatter in detail considerably; in particular the monotonous decrease in binding energy with rising pair separation is by no means reflected (c.f. Fig. 2 of [10]). Further, Baldareschi [14] pointed out that the lattice relaxation modifies the impurity potential significantly. At the present stage of knowledge it seems to be rather hopeless to include these contributions into the pseudopotential and to carry out an ab initio calculation of binding energies. In this paper we make an attempt to find a reliable model potential for the isoelectronic N impurity in GaP which will be adjusted to the experimental electron energies of the NN pairs by solving the Schrodinger equation with a variational Ansatz. This is favoured by the successachieved when employing model potentials for problems in the mixed crystals GaAs1_~P~ [15]. To get an idea of the behaviour of the model potential we use the proposition of Allen [11] that a considerable part of the exciton binding energy comes from the strain field around the N impurity as a consequence of its size difference with respect to the replaced lattice atom. He approximates the crystal in the vicinity of the impurity as an elastic isotropic
.
563
564
ENERGIES OF ELECTRONS BOUND TO NITROGEN PAIRS IN GaP
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exciton counted binding energy from the A line 10
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.
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____ I
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3456 810 1520 ~ NN PAIR DISTANCE IA
Fig. I. Triangles: Experimental binding energy differences E1’I between excitons bound to NN pairs and the exciton bound to isolated nitrogen. The energies are plotted as a function of pair separation as suggested by Allen [111. Full circles: Energies of NN electrons Ef as a function of pair separation. The values stem from Cohen and Sturge [8]. The straight line (slope 3) in the double-logarithmic dence of both energies. scale indicates the r depen—
—
continuum which resultscontributions). in a band-edge Assuming shift varying 3 (apart from angular the as r wavefunction of the electron and the hole to be strongly localized the difference between the observed NN1 exciton level E7 and the A/B line of isolated nitrogen should be identified with the strain contribution of the additional N atom in the pair, i.e. 3 I E7 E~I R~ (R 1 is the pair spacing). This law is almost obeyed by experiment as shown by the triangles in Fig. L The shortcoming of isthe Allen model in the fact that the strain field attractive bothconsists for electron and holes, therefore the hole as well as the electron should be strongly localized. The results of excitation spectroscopy of Cohen and Sturge [81 however, confirm an alternative picture first proposed by HopfIeld, Thomas, and Lynch (HTL) [2]: In the HTL model only the electron is strongly localized due to the electronegative nitrogen whereas the hole is bound in the acceptor-like Coulomb field of the electron. This model —
,
Co
f
iø~ -
I
I
I ~ ~ 5 6 10 20 • i Fig. 2. Eigenvalues of the NN electron problem obtained —
•
with a variational Ansatz for the wave function in the strain-like potential. The energies are plotted as a function of pair separation in relative meaning of the potential(both parameters V units). The 0 and a0 is explained in the text. Note the logarithmic scales! explains very well the differences observed between electron and exciton binding energies at NN pairs but not the origin of the electron binding energy. There is nothing which prevents the Allen mechanism to be responsible for the localization of the electron. lndeed, if its binding energy Ef is plotted as a function of the NN pair separation (solid circles in Fig. 1) these 3. energies are comparable with I E7 the E~I and varytoasbe R bound in a In this paper we argue electron potential consisting of two parts: —
(I) a short.range part originating from pseudopotential differences and some other contributions; (ii) a medium ranging tail caused by strain effects. Whereas the short-range part will not be specified further and approximated with an appropriate Ansatz the tail should behave as r3 on the assumption of the
ENERGIES OF ELECTRONS BOUND TO NITROGEN PAIRS IN GaP
Vol. 36, No.6
~
NH PAIR DISTANCE 10
U
(M
—
.3
NN~
C
~
NN
N~
8
fIt wifh (square welt. r~) potential
potential -150 ‘NN ~NN~
10
IS
6
L
R
______
a ‘4~3.O8eV
Fig. 4. Superposition of the two nitrogen model potentials in the case of NN3 pair (straight line). The dotted line demonstrates the shape of the equivalent squarewell Potentialvalues parameters which provided the best potentials. fIt to experimental are given. art effective potential to be chosen so as to include all its intervalley and umklapp contributions. This practice has been suggested in the paper of Kirczenow eta!. [22]. Our model potential produced by a single N atom will be set in the form
.~5
0. ml
0
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v~v4~
———V a 0.1.88eV 2X1.185A
fit withwell square
—
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~
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Fig. 3. Upper part: Fit of the theoretical electron binding energies by use of the potential of equation (1) (straight line) to the experimental ones (labelled NN~). The experimental exciton binding energies (NN7) are also shown for comparison. Experimental points origmate from Cohen and Sturge [8]. The dotted line is the fit with part: Corresponding the simplifiedvariational square-well Bohr-radii potential.forLower both strain-like (straight line) and square-well (dotted line) potential as a function of NN-pair distance. The bound state vanishes above NN~Ousing the strain-like and above NN~the square-well potential. angular dependence to be smeared out due to the spreading of the wavefunction. The question arises of whether the parameters of this potential can be chosen in such a way as to produce the observed electron binding energies of the NN electrons in form of solutions of the proper one-particle Schrodinger equations. In contrast to previous papers dealing with analogous problems [9, 10, 15] we use the effective mass (EM) formalism. Recently several authors [16—22] showed that the EM formalism works well also for potentials with strong central-cell, i.e. short-ranging part. In this case we are concerned with intervalley and umklapp termsofofour thebare waveshort-ranging functions. Due lack of knowledge parttowetherequire
— —
t
— —
3 V0 V0(a0/r)
(r (r >< aa0) (square-well part), 0) (strain-like part), (1)
where the parameter V0 must be fitted to the experimental values of the electron energies E7 and a0 will piesumably lie in the region of half the nearest-neighbour distance (~1.185 A) in GaP. Clearly, this potential 3 tail. In the above shape it has exhibits the advantage the required of being r very smooth in Fourier representation a fact which additionally recommends the applicability of the EM theory. To avoid the problems related with intervalley terms of the kinetic energy and the detailed GaP band structure [23) we assume an isotropic parabolic dispersion law. The choice of an appropriate effective mass m5 will be discussed below. The 1-lamiltonian for one electron bound to an NN pair with separation R~is H = (— h2/2,n5) ~2 + V(r) + V(r + R,). (2) This problem corresponds to that of the H~ion with the exception that a term describing the repelling of the two nuclei is missed here. Such forces are held by the lattice. For the wave function we make the Ansatz [2(1 + ~)J~ {u 1(r) + u1(r + R,)), i~ =
u1(r)
(I + a,R1 + a?R~/3) exp (—~1R,). 1/x)”2 exp (— c~r) (cr~ °
=
(3)
(4) (5)
566
ENERGIES OF ElECTRONS BOUND TO NITROGEN PAIRS IN GaP
is a hydrogenic function with a1 = a~jas variational parameter. The determination of the eigenvalues of equation (2) is standard. As a result of the variation the binding energies in units of the potential strength V0 are plotted in Fig. 2 as a function of the pair2(2ma~ distance.V The best fit to a — 3 slope is obtained with h 0)’ = 0.88. If for simplicity a0 is set exactly equal to 1.185 A then the depth V0 which can now be extracted from experiment, turns out to be 3.08 eV. In our procedure the effective mass m~is actually not explicitly needed because it appears in the cornputation only via the product m’a~V0. Our fit corre sponds to an effective mass approximately equal to unity. This large value which differs from the distinctly smaller one used for more extended wave functions, should be attributed to the real band structure effects discussed before. It is of interest to note that an isotropic effective mass value of unity together with the medium dielectric constant of 11.19 [24] just yields the binding energy of the isocoric sulphur donor E5 = 107 meV [25] in single-valley approximation. The fit to the absolute values of experimental electron binding energies is illustrated by the straight line in Fig. 3. For the isolated nitrogen atom (corresponding to NNO.) no bound state occurs. The outcome of our calculations is a model potential for the electrons which may be useful for further problems. In some cases, however, the potential might be still too complicated to apply it. This situation can occur when needing its Fourier transform which is no elementary function. For this reason we have carried out a calculation with a simpler square-well potential. Best fIt is obtained with parameters indicated in Fig. 4 in comparison with the strain-like potential. The resulting binding energies are characterized by the dotted curve in Fig. 3. It is evident that the agreement is not as good as before but for some purposes it might be satisfactory. This can happen, e.g. where nitrogen acts as a scattering potential for free carriers as in electron— hole liquids. The problem next to be tackled is the binding of the exciton to nitrogen or its pairs. Theoretical work on this field is under way. Acknowledgements — We thank Prof. K. Unger and Dr. R. Schwabe for many stimulating discussions and Mrs. 1-I. Seewitz for carefully preparing the manuscript. REFERENCES 1.
D.G. Thomas & J.J. Hopfield, Phys. Rev. 150, 680 (1966).
2. 3. 4.
5. 6. 7. 8 9. 10. 11. 11
Vol. 36, No.6
J.J. Hopfield, D.G. Thomas & R.T. Lynch, Ph vs. Rev. Lett. 17,312 (1966). For a review see also W. Czaja, FestkOrperproblerne 11,65 (1971). R. Schwabe, F. Thuselt, R. Bindemann, W. Seifert & K. Jacobs, Phys. Lett. 64A, 226 (1977). R. Schwabe, F. Thuselt, H. Weinert & R. Bindemann, Phys. Status Solidi (b) 95, 571 (1979). J.E. Kardontchik & E. Cohen, Pvs. Rev. Bl9, 3181 (1979). R. Schwabe, F. Thuselt, H. Weinert & R. Bindemann, Proc. tnt. Conf RECON ‘79, p. 191. Prague (1979). E. Cohen & M.D. Sturge, P/n’s. Rev. BIS. 1039 (1977). R.A. Faulkner, P/it’s. Rev. 175, 991 (1968). M. Brand & M. Jaros,J. P/it’s. Cl2, 2789 (1979). J.W. J.W. Allen,J. Allen,J. P/it’s. Phvs. Cl, C4, 1136(1968). 1936 (1971).
—.
13. 14 15:
16 17. 18. 19 20. 2!. 2’ —.
23.
24. 25.
J.C. Phillips, P/n’s. Rev. Left. 22, 285 (1968). A. Baldareschi,J. Lwninesc. 7, 79 (1973). W.Y. Ilsu, J.D. Dow, D.J. Wolford & B.G. Streetman, Phys. Rev. B 16, 1597 (1977); G. Klcirnan, Phys. Rev. Bl9, 3198 (1979) (and earlier work cited therein). K. Shindo & U. Nara, J. Phys. Soc. Japan 40, 1640 (1976). R. Resta,J. Pkvs. dO, L179 (1977). M. Altarelli, W.Y. Ilsu & R.A. Sabatini, J. l’hys. ClO. L605 (1977). D.C. Herbert & J. Inkson,J. P/n’s. dO, L695 (1977). L. Resca & R. Resta, Solid State commun. 29, 275 (1979). S.T. Pantelides, Solid State com,nun. 30, 65 (1979). G. Kirczenow, R. Barrie & B. Bergersen, P/:ys. Rev. Bl9, 2139 (1979). Complications in kinetic-energy computations for wave functions which are not too localized in k-space arise from (I) camel’s back in (100). direction, (ii) the fact that E(k) generally raises only in the vicinity of a given minimum in the Brillouin zone but falls when coming to another minimum (see the discussion in 1191), and (iii) deviations from rotational symmetry around the X point in the direction perpendicular to (100> [especially in the XW direction the band structure becomes non-parabolic. & T.K. Bergstresser, Phys. Rev. C.f. 141,M.L. 789 Cohen (1966)1. R. Bindemann, Ii. Hempel & K. Kreher, P/n’s. Status Solidi (a) 52, K2OI (1979). A.A. Kopylov & AN. Pikh tin, Fiz. Tekh. PaInpray. II, 867 (1977).
The binding energies of electrons bound to nitrogen pairs in GaP are evaluated by use of the effective mass theory with a suitable model potential. The variational procedure yields (I) satisfactory agreement with previous experimental values for nitrogen pairs and (ii) no bound state for the isolated nitrogen atom. Our results suggest the electron to be bound in a potential with strain.likc contributions which approaches earlier ideas of Allen. NITROGEN represents an isoelectronic impurity in GaP which has been studied intensively because of the fundamental scientific and practical aspects of its optical properties. With nitrogen doping the lumin. csccnce spectra in Gal’ crystals are dramatically altered both at low [1—3Jand high [4—7] excitation intensities. Sharp absorption and emission zero phonon lines give evidence for bound exciton states produced by the impurity potential. Excitons can also be bound by pairs of nitrogen labelled NN, (1 = 1, 2, ) where i indicates the separation of atoms in the phosphorus sublattice. These NN1 lines form a series raising in energy with increasing pair distance. This series converges to the A/B.line of the isolated nitrogen exciton [I, 8]. Due to precise excitation.spectroscopic measurements of Cohen and Sturge [8] also the existence of excited exciton states for near-neighbour NN-pairs is proved. These states reveal a hydrogenic series of acceptor-like transitions with a continuum resulting from bound NN-electron—free hole states. In contrast to the case of hydrogen-like impurities, the experimental features of both exciton and electron nitrogen states are beyond any successful explanation. As a reason for this failure the lack of knowledge of the real impurity potential within the host may be respon. sible. A generally accepted fact is its short-ranging caused either by: (i) differences in the electron affinities (electronegativities) between impurity and host atom [9, 10] ; . .
(ii) strain fields around the impurity [II, 121; or (iii) both (13, 141. Obviously, the explanation of the binding energies of the NN electron states in terms of screened pseudopotential differences is a hard task. Although we know results which are in broad agreement with experiment [101 the calculated values scatter in detail considerably; in particular the monotonous decrease in binding energy with rising pair separation is by no means reflected (c.f. Fig. 2 of [10]). Further, Baldareschi [14] pointed out that the lattice relaxation modifies the impurity potential significantly. At the present stage of knowledge it seems to be rather hopeless to include these contributions into the pseudopotential and to carry out an ab initio calculation of binding energies. In this paper we make an attempt to find a reliable model potential for the isoelectronic N impurity in GaP which will be adjusted to the experimental electron energies of the NN pairs by solving the Schrodinger equation with a variational Ansatz. This is favoured by the successachieved when employing model potentials for problems in the mixed crystals GaAs1_~P~ [15]. To get an idea of the behaviour of the model potential we use the proposition of Allen [11] that a considerable part of the exciton binding energy comes from the strain field around the N impurity as a consequence of its size difference with respect to the replaced lattice atom. He approximates the crystal in the vicinity of the impurity as an elastic isotropic
.
563
564
ENERGIES OF ELECTRONS BOUND TO NITROGEN PAIRS IN GaP
~150-
I
-~
•
0.7
—
a•
a. w
_________
~100 w z w
•
Vol. 36, No.6 ~•
••.•
I
0.
083
-‘ -~
0.8
~
0.68
I
z
150
exciton counted binding energy from the A line 10
a.
-
0 IUI
1.0
io-2
a ~
.
4/)
(EA_~) ~.etectron energy binding
= parameter~ 2rri’a~V 0
> 0
z ~
____ I
2
-
I.’
0
z
11~1.111..
3456 810 1520 ~ NN PAIR DISTANCE IA
Fig. I. Triangles: Experimental binding energy differences E1’I between excitons bound to NN pairs and the exciton bound to isolated nitrogen. The energies are plotted as a function of pair separation as suggested by Allen [111. Full circles: Energies of NN electrons Ef as a function of pair separation. The values stem from Cohen and Sturge [8]. The straight line (slope 3) in the double-logarithmic dence of both energies. scale indicates the r depen—
—
continuum which resultscontributions). in a band-edge Assuming shift varying 3 (apart from angular the as r wavefunction of the electron and the hole to be strongly localized the difference between the observed NN1 exciton level E7 and the A/B line of isolated nitrogen should be identified with the strain contribution of the additional N atom in the pair, i.e. 3 I E7 E~I R~ (R 1 is the pair spacing). This law is almost obeyed by experiment as shown by the triangles in Fig. L The shortcoming of isthe Allen model in the fact that the strain field attractive bothconsists for electron and holes, therefore the hole as well as the electron should be strongly localized. The results of excitation spectroscopy of Cohen and Sturge [81 however, confirm an alternative picture first proposed by HopfIeld, Thomas, and Lynch (HTL) [2]: In the HTL model only the electron is strongly localized due to the electronegative nitrogen whereas the hole is bound in the acceptor-like Coulomb field of the electron. This model —
,
Co
f
iø~ -
I
I
I ~ ~ 5 6 10 20 • i Fig. 2. Eigenvalues of the NN electron problem obtained —
•
with a variational Ansatz for the wave function in the strain-like potential. The energies are plotted as a function of pair separation in relative meaning of the potential(both parameters V units). The 0 and a0 is explained in the text. Note the logarithmic scales! explains very well the differences observed between electron and exciton binding energies at NN pairs but not the origin of the electron binding energy. There is nothing which prevents the Allen mechanism to be responsible for the localization of the electron. lndeed, if its binding energy Ef is plotted as a function of the NN pair separation (solid circles in Fig. 1) these 3. energies are comparable with I E7 the E~I and varytoasbe R bound in a In this paper we argue electron potential consisting of two parts: —
(I) a short.range part originating from pseudopotential differences and some other contributions; (ii) a medium ranging tail caused by strain effects. Whereas the short-range part will not be specified further and approximated with an appropriate Ansatz the tail should behave as r3 on the assumption of the
ENERGIES OF ELECTRONS BOUND TO NITROGEN PAIRS IN GaP
Vol. 36, No.6
~
NH PAIR DISTANCE 10
U
(M
—
.3
NN~
C
~
NN
N~
8
fIt wifh (square welt. r~) potential
potential -150 ‘NN ~NN~
10
IS
6
L
R
______
a ‘4~3.O8eV
Fig. 4. Superposition of the two nitrogen model potentials in the case of NN3 pair (straight line). The dotted line demonstrates the shape of the equivalent squarewell Potentialvalues parameters which provided the best potentials. fIt to experimental are given. art effective potential to be chosen so as to include all its intervalley and umklapp contributions. This practice has been suggested in the paper of Kirczenow eta!. [22]. Our model potential produced by a single N atom will be set in the form
.~5
0. ml
0
r(AI
v~v4~
———V a 0.1.88eV 2X1.185A
fit withwell square
—
5
~
‘,
:1
___________
1NN~ II I
3456
1
—
NN
2
••
NN
NN~ I1NN~
f~IOO
•
‘~~‘
-
: ~
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~.
—A
I
• Nr4~
~1
______________
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___~-___4
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ENERGY I.V)
-6-5-4-3-2-1
—~ ~
—
565
5
10 NN PAIR DISTAIICE
IS
V(r)
~
Fig. 3. Upper part: Fit of the theoretical electron binding energies by use of the potential of equation (1) (straight line) to the experimental ones (labelled NN~). The experimental exciton binding energies (NN7) are also shown for comparison. Experimental points origmate from Cohen and Sturge [8]. The dotted line is the fit with part: Corresponding the simplifiedvariational square-well Bohr-radii potential.forLower both strain-like (straight line) and square-well (dotted line) potential as a function of NN-pair distance. The bound state vanishes above NN~Ousing the strain-like and above NN~the square-well potential. angular dependence to be smeared out due to the spreading of the wavefunction. The question arises of whether the parameters of this potential can be chosen in such a way as to produce the observed electron binding energies of the NN electrons in form of solutions of the proper one-particle Schrodinger equations. In contrast to previous papers dealing with analogous problems [9, 10, 15] we use the effective mass (EM) formalism. Recently several authors [16—22] showed that the EM formalism works well also for potentials with strong central-cell, i.e. short-ranging part. In this case we are concerned with intervalley and umklapp termsofofour thebare waveshort-ranging functions. Due lack of knowledge parttowetherequire
— —
t
— —
3 V0 V0(a0/r)
(r (r >< aa0) (square-well part), 0) (strain-like part), (1)
where the parameter V0 must be fitted to the experimental values of the electron energies E7 and a0 will piesumably lie in the region of half the nearest-neighbour distance (~1.185 A) in GaP. Clearly, this potential 3 tail. In the above shape it has exhibits the advantage the required of being r very smooth in Fourier representation a fact which additionally recommends the applicability of the EM theory. To avoid the problems related with intervalley terms of the kinetic energy and the detailed GaP band structure [23) we assume an isotropic parabolic dispersion law. The choice of an appropriate effective mass m5 will be discussed below. The 1-lamiltonian for one electron bound to an NN pair with separation R~is H = (— h2/2,n5) ~2 + V(r) + V(r + R,). (2) This problem corresponds to that of the H~ion with the exception that a term describing the repelling of the two nuclei is missed here. Such forces are held by the lattice. For the wave function we make the Ansatz [2(1 + ~)J~ {u 1(r) + u1(r + R,)), i~ =
u1(r)
(I + a,R1 + a?R~/3) exp (—~1R,). 1/x)”2 exp (— c~r) (cr~ °
=
(3)
(4) (5)
566
ENERGIES OF ElECTRONS BOUND TO NITROGEN PAIRS IN GaP
is a hydrogenic function with a1 = a~jas variational parameter. The determination of the eigenvalues of equation (2) is standard. As a result of the variation the binding energies in units of the potential strength V0 are plotted in Fig. 2 as a function of the pair2(2ma~ distance.V The best fit to a — 3 slope is obtained with h 0)’ = 0.88. If for simplicity a0 is set exactly equal to 1.185 A then the depth V0 which can now be extracted from experiment, turns out to be 3.08 eV. In our procedure the effective mass m~is actually not explicitly needed because it appears in the cornputation only via the product m’a~V0. Our fit corre sponds to an effective mass approximately equal to unity. This large value which differs from the distinctly smaller one used for more extended wave functions, should be attributed to the real band structure effects discussed before. It is of interest to note that an isotropic effective mass value of unity together with the medium dielectric constant of 11.19 [24] just yields the binding energy of the isocoric sulphur donor E5 = 107 meV [25] in single-valley approximation. The fit to the absolute values of experimental electron binding energies is illustrated by the straight line in Fig. 3. For the isolated nitrogen atom (corresponding to NNO.) no bound state occurs. The outcome of our calculations is a model potential for the electrons which may be useful for further problems. In some cases, however, the potential might be still too complicated to apply it. This situation can occur when needing its Fourier transform which is no elementary function. For this reason we have carried out a calculation with a simpler square-well potential. Best fIt is obtained with parameters indicated in Fig. 4 in comparison with the strain-like potential. The resulting binding energies are characterized by the dotted curve in Fig. 3. It is evident that the agreement is not as good as before but for some purposes it might be satisfactory. This can happen, e.g. where nitrogen acts as a scattering potential for free carriers as in electron— hole liquids. The problem next to be tackled is the binding of the exciton to nitrogen or its pairs. Theoretical work on this field is under way. Acknowledgements — We thank Prof. K. Unger and Dr. R. Schwabe for many stimulating discussions and Mrs. 1-I. Seewitz for carefully preparing the manuscript. REFERENCES 1.
D.G. Thomas & J.J. Hopfield, Phys. Rev. 150, 680 (1966).
2. 3. 4.
5. 6. 7. 8 9. 10. 11. 11
Vol. 36, No.6
J.J. Hopfield, D.G. Thomas & R.T. Lynch, Ph vs. Rev. Lett. 17,312 (1966). For a review see also W. Czaja, FestkOrperproblerne 11,65 (1971). R. Schwabe, F. Thuselt, R. Bindemann, W. Seifert & K. Jacobs, Phys. Lett. 64A, 226 (1977). R. Schwabe, F. Thuselt, H. Weinert & R. Bindemann, Phys. Status Solidi (b) 95, 571 (1979). J.E. Kardontchik & E. Cohen, Pvs. Rev. Bl9, 3181 (1979). R. Schwabe, F. Thuselt, H. Weinert & R. Bindemann, Proc. tnt. Conf RECON ‘79, p. 191. Prague (1979). E. Cohen & M.D. Sturge, P/n’s. Rev. BIS. 1039 (1977). R.A. Faulkner, P/it’s. Rev. 175, 991 (1968). M. Brand & M. Jaros,J. P/it’s. Cl2, 2789 (1979). J.W. J.W. Allen,J. Allen,J. P/it’s. Phvs. Cl, C4, 1136(1968). 1936 (1971).
—.
13. 14 15:
16 17. 18. 19 20. 2!. 2’ —.
23.
24. 25.
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