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Optical transitions at deep impurity centres in an electric field
Journal of Luminescence 24/25 (1981) 325—328 North-Holland Publishing Company
325
OPTICAL TgANSITIONS AT DEEP IMPURITY CENTRES IN AN ELECTRIC FIELD P..
Enderlein,
R. Fidclicke,
F. Rechotedt
K. Peuker
Sektion Physik, I-Iumboldt—Univarsitht zu Carlin, G.D.P~. P.. S. 3~uer, Xerox Palo Alto P.eoearch Center, Palo Alto, U.S.A.
The effect of an external electric field on deep impurity states is investigated theoretically for a 3— dimensional pseudo— 6—pot ent ial folloning Vinogradov [1]. Electric field modified aboorption spectra are calculated ~ith direct or indirect gap and with the renard of multiphonon pr000soos. I NTP.ODUCTION Deep impurity centres are exposed to external or internal electric fields in various experimental situations as, e.g., in measurements using pn—junctions, surface barriers or in electroabaorption experiments. On the other hand, the theoretical understanding of the electric field effect on deep impurities turns out to be rather poor. The reason is obvious. As is well—knovin from the hydrogen atom or from shallow impuritiem and /annier—Mott excitons, serious difficulties arise in calculating the influence of an electric field on localized states. Perturbation theory can be used only for weak electric fields whereas in experiment often much higher fields occure. For such field strengths the continous nature of the energy mpectrum cannot be disregarded. The problem is no longer to calculate the electric field induced change of binding energy but the dansity of states as a function of energy and electric field strength. If radiative or non—radiative transitions are considered one needs in addition non—perturbative results for impurity wave functions to get the field dependent transition probabilities. Dut the calculation of binding energies of deep impurity centres without any external perturbations forms a difficult problem by itself which has attracted the theoretical activities during the last few years. For many deep centres a strong lattice coupling occurs. The line shape for optical transitions is essentially influenced by multiphonon excitation and deexcitation processes. These processes are also important for electric field induced changes of deep impurity properties. Thus, the calculation of the electric field effect becomes a rather complex problem. The complexity can be reduced by adopting an impurity model which can be solved if no electric field or no lattice coupling is present. Vinogradov demonstrated the exact analytical eolubility of the 3—dimensional 6—potential in the presence of an electric field [lJ. He calculated the absorption spectrum for the forbidden transition from the impurity level to the energy band From vthich the impurity level splits up. t/e will use the 3—dimensional 6—potential for a more detailed analysis of the electric field effect on deep impurity states and their optical spectra within a two band model and with the regard of multiphonon processes. In section 2 the Schrb— dinger equation mill be solved without lattice coupling. The 0 022—2313/81 /0000—0000/$02.75 © North-Holland
326
R Em/er/tin ci a/
/ Optical
transitions at c/cop i,npurits centres
absorption spectrum for band to impurtty tranoitiono is colcuisted. In section 3 these results crc sxtcnded to multiphonon esoiotsd rs— diat ivs poocesems.
‘AVE FUNCTIONS AID A3SORPTIO’I
SPFCTNUM
A t~o band model -jith an nootropic conduction band Ec(~) and en iootrop±c valence hand Ev(~) miii be conmidermd. The energy gap soy be direct on indirect, opticol trenmitionm et r ore emnumed to be dipole allonjed. For the impurity potential V(x) ~o une a modified 3—dimensional 6—potential. The modification in nec003sry mince en attractive 6—potential in 3 diomnoionm rsoultn in too large (infinite) binding energiom end too otrong localization of the ground state which shows a 1/r singularity at x6—potentisl = 0 (i ~ = r). the The folio— singuby larity can be removed by replacing the ning pseudo—6—potentiel V(~)
=
d~(~) (1
V
-
The impurity
+
is
problem
~)
.
(1)
.
tree ted nit him the one ha
0d epprocimot ion. The to envelope ‘cove functiono are ~ ~ and ~ ~ The impuritc’ state is attached to the conduction hone. Tho minimum of E(~) is located at k0, that of—E (~) stE. The effective memo of Ec(~) at ~ is m~. t/e define the ~ollooing natural unito of energy, E,
i~npth, L, E
2m~O
=
The
and field ,
electric
L~
field
strsnr-th,
F: 2 , F
~c)Y2
=
E3/2
(2)
c/Eo)/
=
strenqt~ 1~n units
of
F
nih
he denoted
by
~,
the energy
in units of E F ‘ by S and th~ quasimomentum in units of L~F/° by ~. Then the°set of Schrddinner equations for the two enve~ope save functions readtc ~ith parmilel to the z—axeo
r
(E(~)
-
E
-
~ ~)
~(~)
=
(2)
(E(~)
-
E
-
~
~(~)
=
0
~-_)
3Jd3~(i
+
~
~(~)
(3)
,
(4)
.
t/e introduce cylindrical coordinates k kr~ ~f .The relevant ortho— normalized solution of (3) mith the msnnetlc quentuni number m = 0, ~(k, k) can be written am
Ac(k
) =
k z
exp
i fzdk
r
(E (k~ z
k xjzdk~ exp i fZdl<., Here B(E) means functions Ai(x)
Z
(E(k’,
)
k
—
E) (1 + 3(E)
x
r
‘
—
E))
.
(5)
a coefficient which can be expressed by Airy ~nd
3i(x) ~
3(E)
C
in the
following way
‘E daAi(s)
(6)
2(—E)+Ai~2(—Efl+ —i(4F~~+EAi(_E)Bi(_E)+Ai’(-E)Bi’(—E)) (EAi
R Enderlein et al
/ Optical
transitions at deep impurity centres
327
go,: optical transitions from the valence band to the impurity level in the vicinity of ‘sill be considered. The parabolic appro— x isa t ion 2 2 = A + (mc/mcr)k E~~(t)= — Eg — (m/m~)k (7) with A = E ( ) — E~(i~ 0) can be applied in this case. For the absorption ~oefficient ~ (w ) it follows, apart from a certain field independent factor oc(cs)
o
SdE I 3(E)
~
E x = Eg
+
A
j 2 ~ E(W) . 1Ai2(x~) (Ai2(y+r)+3i2(y+r))
3~
(~)~ F~~ —
~
~
11C~J
‘I
=
A
—
x > y
(Ai2(x+r)+3i2(x+r)) x < y E .
(it)
(ho)
Expression (B) Foroc allows a simple physical interpretation: 9 (~) means the optical transition probability from the valence ba~d into an impurity level at energy E caused by a photon ~w F—2/31 3(E)12 represents the probability of finding an impurity state at energy E in the presence of an electric field. In (3) a term due to the Franz—Keldyeh effect of the direct edge at E ham been omitted. In the case of indirect materials with A >~ 1 it is exponentially small, and ~E~°-~) can be replaced by a 6— function. For~ it follows approximately the simple expression 3 (~o~- Eg)
2
(11)
In Figure 1 we shot: the absorption spectrum (11) with 3(E) from (6) for various field strengths. Below the gap one observes the 2 broadened impurity peak close to the zero field position — 16u (in units of E ). For weak fields it shifts according to tile quadratic Stark e~fcct.The broadening increases with risiog field strength. Above the gap the absorption spectrum shows oscillations with a field dependent :-,idth, as usual in the case of a Continuous absorption spectrum. Note, however, that in our case the absorption is due to transitions into impurity instead of band states at E In Table 1 we show the energy Unit E and the electric 2field unit F0 for diFferent impurity binding en~rgias E3 — 16T1 E0. The value E5 = 10 meV corresponds to the isolated N—impurity centre and E 7meV B 10 160 500 1000
E /meV 0
0.064 1.013 1.266 6.333
F /Vcm~ 0 1 137.10 873 123102 137.10
Table 1: Characteristic Energies E and field stre~gths F for various bindin~ energies
E6 = 160 meV to the NN1—pair in GaP. Although our theory does not account for electron—hole interaction, life time broadening, effective mass an— isotropy or the finit range of the im— purity potential, the results fit quite well to electroabsorption data for GaP:N obtained earlier by one of us t2J (see inset of Figure 1, &x is shown instead of ~ as in the main part of Figure 1). Our theory predicts a field strength of “-‘ 10~V/cm for a remarkable field effect on the N— centre mhich was also the field strength used in [2] . The spectral shapes are similar in theory and experiment.
328
I-i Inc/crlcin ci at
-200 Figure
LiTTICE
0 1
200
Absorption
Optical transitions at c/cop i,npuriti ce,:trc-o
cOO
600
-200
213
em
onectrum
200
~00
Figure 2 Electroabsorm— tion spectrum :‘,i.th lattice c o uolin n
COUPLTN3
For the calculation Of tho absorption line shape in the presence of lattice coupiinn it is Convenient to start from the inspection of the retarded too—particle Croen’o function 8(t) needed for the evaluation of I
=
exp(-3
+
iAt + ~
u
2 ((HR 3I
+
1)q
(12)
rouhtiphonon parameter, A — lattice relaxation energy, N— — phonon occupation numbers, U~ — coupling constant). The func~ton o(t) can be shov-:n to eatisfythe equstion of motion “ithout dcc— tron—phonon interactnon. The cslculp~ion of the absorption coefficient -sith multiphonon procesmeo ce (cu), results in the convolution integrml —
~M?()
=J~(w
—
Ca)’) ~ (vi’) dvi’
,
(13)
here ~‘ (cv ) is given by (: ). In Figure 2 we shoe: the field iridu— ced change of ~ ,Ace~, for various field strengths. The function ~ vies calculated within the Einstein model, and the field dependence of U~ -las omitted. The following parameters are used: S = 1, = hOD E , T = 0 1<. Tho spectra ohow several peaks. Those for positive °mnmrgiem are due to the oscillatory behaviour of the opectrum coithout lattice coupling, the peaks for negative energies are related to the field induced broadening of the multi— phonon impurity spectrum. The main peak is located near the phonon— free absorption maximum. The asymmetry of thie peak follows from the asymmetry of ~(w) and from the non—vanishing impurity eN— sorption above the direct gap. REFER E N CE S
1 2 3
V.3. Vinogradov, Fiz. Tverd. Tale 13, 3266 (1871) P.. 3. 3auer, 0. El. Materials 4, 1067 (1875) K. Peuker, D. Suisky, R. 3oyn, and P.. Enderlein, 12. Phys. Oapan 4f, Suppi. A, 633 (18fl0)
Soc.
325
OPTICAL TgANSITIONS AT DEEP IMPURITY CENTRES IN AN ELECTRIC FIELD P..
Enderlein,
R. Fidclicke,
F. Rechotedt
K. Peuker
Sektion Physik, I-Iumboldt—Univarsitht zu Carlin, G.D.P~. P.. S. 3~uer, Xerox Palo Alto P.eoearch Center, Palo Alto, U.S.A.
The effect of an external electric field on deep impurity states is investigated theoretically for a 3— dimensional pseudo— 6—pot ent ial folloning Vinogradov [1]. Electric field modified aboorption spectra are calculated ~ith direct or indirect gap and with the renard of multiphonon pr000soos. I NTP.ODUCTION Deep impurity centres are exposed to external or internal electric fields in various experimental situations as, e.g., in measurements using pn—junctions, surface barriers or in electroabaorption experiments. On the other hand, the theoretical understanding of the electric field effect on deep impurities turns out to be rather poor. The reason is obvious. As is well—knovin from the hydrogen atom or from shallow impuritiem and /annier—Mott excitons, serious difficulties arise in calculating the influence of an electric field on localized states. Perturbation theory can be used only for weak electric fields whereas in experiment often much higher fields occure. For such field strengths the continous nature of the energy mpectrum cannot be disregarded. The problem is no longer to calculate the electric field induced change of binding energy but the dansity of states as a function of energy and electric field strength. If radiative or non—radiative transitions are considered one needs in addition non—perturbative results for impurity wave functions to get the field dependent transition probabilities. Dut the calculation of binding energies of deep impurity centres without any external perturbations forms a difficult problem by itself which has attracted the theoretical activities during the last few years. For many deep centres a strong lattice coupling occurs. The line shape for optical transitions is essentially influenced by multiphonon excitation and deexcitation processes. These processes are also important for electric field induced changes of deep impurity properties. Thus, the calculation of the electric field effect becomes a rather complex problem. The complexity can be reduced by adopting an impurity model which can be solved if no electric field or no lattice coupling is present. Vinogradov demonstrated the exact analytical eolubility of the 3—dimensional 6—potential in the presence of an electric field [lJ. He calculated the absorption spectrum for the forbidden transition from the impurity level to the energy band From vthich the impurity level splits up. t/e will use the 3—dimensional 6—potential for a more detailed analysis of the electric field effect on deep impurity states and their optical spectra within a two band model and with the regard of multiphonon processes. In section 2 the Schrb— dinger equation mill be solved without lattice coupling. The 0 022—2313/81 /0000—0000/$02.75 © North-Holland
326
R Em/er/tin ci a/
/ Optical
transitions at c/cop i,npurits centres
absorption spectrum for band to impurtty tranoitiono is colcuisted. In section 3 these results crc sxtcnded to multiphonon esoiotsd rs— diat ivs poocesems.
‘AVE FUNCTIONS AID A3SORPTIO’I
SPFCTNUM
A t~o band model -jith an nootropic conduction band Ec(~) and en iootrop±c valence hand Ev(~) miii be conmidermd. The energy gap soy be direct on indirect, opticol trenmitionm et r ore emnumed to be dipole allonjed. For the impurity potential V(x) ~o une a modified 3—dimensional 6—potential. The modification in nec003sry mince en attractive 6—potential in 3 diomnoionm rsoultn in too large (infinite) binding energiom end too otrong localization of the ground state which shows a 1/r singularity at x6—potentisl = 0 (i ~ = r). the The folio— singuby larity can be removed by replacing the ning pseudo—6—potentiel V(~)
=
d~(~) (1
V
-
The impurity
+
is
problem
~)
.
(1)
.
tree ted nit him the one ha
0d epprocimot ion. The to envelope ‘cove functiono are ~ ~ and ~ ~ The impuritc’ state is attached to the conduction hone. Tho minimum of E(~) is located at k0, that of—E (~) stE. The effective memo of Ec(~) at ~ is m~. t/e define the ~ollooing natural unito of energy, E,
i~npth, L, E
2m~O
=
The
and field ,
electric
L~
field
strsnr-th,
F: 2 , F
~c)Y2
=
E3/2
(2)
c/Eo)/
=
strenqt~ 1~n units
of
F
nih
he denoted
by
~,
the energy
in units of E F ‘ by S and th~ quasimomentum in units of L~F/° by ~. Then the°set of Schrddinner equations for the two enve~ope save functions readtc ~ith parmilel to the z—axeo
r
(E(~)
-
E
-
~ ~)
~(~)
=
(2)
(E(~)
-
E
-
~
~(~)
=
0
~-_)
3Jd3~(i
+
~
~(~)
(3)
,
(4)
.
t/e introduce cylindrical coordinates k kr~ ~f .The relevant ortho— normalized solution of (3) mith the msnnetlc quentuni number m = 0, ~(k, k) can be written am
Ac(k
) =
k z
exp
i fzdk
r
(E (k~ z
k xjzdk~ exp i fZdl<., Here B(E) means functions Ai(x)
Z
(E(k’,
)
k
—
E) (1 + 3(E)
x
r
‘
—
E))
.
(5)
a coefficient which can be expressed by Airy ~nd
3i(x) ~
3(E)
C
in the
following way
‘E daAi(s)
(6)
2(—E)+Ai~2(—Efl+ —i(4F~~+EAi(_E)Bi(_E)+Ai’(-E)Bi’(—E)) (EAi
R Enderlein et al
/ Optical
transitions at deep impurity centres
327
go,: optical transitions from the valence band to the impurity level in the vicinity of ‘sill be considered. The parabolic appro— x isa t ion 2 2 = A + (mc/mcr)k E~~(t)= — Eg — (m/m~)k (7) with A = E ( ) — E~(i~ 0) can be applied in this case. For the absorption ~oefficient ~ (w ) it follows, apart from a certain field independent factor oc(cs)
o
SdE I 3(E)
~
E x = Eg
+
A
j 2 ~ E(W) . 1Ai2(x~) (Ai2(y+r)+3i2(y+r))
3~
(~)~ F~~ —
~
~
11C~J
‘I
=
A
—
x > y
(Ai2(x+r)+3i2(x+r)) x < y E .
(it)
(ho)
Expression (B) Foroc allows a simple physical interpretation: 9 (~) means the optical transition probability from the valence ba~d into an impurity level at energy E caused by a photon ~w F—2/31 3(E)12 represents the probability of finding an impurity state at energy E in the presence of an electric field. In (3) a term due to the Franz—Keldyeh effect of the direct edge at E ham been omitted. In the case of indirect materials with A >~ 1 it is exponentially small, and ~E~°-~) can be replaced by a 6— function. For~ it follows approximately the simple expression 3 (~o~- Eg)
2
(11)
In Figure 1 we shot: the absorption spectrum (11) with 3(E) from (6) for various field strengths. Below the gap one observes the 2 broadened impurity peak close to the zero field position — 16u (in units of E ). For weak fields it shifts according to tile quadratic Stark e~fcct.The broadening increases with risiog field strength. Above the gap the absorption spectrum shows oscillations with a field dependent :-,idth, as usual in the case of a Continuous absorption spectrum. Note, however, that in our case the absorption is due to transitions into impurity instead of band states at E In Table 1 we show the energy Unit E and the electric 2field unit F0 for diFferent impurity binding en~rgias E3 — 16T1 E0. The value E5 = 10 meV corresponds to the isolated N—impurity centre and E 7meV B 10 160 500 1000
E /meV 0
0.064 1.013 1.266 6.333
F /Vcm~ 0 1 137.10 873 123102 137.10
Table 1: Characteristic Energies E and field stre~gths F for various bindin~ energies
E6 = 160 meV to the NN1—pair in GaP. Although our theory does not account for electron—hole interaction, life time broadening, effective mass an— isotropy or the finit range of the im— purity potential, the results fit quite well to electroabsorption data for GaP:N obtained earlier by one of us t2J (see inset of Figure 1, &x is shown instead of ~ as in the main part of Figure 1). Our theory predicts a field strength of “-‘ 10~V/cm for a remarkable field effect on the N— centre mhich was also the field strength used in [2] . The spectral shapes are similar in theory and experiment.
328
I-i Inc/crlcin ci at
-200 Figure
LiTTICE
0 1
200
Absorption
Optical transitions at c/cop i,npuriti ce,:trc-o
cOO
600
-200
213
em
onectrum
200
~00
Figure 2 Electroabsorm— tion spectrum :‘,i.th lattice c o uolin n
COUPLTN3
For the calculation Of tho absorption line shape in the presence of lattice coupiinn it is Convenient to start from the inspection of the retarded too—particle Croen’o function 8(t) needed for the evaluation of I
=
exp(-3
+
iAt + ~
u
2 ((HR 3I
+
1)q
(12)
rouhtiphonon parameter, A — lattice relaxation energy, N— — phonon occupation numbers, U~ — coupling constant). The func~ton o(t) can be shov-:n to eatisfythe equstion of motion “ithout dcc— tron—phonon interactnon. The cslculp~ion of the absorption coefficient -sith multiphonon procesmeo ce (cu), results in the convolution integrml —
~M?()
=J~(w
—
Ca)’) ~ (vi’) dvi’
,
(13)
here ~‘ (cv ) is given by (: ). In Figure 2 we shoe: the field iridu— ced change of ~ ,Ace~, for various field strengths. The function ~ vies calculated within the Einstein model, and the field dependence of U~ -las omitted. The following parameters are used: S = 1, = hOD E , T = 0 1<. Tho spectra ohow several peaks. Those for positive °mnmrgiem are due to the oscillatory behaviour of the opectrum coithout lattice coupling, the peaks for negative energies are related to the field induced broadening of the multi— phonon impurity spectrum. The main peak is located near the phonon— free absorption maximum. The asymmetry of thie peak follows from the asymmetry of ~(w) and from the non—vanishing impurity eN— sorption above the direct gap. REFER E N CE S
1 2 3
V.3. Vinogradov, Fiz. Tverd. Tale 13, 3266 (1871) P.. 3. 3auer, 0. El. Materials 4, 1067 (1875) K. Peuker, D. Suisky, R. 3oyn, and P.. Enderlein, 12. Phys. Oapan 4f, Suppi. A, 633 (18fl0)
Soc.