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Energy levels of bound polarons
0038-1098/82/461435 -04503.00/0 Pergamon Press Ltd.
Solid State Communications, Vol. 44, No. 10, pp. 1435-1438, 1982 Printed in Great Britain.
ENERGY LEVELS OF BOUND POLARONS j. Devreese,a, b R. Evrard, e E. Kartheuser c and F. Brosensa, d aphysics Department, University of Antwerpen (U.I.A.), Wilrijk, Belgium blnstitute of Applied Mathematics, University of Antwerpen (R.U.C.A.), Antwerpen, Belgium eUniversity of Li6ge, Sart Tilman, Liege, Belgium d Research Associate of the National Fund of Scientific Research, Belgium
(Received 26 July 1982 by S. Lundqvist) A new variational wave function to describe the ground state and the excited states of a bound polaron is proposed. It is of the form I~) = clO)lCn)+ ~ g~ V~* (e -ik-r -- P~)aklO)lg~n). • + k
It is argued that this form is reasonable for all electron-phonon coupling a and all strengths/3 of the Coulomb potential. Numerical and analytical results are derived for the energy of the ground state and compared to existing results. Results for the energy of the lowest p-type excited state of the bound polaron are obtained. 1. INTRODUCTION THE STUDY of bound polarons is of interest for a better understanding of centers consisting of an electron bound to a charged impurity or a vacancy (hole) in a polar semiconductor or an ~onic crystal. For example, the spectra of shallow impurities in polar semiconductors ( I l l - V , I I - V I compounds) are influenced by the Fr6hlich polaron interaction. The bound polaron is also of some interest to the exciton problem as a limiting case where one of the masses tends to infinity. Several approximations have been developed to study bound polarons. Approximate weak coupling schemes [1] were followed by the "exact" calculation of the propagator for the bound polaron in the weak coupling limit [2]. A conventional variational scheme was developed for the ground state energy of the bound polaron [3]. Also an "effective" potential theory was developed using Path-integrals, to analyse the ground state energy [4]. For "deep" impurities (for which the ionization energy is considerably larger than the L.O. phonon energy) an adiabatic approximation was introduced. [5] Nevertheless, despite these developments, the bound polaron is far from completely solved. For example, putting aside questions regarding the accuracy of the results obtained so far, it may be remarked that the excited states of polarons have only been studied in the asymptotic limit of weak coupling. As far as we know, no treatment of the excited states of bound polarons
exists which is valid for strongly polar materials when the ionization energy of the impurity center is of the order of magnitude or larger than the energy of an L.O. phonon. In the present letter we present a method to calculate the spectrum of a bound polaron which is applicable both to the ground state and the excited states and which gives a precision comparable to the best existing ground state calculations. The validity of this method is not restricted to weak coupling. 2. TRIAL WAVE FUNCTION Consider the Hamiltonian for an electron (or hole) in the Coulomb field of a donor (acceptor) impurity center, interacting with the polar L.O. modes of vibration of the lattice (Frohlich-interaction): (units are chosen with 2m = Ia = O~Lo = 1).
H = p2
#__+ ~.af~ak+He, ph, F
(la)
k
where
He, ph = ~, (V~ak e ik'r + V~a~ e-lk'r),
(lb)
k
with
e2 /3-Eo
and
/4~r~1/2 1 Vk = i ~ v ) ~1/2_~.
ot is Fr61ich's coupling constant. In what follows we construct a trial wave function for all a and/3. For/3 >> 1 the adiabatic approximation is
1435
1436
ENERGY LEVELS OF BOUND POLARONS
valid, i.e. 1~) = [¢ph)ldPn),
(2)
where ICph) is the phonon wave funcaon, ICn) the electron wave function. The variational procedure then give s:
]~ph)
=
(3a)
UI0)
with U = exp
[
~ (VkPka k --V{tPkak
(3b)
k
Vol. 44, No. 10
of/L Therefore the only case left to consider is ct <~ 1, /5 ~< I simultaneously, i.e. the region where perturbation theory is valid. But if/3 -+ 0 the radius of the orbit becomes large. Then Pk = (¢,1 e~k"lCn ) ~ 0 and the Hamlltonian (5) reduces to the starting bound polaron Hamiltonian whereas the wave function (9) tends to the ordinary first-order perturbation wave function. Therefore it is expected that a trial wave function of the type (10) is a reasonable approximation for all values of a and In summary we therefore propose the following type of trial wave function for the bound polaron:
and Pk =
<~,leik"/¢,).
(3C)
Therefore we first perform the transformation
I¢) = c l 0 > I G > +
~
-V*k g k*" ( e -,k., - - P k*) a k + lO)lCn),
(12)
k
where c is a normalization constant c = c* and
H' = U-1HU
c 2 4- ~
Igkl2lVkl2(1 --IRk/2) = 1.
(13)
k
= He + ~ aftak + Hep,
(4)
k
3. ANALYTICAL RESULTS OF THE VARIATIONAL PROCEDURE
where
He
From the vanaUonal principle we obtain
I Vkl21Pk[2 -
__ p2___~ 4- Z r k
-- ~ ]VkI2(P~ eik'r + Pk e-ik'r) •
(5)
gk = c
I --Ipk[ 2 O f f k ) + O2(k) -- (1 -- IPkl2)X/2 '
(14)
k
with
Now consider in H ' the term:
1 X = - ~
Hep : E {Vk(elk'r--Pk)ak q- Vl~(e - i k ' r --p~)a~}
C
k
(6)
as a perturbation. The unperturbed state is lff.) = IO)I¢.),
IVkl2(1 -- IPkl2)fgk + g ~ ) ,
(15a)
k
D~O0 -- (¢,J(eik'r-- PQHe(e-ek'r-- P~)lCn),
(15b)
D=(k) = (1 -- IPkl2)(1 -- (¢nlaelCn))
(15c)
(7)
and the following expression for the energy
(8)
E = <~.lnel¢.> + x . 2
where Io)is the phonon ground state and
Hel¢.) = e.[¢.).
(16)
The first-order perturbation wave function is: 4. GROUND STATE ENERGY
I~,) = [O)lCn)+ ~ V~ ~ I1k)l¢./) k
X
We choose a hydrogen like behaviour
1
K3/2
(lk1(¢i l( e - ' k ' r -- p~)ag I0)1¢.) e,--%-- 1
(9)
Now averaging by replacing ej by ¢(k) and using completeness, it follows that I~,) "" 10)lCn)+ Y- V~g~(e-*k'~-pDa~lO>l¢,,) k
(10)
with
g~ = [en--g(k ) - 1] -1.
(11)
(17)
kbn) = 2a/=Xf~ e -Kr/2
for the electron ground state trial wave function, where r is a variational parameter. Equation (15a) becomes X 2
20t ]"
(1 - - [pkl2) 2
dk
-~ J D~(k) + O2(k) -- (1 --Ipkl2)x/2 '
(18a)
where K4
This adiabatic wave function with its first order correction is also valid at strong coupling (a >> 1) independently
Pk
(r 2 + k2)2
(18b)
and
E -
[K 2 K13]_ 2
Dl(k) =
]
(19)
1--~-
2
]6
"
(20)
The solution for X [equation (18)] must then be introduced into the energy expression (16) with the wave function (17). It follows: ~:2
13~:
5
2
2
~ax
¢13 4
501ex X +29 5 2
(26)
6. NUMERICAL RESULTS
-~Pk
x ~ d3k' Pk'Pk'+k k¢2 D:(k) = (1--02)
K2 4
which has to be minimized with respect to r.
X 2 (K2-i-k2)2--K24-k --
E-
1437
ENERGY LEVELS OF BOUND POLARONS
Vol. 44, No. 10
+ X__
In Table 1 we present results for the ground state energy of the bound polaron as obtained here for several a and 13and compare them to the lowest available vanational results [3]. Our results are very close to these of Larsen for the ground state. Table 1. Ground state energy/in units hwLo) o f the bound polaron for several values o f a and 13as obtained from the present calculation, compared to the variational results o f [3]
(21)
2
13 = 6.32
which is to be minimized with respect to r. 5. FIRST EXCITED STATE For the first excited state a 2p-like electron wave function is chosen:
13 = 4.47
ot
E (present)
E [3]
E (present)
E [3]
2 5 7 11
--14.66 --23.0 --29.41 --44.6
--14.69 --23.0 --29.47 --44.6
--8.60 --15.21 --20.52 --33.4
--8.64 --15.30 -20.61 --33.4
K5/2
1¢-) =
3/2 ~ r e-Kr/2cos0, 2 X/~r
(22)
(0 is measured with respect to some fixed z-direction). From equation (3c) one can calculate Ok using equation (22):
--
1{6
Ok
(1
6kz ,~
K2 +
(1¢2 + k2)3
1
(23)
r2 +Jr13+ 501o~ 1 4 4 29 5 ] '
D,(k) = k 2 + ( 1 - - p ~ ) ( 4 2 {2 [KK 2_ +~2 k 2Ok 4 (~2 + k2)3 13Ks[
1
(24)
1 3 5 7 9 11
where G(k) =
13 = 1
13 = 2
--0.7626 --1.971 --3.207 --4.600 --6.199 --8.029
--0.8775 --2.237 --3.644 --5.214 --6.996 --9.014
-- 1.113 --2.628 --4.205 --5.955 --7.922 --10.13
In Table 2 the energy of the lowest p-type relaxedexcited state of the polaron as obtained here is presented for several a and 13.As far as we know results for excited states of the bound polaron for arbitrary a and/3 are obtained here for the first time.
]
501
/ -- oacG(k) }
13=0
K134 529 015)
2kz2 " 2 + 4 k 2 ] (K2 + k2)4 4k~
E2p
a
k2]
and the expressions (15b) and (15c) for Dl(k) and D2(k) become: O2(k) = (1--p~)
Table 2. Calculated energy o f first excited 2p state (in units hcoLo) of the bound polaron for several values of a and 13
7. ANALYTICAL RESULTS
(25a)
/
We obtained several analytical results in limiting cases. For the ground state it follows dak , Pk'Pk'÷k k'2
(25b)
which we have evaluated analytically. Again equation (15a) has to be solved, and the energy of the first excited state becomes:
E~s -
132 5o~ 4
16
~+...
Eu = --¼(13 + ~ ) 2 + . . . and of course,
if a + 0 and13 -* 0,
(27)
ifa~
(28)
1438
ENERGY LEVELS OF BOUND POLARONS
Vol. 44, No. 10
REFERENCES Ets -
4
if a = 0.
(29) 1.
For the p-type excited states one finds E2p = - - a -- 0.0625/32 - - 0.097 851o~ + . . . if a ~ 0 and/3 ~ 0
(30) 2.
E2p = - - 0.0625/32 -- 0.03829a 2 -- 0.0978o43 if ~-~ oo.
(31)
Note that - - 0 . 0 3 8 2 9 a 2 is the result for the relaxed excited state o f the free polaron. Again, if a = 0 the result obtained is of course: E -
4
i f ~ = 0.
(32)
3. 4. 5.
K.K. Baja1 & T.D. Clark, Solid State Commun. 8, 1419 (1970); K.K. Bajaj & T.D. Clark, Solid State Commun. 11, 1135 (1972); K.K. Bajaj & T.D. Clark, Phys. Status Solidi (b) 52, 195 (1972); T.D. Clark & K.K. Bajaj, Phys. Status Solidi (b) 56,211 (1973). M.H. Engineer & N. Tzoar, Phys. Rev. BS, 3029 (1972). D.M. Larsen, Phys. Rev. 187, 1147 (1969). M. Matsuura, Can. Y. Phys. 52, 1 (1974); see also P.M. Platzmann, Phys. Rev. 125, 1961 (1962). S.I. Pekar & M. Deigen, Zh. Eksp. Theor. Fiz. 18, 481 (1948).
Solid State Communications, Vol. 44, No. 10, pp. 1435-1438, 1982 Printed in Great Britain.
ENERGY LEVELS OF BOUND POLARONS j. Devreese,a, b R. Evrard, e E. Kartheuser c and F. Brosensa, d aphysics Department, University of Antwerpen (U.I.A.), Wilrijk, Belgium blnstitute of Applied Mathematics, University of Antwerpen (R.U.C.A.), Antwerpen, Belgium eUniversity of Li6ge, Sart Tilman, Liege, Belgium d Research Associate of the National Fund of Scientific Research, Belgium
(Received 26 July 1982 by S. Lundqvist) A new variational wave function to describe the ground state and the excited states of a bound polaron is proposed. It is of the form I~) = clO)lCn)+ ~ g~ V~* (e -ik-r -- P~)aklO)lg~n). • + k
It is argued that this form is reasonable for all electron-phonon coupling a and all strengths/3 of the Coulomb potential. Numerical and analytical results are derived for the energy of the ground state and compared to existing results. Results for the energy of the lowest p-type excited state of the bound polaron are obtained. 1. INTRODUCTION THE STUDY of bound polarons is of interest for a better understanding of centers consisting of an electron bound to a charged impurity or a vacancy (hole) in a polar semiconductor or an ~onic crystal. For example, the spectra of shallow impurities in polar semiconductors ( I l l - V , I I - V I compounds) are influenced by the Fr6hlich polaron interaction. The bound polaron is also of some interest to the exciton problem as a limiting case where one of the masses tends to infinity. Several approximations have been developed to study bound polarons. Approximate weak coupling schemes [1] were followed by the "exact" calculation of the propagator for the bound polaron in the weak coupling limit [2]. A conventional variational scheme was developed for the ground state energy of the bound polaron [3]. Also an "effective" potential theory was developed using Path-integrals, to analyse the ground state energy [4]. For "deep" impurities (for which the ionization energy is considerably larger than the L.O. phonon energy) an adiabatic approximation was introduced. [5] Nevertheless, despite these developments, the bound polaron is far from completely solved. For example, putting aside questions regarding the accuracy of the results obtained so far, it may be remarked that the excited states of polarons have only been studied in the asymptotic limit of weak coupling. As far as we know, no treatment of the excited states of bound polarons
exists which is valid for strongly polar materials when the ionization energy of the impurity center is of the order of magnitude or larger than the energy of an L.O. phonon. In the present letter we present a method to calculate the spectrum of a bound polaron which is applicable both to the ground state and the excited states and which gives a precision comparable to the best existing ground state calculations. The validity of this method is not restricted to weak coupling. 2. TRIAL WAVE FUNCTION Consider the Hamiltonian for an electron (or hole) in the Coulomb field of a donor (acceptor) impurity center, interacting with the polar L.O. modes of vibration of the lattice (Frohlich-interaction): (units are chosen with 2m = Ia = O~Lo = 1).
H = p2
#__+ ~.af~ak+He, ph, F
(la)
k
where
He, ph = ~, (V~ak e ik'r + V~a~ e-lk'r),
(lb)
k
with
e2 /3-Eo
and
/4~r~1/2 1 Vk = i ~ v ) ~1/2_~.
ot is Fr61ich's coupling constant. In what follows we construct a trial wave function for all a and/3. For/3 >> 1 the adiabatic approximation is
1435
1436
ENERGY LEVELS OF BOUND POLARONS
valid, i.e. 1~) = [¢ph)ldPn),
(2)
where ICph) is the phonon wave funcaon, ICn) the electron wave function. The variational procedure then give s:
]~ph)
=
(3a)
UI0)
with U = exp
[
~ (VkPka k --V{tPkak
(3b)
k
Vol. 44, No. 10
of/L Therefore the only case left to consider is ct <~ 1, /5 ~< I simultaneously, i.e. the region where perturbation theory is valid. But if/3 -+ 0 the radius of the orbit becomes large. Then Pk = (¢,1 e~k"lCn ) ~ 0 and the Hamlltonian (5) reduces to the starting bound polaron Hamiltonian whereas the wave function (9) tends to the ordinary first-order perturbation wave function. Therefore it is expected that a trial wave function of the type (10) is a reasonable approximation for all values of a and In summary we therefore propose the following type of trial wave function for the bound polaron:
and Pk =
<~,leik"/¢,).
(3C)
Therefore we first perform the transformation
I¢) = c l 0 > I G > +
~
-V*k g k*" ( e -,k., - - P k*) a k + lO)lCn),
(12)
k
where c is a normalization constant c = c* and
H' = U-1HU
c 2 4- ~
Igkl2lVkl2(1 --IRk/2) = 1.
(13)
k
= He + ~ aftak + Hep,
(4)
k
3. ANALYTICAL RESULTS OF THE VARIATIONAL PROCEDURE
where
He
From the vanaUonal principle we obtain
I Vkl21Pk[2 -
__ p2___~ 4- Z r k
-- ~ ]VkI2(P~ eik'r + Pk e-ik'r) •
(5)
gk = c
I --Ipk[ 2 O f f k ) + O2(k) -- (1 -- IPkl2)X/2 '
(14)
k
with
Now consider in H ' the term:
1 X = - ~
Hep : E {Vk(elk'r--Pk)ak q- Vl~(e - i k ' r --p~)a~}
C
k
(6)
as a perturbation. The unperturbed state is lff.) = IO)I¢.),
IVkl2(1 -- IPkl2)fgk + g ~ ) ,
(15a)
k
D~O0 -- (¢,J(eik'r-- PQHe(e-ek'r-- P~)lCn),
(15b)
D=(k) = (1 -- IPkl2)(1 -- (¢nlaelCn))
(15c)
(7)
and the following expression for the energy
(8)
E = <~.lnel¢.> + x . 2
where Io)is the phonon ground state and
Hel¢.) = e.[¢.).
(16)
The first-order perturbation wave function is: 4. GROUND STATE ENERGY
I~,) = [O)lCn)+ ~ V~ ~ I1k)l¢./) k
X
We choose a hydrogen like behaviour
1
K3/2
(lk1(¢i l( e - ' k ' r -- p~)ag I0)1¢.) e,--%-- 1
(9)
Now averaging by replacing ej by ¢(k) and using completeness, it follows that I~,) "" 10)lCn)+ Y- V~g~(e-*k'~-pDa~lO>l¢,,) k
(10)
with
g~ = [en--g(k ) - 1] -1.
(11)
(17)
kbn) = 2a/=Xf~ e -Kr/2
for the electron ground state trial wave function, where r is a variational parameter. Equation (15a) becomes X 2
20t ]"
(1 - - [pkl2) 2
dk
-~ J D~(k) + O2(k) -- (1 --Ipkl2)x/2 '
(18a)
where K4
This adiabatic wave function with its first order correction is also valid at strong coupling (a >> 1) independently
Pk
(r 2 + k2)2
(18b)
and
E -
[K 2 K13]_ 2
Dl(k) =
]
(19)
1--~-
2
]6
"
(20)
The solution for X [equation (18)] must then be introduced into the energy expression (16) with the wave function (17). It follows: ~:2
13~:
5
2
2
~ax
¢13 4
501ex X +29 5 2
(26)
6. NUMERICAL RESULTS
-~Pk
x ~ d3k' Pk'Pk'+k k¢2 D:(k) = (1--02)
K2 4
which has to be minimized with respect to r.
X 2 (K2-i-k2)2--K24-k --
E-
1437
ENERGY LEVELS OF BOUND POLARONS
Vol. 44, No. 10
+ X__
In Table 1 we present results for the ground state energy of the bound polaron as obtained here for several a and 13and compare them to the lowest available vanational results [3]. Our results are very close to these of Larsen for the ground state. Table 1. Ground state energy/in units hwLo) o f the bound polaron for several values o f a and 13as obtained from the present calculation, compared to the variational results o f [3]
(21)
2
13 = 6.32
which is to be minimized with respect to r. 5. FIRST EXCITED STATE For the first excited state a 2p-like electron wave function is chosen:
13 = 4.47
ot
E (present)
E [3]
E (present)
E [3]
2 5 7 11
--14.66 --23.0 --29.41 --44.6
--14.69 --23.0 --29.47 --44.6
--8.60 --15.21 --20.52 --33.4
--8.64 --15.30 -20.61 --33.4
K5/2
1¢-) =
3/2 ~ r e-Kr/2cos0, 2 X/~r
(22)
(0 is measured with respect to some fixed z-direction). From equation (3c) one can calculate Ok using equation (22):
--
1{6
Ok
(1
6kz ,~
K2 +
(1¢2 + k2)3
1
(23)
r2 +Jr13+ 501o~ 1 4 4 29 5 ] '
D,(k) = k 2 + ( 1 - - p ~ ) ( 4 2 {2 [KK 2_ +~2 k 2Ok 4 (~2 + k2)3 13Ks[
1
(24)
1 3 5 7 9 11
where G(k) =
13 = 1
13 = 2
--0.7626 --1.971 --3.207 --4.600 --6.199 --8.029
--0.8775 --2.237 --3.644 --5.214 --6.996 --9.014
-- 1.113 --2.628 --4.205 --5.955 --7.922 --10.13
In Table 2 the energy of the lowest p-type relaxedexcited state of the polaron as obtained here is presented for several a and 13.As far as we know results for excited states of the bound polaron for arbitrary a and/3 are obtained here for the first time.
]
501
/ -- oacG(k) }
13=0
K134 529 015)
2kz2 " 2 + 4 k 2 ] (K2 + k2)4 4k~
E2p
a
k2]
and the expressions (15b) and (15c) for Dl(k) and D2(k) become: O2(k) = (1--p~)
Table 2. Calculated energy o f first excited 2p state (in units hcoLo) of the bound polaron for several values of a and 13
7. ANALYTICAL RESULTS
(25a)
/
We obtained several analytical results in limiting cases. For the ground state it follows dak , Pk'Pk'÷k k'2
(25b)
which we have evaluated analytically. Again equation (15a) has to be solved, and the energy of the first excited state becomes:
E~s -
132 5o~ 4
16
~+...
Eu = --¼(13 + ~ ) 2 + . . . and of course,
if a + 0 and13 -* 0,
(27)
ifa~
(28)
1438
ENERGY LEVELS OF BOUND POLARONS
Vol. 44, No. 10
REFERENCES Ets -
4
if a = 0.
(29) 1.
For the p-type excited states one finds E2p = - - a -- 0.0625/32 - - 0.097 851o~ + . . . if a ~ 0 and/3 ~ 0
(30) 2.
E2p = - - 0.0625/32 -- 0.03829a 2 -- 0.0978o43 if ~-~ oo.
(31)
Note that - - 0 . 0 3 8 2 9 a 2 is the result for the relaxed excited state o f the free polaron. Again, if a = 0 the result obtained is of course: E -
4
i f ~ = 0.
(32)
3. 4. 5.
K.K. Baja1 & T.D. Clark, Solid State Commun. 8, 1419 (1970); K.K. Bajaj & T.D. Clark, Solid State Commun. 11, 1135 (1972); K.K. Bajaj & T.D. Clark, Phys. Status Solidi (b) 52, 195 (1972); T.D. Clark & K.K. Bajaj, Phys. Status Solidi (b) 56,211 (1973). M.H. Engineer & N. Tzoar, Phys. Rev. BS, 3029 (1972). D.M. Larsen, Phys. Rev. 187, 1147 (1969). M. Matsuura, Can. Y. Phys. 52, 1 (1974); see also P.M. Platzmann, Phys. Rev. 125, 1961 (1962). S.I. Pekar & M. Deigen, Zh. Eksp. Theor. Fiz. 18, 481 (1948).