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Size quantization of excitons in quasi-zero-dimensional semiconductor structures
PhysicsLettersAl68(1992)433—436 North-Holland
PHYSICS LETTERS A
Size quantization of excitons in quasi-zero-dimensional semiconductor structures S.I.
Pokutnyi
Department of Physics, State PedagogicalInstitute, 324 086 Krivoi Rog, Ukraine Received 10 May 1992; accepted for publication 13 July 1992 Communicated by V.M. Agranovich
A size quantization theory ofthe exciton energy spectrum in a small semiconductorcrystal is developed for the conditions when the polarizationinteraction of an electron and a hole with the surface of a microcrystal plays an important role.
1. The optical properties of quasi-zero-dimensional structures, semiconductor microcrystals (SMs) of spherical shape of size a 1—10 nm grown in transparent dielectric matrices, are presently the subject of intensive investigations [1,21. The optical properties of such heterophase systems are determined by the energy spectrum of a spatially restricted electron—hole pair (exciton). The observed [1,2] influence of the boundary of a SM of radius a on the spectrum E~1(a)(n, I are the main and the orbital quantum numbers) of a quasi-particle (an electron or an exciton) with effective mass !~was treated in ref. [1] as a quantum dimensional effect associated with a purely spatial restriction of the quantization region [3]: 2 I‘2 ~ 2~ g ~ 2 E ‘ ‘—E ‘h where Eg is the width of the forbidden band in an unrestricted semiconductor, can, are the roots of the Bessel function .11+112 (p,,,) = 0. Formula (1) describes qualitatively the position of the exciton level ,.
(n, 1) in a SM of radius a within the framework of a simple model which takes into account only the Coulomb interaction of an electron with a hole in a simple parabolic zone. The arising polarization interaction of charge carriers with the charge induced on the SM surface [4—il] was not included in (1). The present work investigates theoretically the energy spectrum of an exciton in a small SM and its dependence on the SM radius, the effective mass of Elsevier Science Publishers B.V.
an electron and a hole, the relative dielectric function, under conditions when the polarization interaction of charge carriers with the SM surface plays an important role. 2. Let us consider a simple model: a neutral spherical SM of radius a, with a dielectric function 82, surrounded by a medium with e~.In the bulk of such a SM an electron e and a hole h with effective masses me and mi,, respectively, are moving (re and r~are the distances of the electron and the hole from the SM center); the dielectric functions of the SM and the matrix differappreciably (82 >~er). It is assumed that the bands of the electrons and holes have parabolic shapes. The characteristic dimensions of the problem are the quantities a, a~, ah,radii whereofacthe = elecm~e2, ah=6 2/mhe2 are the Bohr 2h
tron and the hole, respectively, in a semiconductor with 82 (e is the electron charge). The fact that all characteristic dimensions in the problem exceed appreciably the interatomic a 0 (a, a~,ah ~ a0) allows one to consider the electron and hole motion in the effective mass approximation. In the model under study within the scope of the above approximations, the Hamiltonian of the exciton is
Volume 168, number 5,6
PHYSICS LETTERS A
the perturbation theory one can easily obtain the spectrum of an exciton, E~:t~’(a), in the state (fle, ‘e~me; flh, ‘h~mh) (here fle, 1e, me and flh, 4,, m~are
2/2me)i~e—(il2/2mh)L~h
H=—(h
+ Vhh(rh, a)+Eg + Veh(re, + Vee~(1~, a)+
rh)
VCh(r~,re,, a)+
Vhe’(re, re,, a)
,
(2)
where the first two terms determine the kinetic energy of an electron and a hole, Veh(re, rh) is the Coulomb interaction of an electron with a hole, a
e
Veh
7 September 1992
2’
(3)
the main, orbital and magnetic quantum numbers of an electron and a hole): Enh.lb.mh(a)
=
(h2/2m~a2)ca~,, + ~e’
(a)
±~I~’(a)+Eg, (7) n where the first term is the kinetic energy of an electron in an infinite spherical well, and ~e’(a) is the
2a (r~—2rerhcosO+r~)”
0
the angle between re and rh, Vee’ (re, a) and Vhh (rh, a) are the interactions with the self-image of an electron and a hole, respectively, ~h’ (re, ri,, a) and Vhe’ (re, r~,a) are their interactions with “alien” images. At arbitrary values of e~and 62 the terms in is
(2) which describe the energy of the polarization interaction of an electron and a hole with the SM surface, can be written in an analytical form [7] which in the case 62 ~ e~becomes especially simple [81: 2
2e Vhh.=—(_a
2—r~+
2::a(
a2 a2—r~+
62\ —)~
2a \,a
Vce • =
Veh
=
‘
(4)
Y’n,,i~,m,
(2/r)”2 y a
=
ca)
(8)
~1/e+l/2((0ne,1e1~e/~1)
i~,pn~(0,
J,,+ 312 (can,,,,)
(where ~1cm, are the normalized spherical functions, J~(x) are the Bessel functions). The quaptity A ‘fc~”~’ (a) is an eigenvalue of the heavy hole Hamiltonian Hh= —(h2/2mh)L~ + Vhh(rh, a)+ Vfl,,eme(rh, a)
(9)
,
Vne/eme(rh, a)
Vhe
e2 = —
62\
average value of the electron interaction with the selfimage with the infinitely-deep spherical well functions
a 2—2rerh cos 0+a2] 1/2
2e
(5)
= ~“eh(rh, a) + VCh (rh, a) + ~C(rh, a) (10) Here is the average value ofthe Coulomb interactionand ofthe electron and hole interaction with “alien” images on “free” electron states (8). .
~
2a [(rerh/a) 3. We investigate the energy spectrum of an exciton in a small SM in the case when the size of the SM is restricted by the condition
Quantitative results for the exciton spectrum (7) are obtained here only for the simple case ‘e= 0. Us rngexpressions (3)—(5), (8)and (10) one can eas-
a 0
(6)
ily obtain T’fl~
When this condition is satisfied the polarization interaction plays an important role in the potential energy of Hamiltonian (2). Inequality (6) also permits consideration of the electron and hole motion in the effective mass approximation. Under conditions (6) the adiabatic approximation (mC”~mh) can be used: in this case the kinetic energy of an electron is assumed to be the greatest quantity and the last four terms in (2) are considered together with the nonadiabaticity operator by a perturbation theory. Then, taking into account only the first order of 434
“eb’
0,0 (X,S)=S~[5~fl(27tfleX)/7tfleX2C~(2flfleX)
+ 2 Ci ( 2ltfle) + 2 ln x— 2] V~°’° (x, S) + V~P’° (x, S) =
,
— 2S —‘,
eh
V~0.0(x, S) cc
S
(Z~~,0 +82/Cl) 2)~’+6
Vhh’(x,S)=S~’[(1—x
where
J I
Z~,,0=
2
0
(11) (12)
dxe ~ l—x~ (EfleXe)
2/eI] ,
(13)
Volume 168, number 5,6
PHYSICS LETTERS A
and Ci(y) is the integral cosine. Here and below the and energy is measured in units Ry~=~2/2m~a~ nondimensional values of the length x= rh/a and S= a/ah are used. Note that in the approximation considered the interaction of an electron with the images (its own and “alien”) (13), (12) and the interaction of a hole with an “alien” image (12) yield a constant addition to the hole energy S’. Taking into account formulae (4) and (10)— (12) we write the potential energy in the Hamiltonian of the heavy hole (9) as follows, C”°’°(x, S) = Vhh’ + V”°’°(x, S) 2)’(x,+ 5) sin(2xn~x) /zn~x = S’ [(1 —x 2Ci(27tfl~X)+2Ci(27tfl~)+2lflX+e
within the framework of the adiabatic approximation, the main contribution to the spectrum is made by the second term (kinetic energy of the electron) which is due to the purely spatial restriction of the quantization region; the last two terms associated with the Coulomb and polarization electron—hole interactions arejust corrections. It should be noted that the exciton spectrum (17) is applicable only for the lowest exciton states (ne, 0, 0; th) for which the inequality: E~, 0,0(S) _Eg << L~.E(S)is fulfilled (where AE(S) is the depth of the potential well for electrons in SMs, e.g. in a CdS SM in the region of sizes (6) iiE=2.3—2.5eV [12]).
4. In refs. [1,2] interband absorption spectra of CdS SMs (82=9.3) with sizes from 10 to l0~A dis-
2/Ci4] (14) The minimum of the potential energy (14), Un~,0.0=Une.0.0x=O, S)=P~e,o/S (where P~,,0= mm 2y+82/81 —1, y=O.5’77 is 2 Ci(2ltfle)2 ln(27tfle) — the Euler constant) is reached at the point x=0. A series expansion of the potential (14) in the parameter x2 z< 1 with accuracy up to the first two terms gives the hole spectrum A~,o,o(S) in oscillator form: A~, 0,0(S) = P,,,,0/S+ w( th + 3) w(S, ~
7 September 1992
(15)
where w(S) is2flr+Ir=O, the frequency hole oscillator 1, of is the the main quantum vibrations, tb= number of the hole (flr and 1,. are, respectively, the radial and the orbital quantum numbers ofthe hole). The validity of the representation of the potential (14) as a potential of a three-dimensional harmonic oscillator is reduced to the requirement (a 08/a) 2 << 1 2=(th+~)”2(~l2/mhw)”2 (where aOS=(
Taking into account formulae (13) and (15) we ohtam the exciton spectrum (7) for SMs whose radii satisfy simultaneously the conditions (6), ao/ah <1 ~zS~aC/ah, and (16): E~.o,o(S)Eg+7v2n~S2mh/me
persed in a transparent dielectric matrix of silicate glass were investigated. The effective masses of the electron and the hole in CdS were, respectively, m~=O.2O5mo and mh= Sm0 (i.e. mC<
[— (1
—
~u) —‘]
f(u)= 2513(u+3)213(~—u)~”3
u=a/ã.<~, f(u)=0,
u>~,
(18)
+S’(ZneO+Pfl~O+C 2/Cl)+&)(th+4)
.
(17)
As the exciton spectrum (17) has been obtained
where a is the average SM radius. By incorporating into formula (15) the SM dispersion by the radii a 435
Volume 168, number 5,6
PHYSICS LETTERS A
(18) an expression is obtained which determines the distance between the equidistant series in the spectrum of the hole: &(~)=2.232(1+ ~,t2n~)’~2S~3”2, S= a/ah.
(19)
From the comparison of formula (19) (for n= 1) with the experimental dependence of the splitting magnitude on the SM size a obtained in ref. [2] it follows that for the region of SM radii 15 ~ a~ 30 A the splitting cD(s) (19) is in good agreement with the experimental data [2] and differs from the latter only slightly (,~ 4%). The author is grateful to Professor V.M. Agranovich, Dr. N.A. Efremov and Dr. V.E. Kravtzov for a critical discussion of the obtained results.
[2] A.I. Ekimov, A.A. Onushchenko and ALL Efros, JETP Lett. 43 (1986) 292. [3]ALL. Efros and A.L. Efros, Soy. Phys. Semicond. 16 (1982) [4] 772. V.M. Agranovich, A.G. Mal’shukov and M.A. Mekhtiev, Soy. Phys. JETP 63 (1972) 2274. [5] V.M. Agranovich, Soy. Phys. Solid State 14 (1972) 3684. [6]V.M. Agranovich and Yu.E. Lozovik, JETP Lett. 17 (1973) 209. [7] N.A. Efremov and S.I. Pokutnyi, Sov. Phys. Solid State 27 (1985) 48. [8]N.A. Efremov and S.!. Pokutnyi, Sov. Phys. Solid State 32 (1990) 1637. [9] N.A. Efremov and S.I. Pokutnyi, Sov. Phys. Solid State 32 (1990) 2921. [10]S.!. Pokutnyi, Soy. Phys. Semicond. 25 (1991) 381. [11] S.!. Pokutnyi and N.A. Efremov, Phys. Stat. Sol. (b) 165 (1991)109. [12]V. Grabovskis, Ya. Dzenis and A. Ekimov, Soy. Phys. Solid [13]
State 31(1989)272. I.M. Livshitz and V.V. Slezov, Soy. 479.
References [1] A.!. Ekimov and A.A. Onushchenko, JETP Lett. 40 (1984) 337.
436
7 September 1992
Phys. JETP 35 (1958)
PHYSICS LETTERS A
Size quantization of excitons in quasi-zero-dimensional semiconductor structures S.I.
Pokutnyi
Department of Physics, State PedagogicalInstitute, 324 086 Krivoi Rog, Ukraine Received 10 May 1992; accepted for publication 13 July 1992 Communicated by V.M. Agranovich
A size quantization theory ofthe exciton energy spectrum in a small semiconductorcrystal is developed for the conditions when the polarizationinteraction of an electron and a hole with the surface of a microcrystal plays an important role.
1. The optical properties of quasi-zero-dimensional structures, semiconductor microcrystals (SMs) of spherical shape of size a 1—10 nm grown in transparent dielectric matrices, are presently the subject of intensive investigations [1,21. The optical properties of such heterophase systems are determined by the energy spectrum of a spatially restricted electron—hole pair (exciton). The observed [1,2] influence of the boundary of a SM of radius a on the spectrum E~1(a)(n, I are the main and the orbital quantum numbers) of a quasi-particle (an electron or an exciton) with effective mass !~was treated in ref. [1] as a quantum dimensional effect associated with a purely spatial restriction of the quantization region [3]: 2 I‘2 ~ 2~ g ~ 2 E ‘ ‘—E ‘h where Eg is the width of the forbidden band in an unrestricted semiconductor, can, are the roots of the Bessel function .11+112 (p,,,) = 0. Formula (1) describes qualitatively the position of the exciton level ,.
(n, 1) in a SM of radius a within the framework of a simple model which takes into account only the Coulomb interaction of an electron with a hole in a simple parabolic zone. The arising polarization interaction of charge carriers with the charge induced on the SM surface [4—il] was not included in (1). The present work investigates theoretically the energy spectrum of an exciton in a small SM and its dependence on the SM radius, the effective mass of Elsevier Science Publishers B.V.
an electron and a hole, the relative dielectric function, under conditions when the polarization interaction of charge carriers with the SM surface plays an important role. 2. Let us consider a simple model: a neutral spherical SM of radius a, with a dielectric function 82, surrounded by a medium with e~.In the bulk of such a SM an electron e and a hole h with effective masses me and mi,, respectively, are moving (re and r~are the distances of the electron and the hole from the SM center); the dielectric functions of the SM and the matrix differappreciably (82 >~er). It is assumed that the bands of the electrons and holes have parabolic shapes. The characteristic dimensions of the problem are the quantities a, a~, ah,radii whereofacthe = elecm~e2, ah=6 2/mhe2 are the Bohr 2h
tron and the hole, respectively, in a semiconductor with 82 (e is the electron charge). The fact that all characteristic dimensions in the problem exceed appreciably the interatomic a 0 (a, a~,ah ~ a0) allows one to consider the electron and hole motion in the effective mass approximation. In the model under study within the scope of the above approximations, the Hamiltonian of the exciton is
Volume 168, number 5,6
PHYSICS LETTERS A
the perturbation theory one can easily obtain the spectrum of an exciton, E~:t~’(a), in the state (fle, ‘e~me; flh, ‘h~mh) (here fle, 1e, me and flh, 4,, m~are
2/2me)i~e—(il2/2mh)L~h
H=—(h
+ Vhh(rh, a)+Eg + Veh(re, + Vee~(1~, a)+
rh)
VCh(r~,re,, a)+
Vhe’(re, re,, a)
,
(2)
where the first two terms determine the kinetic energy of an electron and a hole, Veh(re, rh) is the Coulomb interaction of an electron with a hole, a
e
Veh
7 September 1992
2’
(3)
the main, orbital and magnetic quantum numbers of an electron and a hole): Enh.lb.mh(a)
=
(h2/2m~a2)ca~,, + ~e’
(a)
±~I~’(a)+Eg, (7) n where the first term is the kinetic energy of an electron in an infinite spherical well, and ~e’(a) is the
2a (r~—2rerhcosO+r~)”
0
the angle between re and rh, Vee’ (re, a) and Vhh (rh, a) are the interactions with the self-image of an electron and a hole, respectively, ~h’ (re, ri,, a) and Vhe’ (re, r~,a) are their interactions with “alien” images. At arbitrary values of e~and 62 the terms in is
(2) which describe the energy of the polarization interaction of an electron and a hole with the SM surface, can be written in an analytical form [7] which in the case 62 ~ e~becomes especially simple [81: 2
2e Vhh.=—(_a
2—r~+
2::a(
a2 a2—r~+
62\ —)~
2a \,a
Vce • =
Veh
=
‘
(4)
Y’n,,i~,m,
(2/r)”2 y a
=
ca)
(8)
~1/e+l/2((0ne,1e1~e/~1)
i~,pn~(0,
J,,+ 312 (can,,,,)
(where ~1cm, are the normalized spherical functions, J~(x) are the Bessel functions). The quaptity A ‘fc~”~’ (a) is an eigenvalue of the heavy hole Hamiltonian Hh= —(h2/2mh)L~ + Vhh(rh, a)+ Vfl,,eme(rh, a)
(9)
,
Vne/eme(rh, a)
Vhe
e2 = —
62\
average value of the electron interaction with the selfimage with the infinitely-deep spherical well functions
a 2—2rerh cos 0+a2] 1/2
2e
(5)
= ~“eh(rh, a) + VCh (rh, a) + ~C(rh, a) (10) Here is the average value ofthe Coulomb interactionand ofthe electron and hole interaction with “alien” images on “free” electron states (8). .
~
2a [(rerh/a) 3. We investigate the energy spectrum of an exciton in a small SM in the case when the size of the SM is restricted by the condition
Quantitative results for the exciton spectrum (7) are obtained here only for the simple case ‘e= 0. Us rngexpressions (3)—(5), (8)and (10) one can eas-
a 0
(6)
ily obtain T’fl~
When this condition is satisfied the polarization interaction plays an important role in the potential energy of Hamiltonian (2). Inequality (6) also permits consideration of the electron and hole motion in the effective mass approximation. Under conditions (6) the adiabatic approximation (mC”~mh) can be used: in this case the kinetic energy of an electron is assumed to be the greatest quantity and the last four terms in (2) are considered together with the nonadiabaticity operator by a perturbation theory. Then, taking into account only the first order of 434
“eb’
0,0 (X,S)=S~[5~fl(27tfleX)/7tfleX2C~(2flfleX)
+ 2 Ci ( 2ltfle) + 2 ln x— 2] V~°’° (x, S) + V~P’° (x, S) =
,
— 2S —‘,
eh
V~0.0(x, S) cc
S
(Z~~,0 +82/Cl) 2)~’+6
Vhh’(x,S)=S~’[(1—x
where
J I
Z~,,0=
2
0
(11) (12)
dxe ~ l—x~ (EfleXe)
2/eI] ,
(13)
Volume 168, number 5,6
PHYSICS LETTERS A
and Ci(y) is the integral cosine. Here and below the and energy is measured in units Ry~=~2/2m~a~ nondimensional values of the length x= rh/a and S= a/ah are used. Note that in the approximation considered the interaction of an electron with the images (its own and “alien”) (13), (12) and the interaction of a hole with an “alien” image (12) yield a constant addition to the hole energy S’. Taking into account formulae (4) and (10)— (12) we write the potential energy in the Hamiltonian of the heavy hole (9) as follows, C”°’°(x, S) = Vhh’ + V”°’°(x, S) 2)’(x,+ 5) sin(2xn~x) /zn~x = S’ [(1 —x 2Ci(27tfl~X)+2Ci(27tfl~)+2lflX+e
within the framework of the adiabatic approximation, the main contribution to the spectrum is made by the second term (kinetic energy of the electron) which is due to the purely spatial restriction of the quantization region; the last two terms associated with the Coulomb and polarization electron—hole interactions arejust corrections. It should be noted that the exciton spectrum (17) is applicable only for the lowest exciton states (ne, 0, 0; th) for which the inequality: E~, 0,0(S) _Eg << L~.E(S)is fulfilled (where AE(S) is the depth of the potential well for electrons in SMs, e.g. in a CdS SM in the region of sizes (6) iiE=2.3—2.5eV [12]).
4. In refs. [1,2] interband absorption spectra of CdS SMs (82=9.3) with sizes from 10 to l0~A dis-
2/Ci4] (14) The minimum of the potential energy (14), Un~,0.0=Une.0.0x=O, S)=P~e,o/S (where P~,,0= mm 2y+82/81 —1, y=O.5’77 is 2 Ci(2ltfle)2 ln(27tfle) — the Euler constant) is reached at the point x=0. A series expansion of the potential (14) in the parameter x2 z< 1 with accuracy up to the first two terms gives the hole spectrum A~,o,o(S) in oscillator form: A~, 0,0(S) = P,,,,0/S+ w( th + 3) w(S, ~
7 September 1992
(15)
where w(S) is2flr+Ir=O, the frequency hole oscillator 1, of is the the main quantum vibrations, tb= number of the hole (flr and 1,. are, respectively, the radial and the orbital quantum numbers ofthe hole). The validity of the representation of the potential (14) as a potential of a three-dimensional harmonic oscillator is reduced to the requirement (a 08/a) 2 << 1 2=(th+~)”2(~l2/mhw)”2 (where aOS=(
Taking into account formulae (13) and (15) we ohtam the exciton spectrum (7) for SMs whose radii satisfy simultaneously the conditions (6), ao/ah <1 ~zS~aC/ah, and (16): E~.o,o(S)Eg+7v2n~S2mh/me
persed in a transparent dielectric matrix of silicate glass were investigated. The effective masses of the electron and the hole in CdS were, respectively, m~=O.2O5mo and mh= Sm0 (i.e. mC<
[— (1
—
~u) —‘]
f(u)= 2513(u+3)213(~—u)~”3
u=a/ã.<~, f(u)=0,
u>~,
(18)
+S’(ZneO+Pfl~O+C 2/Cl)+&)(th+4)
.
(17)
As the exciton spectrum (17) has been obtained
where a is the average SM radius. By incorporating into formula (15) the SM dispersion by the radii a 435
Volume 168, number 5,6
PHYSICS LETTERS A
(18) an expression is obtained which determines the distance between the equidistant series in the spectrum of the hole: &(~)=2.232(1+ ~,t2n~)’~2S~3”2, S= a/ah.
(19)
From the comparison of formula (19) (for n= 1) with the experimental dependence of the splitting magnitude on the SM size a obtained in ref. [2] it follows that for the region of SM radii 15 ~ a~ 30 A the splitting cD(s) (19) is in good agreement with the experimental data [2] and differs from the latter only slightly (,~ 4%). The author is grateful to Professor V.M. Agranovich, Dr. N.A. Efremov and Dr. V.E. Kravtzov for a critical discussion of the obtained results.
[2] A.I. Ekimov, A.A. Onushchenko and ALL Efros, JETP Lett. 43 (1986) 292. [3]ALL. Efros and A.L. Efros, Soy. Phys. Semicond. 16 (1982) [4] 772. V.M. Agranovich, A.G. Mal’shukov and M.A. Mekhtiev, Soy. Phys. JETP 63 (1972) 2274. [5] V.M. Agranovich, Soy. Phys. Solid State 14 (1972) 3684. [6]V.M. Agranovich and Yu.E. Lozovik, JETP Lett. 17 (1973) 209. [7] N.A. Efremov and S.I. Pokutnyi, Sov. Phys. Solid State 27 (1985) 48. [8]N.A. Efremov and S.!. Pokutnyi, Sov. Phys. Solid State 32 (1990) 1637. [9] N.A. Efremov and S.I. Pokutnyi, Sov. Phys. Solid State 32 (1990) 2921. [10]S.!. Pokutnyi, Soy. Phys. Semicond. 25 (1991) 381. [11] S.!. Pokutnyi and N.A. Efremov, Phys. Stat. Sol. (b) 165 (1991)109. [12]V. Grabovskis, Ya. Dzenis and A. Ekimov, Soy. Phys. Solid [13]
State 31(1989)272. I.M. Livshitz and V.V. Slezov, Soy. 479.
References [1] A.!. Ekimov and A.A. Onushchenko, JETP Lett. 40 (1984) 337.
436
7 September 1992
Phys. JETP 35 (1958)