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Electron–hole pairs binding energies in quantum wires and quantum dots
Materials Science and Engineering B74 (2000) 259 – 262 www.elsevier.com/locate/mseb
Electron–hole pairs binding energies in quantum wires and quantum dots A. Simo˜es Baptista *, H. Abreu Santos Centro de Electrotecnia Teo´rica e Medidas Ele´ctricas, Instituto Superior Te´cnico (Lisbon Technical Uni6ersity), A6. Ro6isco Pais, 1049 -001 Lisbon, Portugal
Abstract In a previous paper, we have calculated the binding energies for excitons in quantum wells [A. Simo˜es Baptista, H. Santos, Microelectron. Eng. 43–44 (1998) 481–487]. This was done by solving simultaneously the Schro¨dinger and the Poisson equations, which determine the interaction potential, both in a fractional dimensional space [F.H. Stillinger, J. Math. Phys. 18 (6) (1977) 1224–1234]. Using the same mathematical framework, we now analyse the so-called one-dimensional and zero-dimensional structures. The results we obtain are essential for simulations of optical-electronic devices that use the optical absorption and the refractive index dispersion curves. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Exciton energies; Fractional-dimensional spaces
1. Introduction The physical confinement of electrons and holes is a key feature of several last-generation quantum electronic and opto-electronic devices. Although the confinement is never complete, increasing the confinement improves the characteristics of these devices. The technological evolution of the past 15 years has made it possible to currently use quantum wells, quantum wires and quantum dots, in which the mobile charge carriers are confined, respectively, along one, two and three directions. In particular, as far as quantum dots are concerned, fabrication techniques may use self-ordering phenomena on crystal surfaces [3,4]. This allows the formation of dense arrays of three-dimensional semiconductor ‘islands’ with similar size and shape that have long-range ordering and are imbedded in a different semiconductor. Combining semiconductor pairs with great permittivity differences results in maximum confinement. It is expected that both theory and modelling cope with the aforementioned technological developments, making it possible to coherently describe all these situations. * Corresponding author. Tel.: +351-1-8486676; fax: + 351-18417672.
In the past 10 years, several attempts have been made to develop a theoretical background that unifies the study of the optical properties of quantum wells, quantum wires and quantum dots, dealing with them within the same framework [5]. Our contribution to the aforementioned unification recurs to the fractional dimensional spaces proposed by Stillinger [2] as the common mainframe. By using these, the three-dimensional situations with partial space confinement, which correspond to space anisotropy, are transformed into an analysis in uniform spaces whose dimension is described by a non-integer parameter b. We presented in Ref. [1] results for quantum wells, corresponding to b\ 2, which we hereby extend by considering bB 2. Furthermore, our modelling is expected to be accurate even when the permittivities of the substrate and of the dots or wires are quite different, and also includes the screening of the Coulomb interactions. Although for wells we have b\ 2, for wires b\ 1 and for dots b\ 0, it should be emphasised that b depends both on the confinement and the materials involved. Therefore, the same b value may correspond to wells, wires or dots of different materials. For b B 2, we begin by calculating the evolution of the electron–hole interaction potential with distance r and showing the differences this potential presents
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A. Simo˜es Baptista, H. Abreu Santos / Materials Science and Engineering B74 (2000) 259–262
when compared with the 1/r three-dimensional one. Next, we discuss the different potential terms appearing in the Schro¨dinger equation and calculate the binding energies of the excitons.
2. The electron–hole interaction potential In Ref. [1], we calculated the electron – hole interaction potential and obtained
kS q V(r)= − (b/2) r o(2p)
b−2 2
K b − 2 (kSr)
In Fig. 1, we represent the evolution of the potential in space when the inverse screening length is constant. The dimension of space b is a parameter. It is clear from the graphics that when r 0 for b\ 2, the curves with greater b value intercept all those with a smaller b value. On the contrary, for bB 2, the curves of the interaction potential do not intercept each other. This can be understood from Eq. (1). When r 0, the interaction potential tends asymptotically to (b − 2)
b− 2 2
q 1 (b/2) 4op r
V(r)= −
2− b q k (b − 2)G (b/2) S 2 o(4p)
(1)
2
where kS is the inverse screening length and r the distance between the electron and the hole.
V(r)= −
G
2Bb53 05 bB2
where G is the gamma function. From this result, it is clear that for bB2, the potential is constant near r=0 and decreases with the decrease of b. For b \2, the value of the potential tends to infinity at r=0 and depends on r as r − (b − 2).
3. The Schro¨dinger equation The Schro¨dinger equation is given by −
'2 2 (C 9 C+ WC= j 2m (t
(2)
and has as its solution C=% Ci exp(− jEi t)
(3)
i
in which the wave functions in space Ci satisfy the wave equation −
h2 2 9 Ci + (W− E)Ci = 0 2m
(4)
The excitonic problem has been considered in the Wannier–Mott effective mass model. In a space of dimension b and using spherical co-ordinates, the previous equation is then equivalent to the following system of non-coupled equations [5].
!
"
l(l +b− 2) ( 2f (b− 1) (f + − W(r)− 2E− f=0 2 (r r (r r2 (5a) ( 2U (U + (b− 1) cot gu −l(l +b− 2)u = 0 (u 2 (u
Fig. 1. Evolution of the interaction potential with r. b is the variable parameter and kS is constant, kS = 0.1. (a) 1 … 6, b =3.0, 2.9, 2.7, 2.5, 2.3 and 2.1; (b) 1 … 6, b =2.1, 1.9, 1.7, 1.5, 1.3 and 1.1; (c) 1 … 6, b =1.1, 0.9, 0.7, 0.5, 0.3 and 0.1.
(5b)
in which we have admitted that Ci = f(r)U(u), W(r)= qV(r) and l is the angular quantum number. Furthermore, the distance r has been normalised to the Bohr radius, aB, and the energy normalised to twice the Rydberg energy, ER. If we make the variable change
A. Simo˜es Baptista, H. Abreu Santos / Materials Science and Engineering B74 (2000) 259–262
261
conductors involved in the heterojunctions, and in particular on their permittivity difference. This term has a minimum when b= 2 and varies with r − 2. Since (3− b)(b −1)50.25, the exciton does not remain confined to the point r= 0 [6]. The value of this term in the confinement energy can be positive or negative. When b is between 1 and 3, the confinement energy will be greater than it would be expected when using the interaction potential for the tridimensional case. The opposite will happen for other values of b.
4. Calculating the eigenvalues Fig. 2. Evolution of the eigenvalue corresponding to the main quantum number n = 0. E normalized to the Rydberg energy ER corresponding to the tridimensional case. 4 × 10 − 1 5b 53.0; 6× 10 − 3 5 ks 5 10 − 1.
Fig. 3. Evolution of the energy eigenvalue for n= 0 with kS. b =2.00 (2), 1.75 () and 1.50 (*).
f=
u r [(b − 1)/2]
>
(6)
n?
then it results from Eq. (5a) that (3 − b)(b − 1) −l(l +b −2) d2u 4 + 2E−W(r)+ dr 2 r2 =0
u
(7)
This radial equation resembles the one-dimensional wave equation of a particle in a potential given by W(r)−
(3− b)(b −1) l(l +b − 2) + r2 4r 2
(8)
The first term describes the Coulomb interaction. The third term is connected with the angular momentum of the exciton. The second term of the potential is negative for 1 Bb B 3 and positive for b \ 3 and b B1. This term may be interpreted as the variation of the confinement energy due to the dependence of the interaction potential on b. It will ultimately depend on the semi-
The eigenvalues associated with Eq. (5a) have been calculated by applying the Pru¨fer transformation. It is assumed that the quantum dots have spherical symmetry. From the results in Fig. 2, we can see that the eigenvalues increase as either b or kS decrease. These results are independent of the materials system used. They only depend on the value of the inverse screening length and on the dimension parameter. Therefore, each point of these curves may correspond to various combinations of material systems, and quantum wires and dots with different radius. When this radius decreases, the inverse screening length also decreases. For values of the radius of the quantum dots r such that raB, we have kS 0. The great majority of structures that have been fabricated up to now have an excitonic energy less than 100ER. It is clear from Fig. 3 that the dimension parameter should therefore be greater than 1.5.
5. Conclusions The electron–hole interaction potential has been established in a fractional dimensional space using the mathematical framework proposed by Stillinger. If the value of the dimension parameter b is greater than two, the curve of the potential for a given b intercepts all those with a smaller b. These curves depend on the interparticle distance r with r − (b − 2) when r tends to zero. For b between zero and two, the curves do not intercept each other and the potential tends to a constant value when r tends to zero. The electron–hole interaction potential thus describes the Coulomb effect between electrons and holes, and the anisotropy of the interaction induced by the anisotropy of the medium. If the radial part of the Schro¨dinger equation is written as in Eq. (7), it resembles the one-dimensional wave equation for a potential with three terms. One is due to the Coulomb effect and the term l is associated with the angular momentum.
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A. Simo˜es Baptista, H. Abreu Santos / Materials Science and Engineering B74 (2000) 259–262
The third term depends on the dimension parameter and on r as r − 2. This term can be considered as a contribution to the confinement potential. It is negative for values of b between one and three, and positive elsewhere. The eigenvalues of the radial part of the Schro¨dinger equation are only functions of the inverse screening length kS and of b. Therefore, the same value can result from systems of different materials and from different radius of quantum dots and wires. The exciton energy grows with the decrease of b and kS. It is clear from our results that for the confined structures fabricated up to now, that have excitons with energies smaller than 100ER, the dimension parameter should be greater than 1.50. It thus seems desirable to look for alternative materials systems with better confinement properties, to further improve even more the quantum and optoelectronic device characteristics.
.
Acknowledgements This work took place at the ‘Centro de Electrotecnia Teo´rica e Medidas Ele´ctricas do Instituto Superior Te´cnico’ and was supported by ‘Fundac¸a˜o para a Cieˆncia e a Tecnologia’ and ‘Fundac¸a˜o Calouste Gulbenkian’. References [1] A. Simo˜es Baptista, H. Santos, Microelectron. Eng. 43–44 (1998) 481 – 487. [2] F.H. Stillinger, J. Math. Phys. 18 (6) (1977) 1224 – 1234. [3] L. Goldstein, F. Glass, J.Y. Marzin, M.N. Charasse, G. Le Roux, Appl. Phys. Lett. 47 (1985) 1099. [4] D. Bimberg, N.N. Ledentsov, M. Girundmann, R. Heitz, J. Bohrer, V.M. Ustinov, J. Lumin. 72 – 74 (1997) 34 – 37. [5] P. Christol, Pierre Lefebvre, Henry Mathieu, IEEE J. Quantum Electron. 30 (10) (1994) 2287 – 2292. [6] L. Landau, E. Lifshitz, Mecaˆnica Quaˆntica. Teoria Na˜o-Relativista, MIR, Moscovo, 1985.
Electron–hole pairs binding energies in quantum wires and quantum dots A. Simo˜es Baptista *, H. Abreu Santos Centro de Electrotecnia Teo´rica e Medidas Ele´ctricas, Instituto Superior Te´cnico (Lisbon Technical Uni6ersity), A6. Ro6isco Pais, 1049 -001 Lisbon, Portugal
Abstract In a previous paper, we have calculated the binding energies for excitons in quantum wells [A. Simo˜es Baptista, H. Santos, Microelectron. Eng. 43–44 (1998) 481–487]. This was done by solving simultaneously the Schro¨dinger and the Poisson equations, which determine the interaction potential, both in a fractional dimensional space [F.H. Stillinger, J. Math. Phys. 18 (6) (1977) 1224–1234]. Using the same mathematical framework, we now analyse the so-called one-dimensional and zero-dimensional structures. The results we obtain are essential for simulations of optical-electronic devices that use the optical absorption and the refractive index dispersion curves. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Exciton energies; Fractional-dimensional spaces
1. Introduction The physical confinement of electrons and holes is a key feature of several last-generation quantum electronic and opto-electronic devices. Although the confinement is never complete, increasing the confinement improves the characteristics of these devices. The technological evolution of the past 15 years has made it possible to currently use quantum wells, quantum wires and quantum dots, in which the mobile charge carriers are confined, respectively, along one, two and three directions. In particular, as far as quantum dots are concerned, fabrication techniques may use self-ordering phenomena on crystal surfaces [3,4]. This allows the formation of dense arrays of three-dimensional semiconductor ‘islands’ with similar size and shape that have long-range ordering and are imbedded in a different semiconductor. Combining semiconductor pairs with great permittivity differences results in maximum confinement. It is expected that both theory and modelling cope with the aforementioned technological developments, making it possible to coherently describe all these situations. * Corresponding author. Tel.: +351-1-8486676; fax: + 351-18417672.
In the past 10 years, several attempts have been made to develop a theoretical background that unifies the study of the optical properties of quantum wells, quantum wires and quantum dots, dealing with them within the same framework [5]. Our contribution to the aforementioned unification recurs to the fractional dimensional spaces proposed by Stillinger [2] as the common mainframe. By using these, the three-dimensional situations with partial space confinement, which correspond to space anisotropy, are transformed into an analysis in uniform spaces whose dimension is described by a non-integer parameter b. We presented in Ref. [1] results for quantum wells, corresponding to b\ 2, which we hereby extend by considering bB 2. Furthermore, our modelling is expected to be accurate even when the permittivities of the substrate and of the dots or wires are quite different, and also includes the screening of the Coulomb interactions. Although for wells we have b\ 2, for wires b\ 1 and for dots b\ 0, it should be emphasised that b depends both on the confinement and the materials involved. Therefore, the same b value may correspond to wells, wires or dots of different materials. For b B 2, we begin by calculating the evolution of the electron–hole interaction potential with distance r and showing the differences this potential presents
0921-5107/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 9 9 ) 0 0 5 7 2 - 3
260
A. Simo˜es Baptista, H. Abreu Santos / Materials Science and Engineering B74 (2000) 259–262
when compared with the 1/r three-dimensional one. Next, we discuss the different potential terms appearing in the Schro¨dinger equation and calculate the binding energies of the excitons.
2. The electron–hole interaction potential In Ref. [1], we calculated the electron – hole interaction potential and obtained
kS q V(r)= − (b/2) r o(2p)
b−2 2
K b − 2 (kSr)
In Fig. 1, we represent the evolution of the potential in space when the inverse screening length is constant. The dimension of space b is a parameter. It is clear from the graphics that when r 0 for b\ 2, the curves with greater b value intercept all those with a smaller b value. On the contrary, for bB 2, the curves of the interaction potential do not intercept each other. This can be understood from Eq. (1). When r 0, the interaction potential tends asymptotically to (b − 2)
b− 2 2
q 1 (b/2) 4op r
V(r)= −
2− b q k (b − 2)G (b/2) S 2 o(4p)
(1)
2
where kS is the inverse screening length and r the distance between the electron and the hole.
V(r)= −
G
2Bb53 05 bB2
where G is the gamma function. From this result, it is clear that for bB2, the potential is constant near r=0 and decreases with the decrease of b. For b \2, the value of the potential tends to infinity at r=0 and depends on r as r − (b − 2).
3. The Schro¨dinger equation The Schro¨dinger equation is given by −
'2 2 (C 9 C+ WC= j 2m (t
(2)
and has as its solution C=% Ci exp(− jEi t)
(3)
i
in which the wave functions in space Ci satisfy the wave equation −
h2 2 9 Ci + (W− E)Ci = 0 2m
(4)
The excitonic problem has been considered in the Wannier–Mott effective mass model. In a space of dimension b and using spherical co-ordinates, the previous equation is then equivalent to the following system of non-coupled equations [5].
!
"
l(l +b− 2) ( 2f (b− 1) (f + − W(r)− 2E− f=0 2 (r r (r r2 (5a) ( 2U (U + (b− 1) cot gu −l(l +b− 2)u = 0 (u 2 (u
Fig. 1. Evolution of the interaction potential with r. b is the variable parameter and kS is constant, kS = 0.1. (a) 1 … 6, b =3.0, 2.9, 2.7, 2.5, 2.3 and 2.1; (b) 1 … 6, b =2.1, 1.9, 1.7, 1.5, 1.3 and 1.1; (c) 1 … 6, b =1.1, 0.9, 0.7, 0.5, 0.3 and 0.1.
(5b)
in which we have admitted that Ci = f(r)U(u), W(r)= qV(r) and l is the angular quantum number. Furthermore, the distance r has been normalised to the Bohr radius, aB, and the energy normalised to twice the Rydberg energy, ER. If we make the variable change
A. Simo˜es Baptista, H. Abreu Santos / Materials Science and Engineering B74 (2000) 259–262
261
conductors involved in the heterojunctions, and in particular on their permittivity difference. This term has a minimum when b= 2 and varies with r − 2. Since (3− b)(b −1)50.25, the exciton does not remain confined to the point r= 0 [6]. The value of this term in the confinement energy can be positive or negative. When b is between 1 and 3, the confinement energy will be greater than it would be expected when using the interaction potential for the tridimensional case. The opposite will happen for other values of b.
4. Calculating the eigenvalues Fig. 2. Evolution of the eigenvalue corresponding to the main quantum number n = 0. E normalized to the Rydberg energy ER corresponding to the tridimensional case. 4 × 10 − 1 5b 53.0; 6× 10 − 3 5 ks 5 10 − 1.
Fig. 3. Evolution of the energy eigenvalue for n= 0 with kS. b =2.00 (2), 1.75 () and 1.50 (*).
f=
u r [(b − 1)/2]
>
(6)
n?
then it results from Eq. (5a) that (3 − b)(b − 1) −l(l +b −2) d2u 4 + 2E−W(r)+ dr 2 r2 =0
u
(7)
This radial equation resembles the one-dimensional wave equation of a particle in a potential given by W(r)−
(3− b)(b −1) l(l +b − 2) + r2 4r 2
(8)
The first term describes the Coulomb interaction. The third term is connected with the angular momentum of the exciton. The second term of the potential is negative for 1 Bb B 3 and positive for b \ 3 and b B1. This term may be interpreted as the variation of the confinement energy due to the dependence of the interaction potential on b. It will ultimately depend on the semi-
The eigenvalues associated with Eq. (5a) have been calculated by applying the Pru¨fer transformation. It is assumed that the quantum dots have spherical symmetry. From the results in Fig. 2, we can see that the eigenvalues increase as either b or kS decrease. These results are independent of the materials system used. They only depend on the value of the inverse screening length and on the dimension parameter. Therefore, each point of these curves may correspond to various combinations of material systems, and quantum wires and dots with different radius. When this radius decreases, the inverse screening length also decreases. For values of the radius of the quantum dots r such that raB, we have kS 0. The great majority of structures that have been fabricated up to now have an excitonic energy less than 100ER. It is clear from Fig. 3 that the dimension parameter should therefore be greater than 1.5.
5. Conclusions The electron–hole interaction potential has been established in a fractional dimensional space using the mathematical framework proposed by Stillinger. If the value of the dimension parameter b is greater than two, the curve of the potential for a given b intercepts all those with a smaller b. These curves depend on the interparticle distance r with r − (b − 2) when r tends to zero. For b between zero and two, the curves do not intercept each other and the potential tends to a constant value when r tends to zero. The electron–hole interaction potential thus describes the Coulomb effect between electrons and holes, and the anisotropy of the interaction induced by the anisotropy of the medium. If the radial part of the Schro¨dinger equation is written as in Eq. (7), it resembles the one-dimensional wave equation for a potential with three terms. One is due to the Coulomb effect and the term l is associated with the angular momentum.
262
A. Simo˜es Baptista, H. Abreu Santos / Materials Science and Engineering B74 (2000) 259–262
The third term depends on the dimension parameter and on r as r − 2. This term can be considered as a contribution to the confinement potential. It is negative for values of b between one and three, and positive elsewhere. The eigenvalues of the radial part of the Schro¨dinger equation are only functions of the inverse screening length kS and of b. Therefore, the same value can result from systems of different materials and from different radius of quantum dots and wires. The exciton energy grows with the decrease of b and kS. It is clear from our results that for the confined structures fabricated up to now, that have excitons with energies smaller than 100ER, the dimension parameter should be greater than 1.50. It thus seems desirable to look for alternative materials systems with better confinement properties, to further improve even more the quantum and optoelectronic device characteristics.
.
Acknowledgements This work took place at the ‘Centro de Electrotecnia Teo´rica e Medidas Ele´ctricas do Instituto Superior Te´cnico’ and was supported by ‘Fundac¸a˜o para a Cieˆncia e a Tecnologia’ and ‘Fundac¸a˜o Calouste Gulbenkian’. References [1] A. Simo˜es Baptista, H. Santos, Microelectron. Eng. 43–44 (1998) 481 – 487. [2] F.H. Stillinger, J. Math. Phys. 18 (6) (1977) 1224 – 1234. [3] L. Goldstein, F. Glass, J.Y. Marzin, M.N. Charasse, G. Le Roux, Appl. Phys. Lett. 47 (1985) 1099. [4] D. Bimberg, N.N. Ledentsov, M. Girundmann, R. Heitz, J. Bohrer, V.M. Ustinov, J. Lumin. 72 – 74 (1997) 34 – 37. [5] P. Christol, Pierre Lefebvre, Henry Mathieu, IEEE J. Quantum Electron. 30 (10) (1994) 2287 – 2292. [6] L. Landau, E. Lifshitz, Mecaˆnica Quaˆntica. Teoria Na˜o-Relativista, MIR, Moscovo, 1985.