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Ekimov ansatz and binding energy of exciton type II in quantum dots

Module by: Long Nguyen

Summary: For investigation the binding energy of exciton type I in quantum dots, Ekimov et al [4] have proposed an ansatz that hole is located at the center of the quantum dot, which gives a good agreement with experimental data and with other theoretical models. Using this proposal, we have computed the binding energy of exciton type II in coated spherical quantum dot and obtained the dependence of energy on the rate between the two radii of the layers of the dot. In the limit of the inner radius tends to zero, our calculated value returns to Ekimov’s result for the case of exciton type I. The method can also be generalized to compute binding energy for multi-layer quantum dot having more than one exciton inside.

  1. Introduction
Semiconductor nanocrystals, with diameters in the range of a few to hundreds nanometer, have been widely studied over the past decade due to their novel luminescence properties. These properties arise from confinement of the electronic states in three dimensions. A elementary electronic excitations near the band gap in semiconductors take the form of electron-hole pairs, called exciton, which are similar to a hydrogen atom with the Bohr radius about ten nanometers. If the physical size of the semiconductor material is smaller than the Bohr radius of the exciton, the exciton is strongly quantum confined. It is for this reason that these nanocrystals are also called quantum dots. In quantum dots the energy of the electronic states can be determined using the model of a particle in a three-dimensional quantum box. In the specific case of spherical semiconductor quantum dots, the energy levels are given as splitting relative to the band energy Eg of the bulk semiconductor,
(1)
where the last term is the Coulombic attraction between the electron and hole of the exciton. Equation (1) has been obtained by many authors using various methods, like exact diagonalization, variation, Monte-Carlo [1, 2]. . . , it implies that as the absorption edge of the quantum dot is shifted to higher energy with decreasing diameter of the quantum dot with a dependence of 1/R2 addition, the absorption spectrum has structure resulting from resonance determined by the principal quantum number n. In previous work [8] we calculate binding energy of exciton in case the electron and the hole located difference position. In this work, we use Akimov ansatz that the hole is located at dot center [4,8] to recalculate (1), which is the binding energy of exciton type I in spherical quantum dots, and the result is quite good. Due to this agreement, we continue to use the ansatz to compute the binding energy of exciton type II in coated quantum dots.
2. Exciton type I in quantum dot
Exciton is called type I if the electron and hole are confined in the same phase space [3]. In this section,we only consider exciton of this type in spherical quantum dot with infinite confining potential
(2)
where r→i(i=e,h)r→i(i=e,h) size 12{ { vec {r}} rSub { size 8{i} } \( i=e,h \) } {}are coordinate of electron and hole, R is radius of the dot (Fig. 1a). The Hamiltonian for exciton type I in the effective-mass description is
H=Eg−ℏ2∇e22me+Ve(r→e)−ℏ2∇h22mh+Vh(r→h)−e2ε∣r→e−r→h∣H=Eg−ℏ2∇e22me+Ve(r→e)−ℏ2∇h22mh+Vh(r→h)−e2ε∣r→e−r→h∣ size 12{H=E rSub { size 8{g} } - { { hbar rSup { size 8{2} } nabla rSub { size 8{e} } rSup { size 8{2} } } over {2m rSub { size 8{e} } rSup { size 8{*} } } } +V rSub { size 8{e} } \( { vec {r}} rSub { size 8{e} } \) - { { hbar rSup { size 8{2} } nabla rSub { size 8{h} } rSup { size 8{2} } } over {2m rSub { size 8{h} } rSup { size 8{*} } } } +V rSub { size 8{h} } \( { vec {r}} rSub { size 8{h} } \) - { {e rSup { size 8{2`} } } over {ε lline { vec {r}} rSub { size 8{e} } - { vec {r}} rSub { size 8{h} } rline } } } {} (3)
where Eg is the energy gap, the next four terms are Hamiltonian which describes non interacting electron-hole pair, the last term is Coulomb potential of electron and hole. In the limit of strong confinement R<<aB, where aB is the Bohr radius of exciton, both the electron and the hole are quantized separately and the Coulomb potential can be treated as a small perturbation. Using Ekimov ansatz that the hole is located at dot center, we estimate correction of Coulomb interaction by perturbation theory. The energy of the lowest state of and exciton in the first order of perturbation is given by
(4)
where correction energy EI is given by
ΔEI=〈ψ(r→e,r→h)∣Heh∣ψ(r→e,r→h)〉=−e2ε∫∣Φ(r→e)∣2∣Φ(r→h)∣2∣r→e−r→h∣dr→edr→hΔEI=〈ψ(r→e,r→h)∣Heh∣ψ(r→e,r→h)〉=−e2ε∫∣Φ(r→e)∣2∣Φ(r→h)∣2∣r→e−r→h∣dr→edr→h size 12{ΔE rSub { size 8{I} } = \langle ψ \( { vec {r}} rSub { size 8{e} } , { vec {r}} rSub { size 8{h} } \) \lline H rSub { size 8{ ital "eh"} } \lline ψ \( { vec {r}} rSub { size 8{e} } , { vec {r}} rSub { size 8{h} } \) \rangle = - { {e rSup { size 8{2} } } over {ε} } Int rSub {} rSup {} { { { lline Φ \( { vec {r}} rSub { size 8{e} } \) rline rSup { size 8{2} } lline Φ \( { vec {r}} rSub { size 8{h} } \) rline rSup { size 8{2} } } over { lline { vec {r}} rSub { size 8{e} } - { vec {r}} rSub { size 8{h} } rline } } d { vec {r}} rSub { size 8{e} } d { vec {r}} rSub { size 8{h} } } } {} (5)
Next, we consider two cases i) when the hole seats at the dot center and ii) when it can moves around in effective potential.
i) The hole seats at the dot center
The wave function at ground states of hole and electron is given by
∣Φ(r→h)∣2=δ(r→h−0→),∣Φ(r→e)∣2=12πRre2sin2(πreR)∣Φ(r→h)∣2=δ(r→h−0→),∣Φ(r→e)∣2=12πRre2sin2(πreR) size 12{ lline Φ \( { vec {r}} rSub { size 8{h} } \) rline rSup { size 8{2} } =δ \( { vec {r}} rSub { size 8{h} } - { vec {0}} \) , lline Φ \( { vec {r}} rSub { size 8{e} } \) rline rSup { size 8{2} } = { {1} over {2π ital "Rr" rSub { size 8{e} } rSup { size 8{2} } } } "sin" rSup { size 8{2} } \( π { {r rSub { size 8{e} } } over {R} } \) } {} (6)
For convenience, we adopt the dot radius R for the unit of length, and e2/R2 {} for unit of energy. In the unitless system and from the expressions (5) and (6) we get
ΔEI=−2∫0R1resin2(πreR)dreΔEI=−2∫0R1resin2(πreR)dre size 12{ΔE rSub { size 8{I} } = - 2 Int cSub { size 8{0} } cSup { size 8{R} } { { {1} over {r rSub { size 8{e} } } } "sin" rSup { size 8{2} } \( π { {r rSub { size 8{e} } } over {R} } \) ital "dr" rSub { size 8{e} } } } {} (7)
Note that the integral in (7) is impossible to be analytically computed. We have employed numerical technique to perform the task and obtained the result
ΔEI=−2.438ΔEI=−2.438 size 12{ΔE rSub { size 8{I} } = - 2 "." "438"} {} (8)
ii) The hole moves in effective potential
Due to the condition that is the real semiconductor the hole are much heavier than electron. Then the electron is moving very fast compare with the hole, them we can treat that the hole is moving in the effective potential of electron
ueff(ρ)=∫−11(r+ρ)2−(r−ρ)2rρsin2(πr)drueff(ρ)=∫−11(r+ρ)2−(r−ρ)2rρsin2(πr)dr size 12{u rSub { size 8{ ital "eff"} } \( ρ \) = Int cSub { size 8{ - 1} } cSup { size 8{1} } { { { sqrt { \( r+ρ \) rSup { size 8{2} } } - sqrt { \( r - ρ \) rSup { size 8{2} } } } over {rρ} } "sin" rSup { size 8{2} } \( πr \) ital "dr"} } {} (9)
ΔEII=2∫01ueffρsin2(πρ)dρΔEII=2∫01ueffρsin2(πρ)dρ size 12{ΔE rSub { size 8{ ital "II"} } =2 Int cSub { size 8{0} } cSup { size 8{1} } {u rSub { size 8{ ital "eff"} } left (ρ right )"sin" rSup { size 8{2} } \( ital "πρ" \) dρ} } {} (10)
We obtain
ΔEI=1.78ΔEI=1.78 size 12{ΔE rSub { size 8{I} } =1 "." "78"} {} (11)
That value of correction energy is in good agreement with other results of other approaches. The difference between (8) and (11) is due to movement of the hole. Using again this way, we calculate binding energy of exciton type II in coated quantum dot.
  1. Exciton type II in quantum dot
In the exciton type II, the electron and hole are located in different regions of the phase space [5, 10]. We consider in this section exciton type II in coated quantum dot (Fig. 1b) where the hole is confining in the internal sphere with the radius R2R2 size 12{R rSub { size 8{2} } } {} and the electron is moving the space between two spheres R1R1 size 12{R rSub { size 8{1} } } {} and R2R2 size 12{R rSub { size 8{2} } } {}. The confined potential for electron and hole are
, (14.a)
. (14.b)
_rerhrerhREgErEgErR1R2_+_+++__(a) (b)
Fig. 1: Models of exciton type I (a) and type II (b) in quantum dot
Similarly we consider the two cases of the movement of the hole and obtain the Coulomb correction energy with respect to them.
i) The hole seats is the center the dot
ΔEII(η)=21−η∫η11rsin2(πx−η1−η)drΔEII(η)=21−η∫η11rsin2(πx−η1−η)dr size 12{ΔE rSub { size 8{ ital "II"} } \( η \) = { {2} over {1 - η} } Int cSub { size 8{η} } cSup { size 8{1} } { { {1} over {r} } "sin" rSup { size 8{2} } \( π { {x - η} over {1 - η} } \) } ital "dr"} {} (15)
ii) The hole moves in effective potential
ueffη,ρ=∫η1(r+ηρ)2−(r−ηρ)2rηρsin2(πr−η1−η)drueffη,ρ=∫η1(r+ηρ)2−(r−ηρ)2rηρsin2(πr−η1−η)dr size 12{u rSub { size 8{ ital "eff"} } left (η,ρ right )= Int cSub { size 8{η} } cSup { size 8{1} } { { { sqrt { \( r+ ital "ηρ" \) rSup { size 8{2} } } - sqrt { \( r - ital "ηρ" \) rSup { size 8{2} } } } over {r ital "ηρ"} } "sin" rSup { size 8{2} } \( π { {r - η} over {1 - η} } \) d} `r} {} (16)
where η=R1/R2η=R1/R2 size 12{η=R rSub { size 8{1} } /R rSub { size 8{2} } } {} Energy correction is given by
ΔEII(η)=21−η∫01ur,ρsin2(πρ)dρΔEII(η)=21−η∫01ur,ρsin2(πρ)dρ size 12{ΔE rSub { size 8{ ital "II"} } \( η \) = { {2} over {1 - η} } Int cSub { size 8{0} } cSup { size 8{1} } {u left (r,ρ right )"sin" rSup { size 8{2} } \( ital "πρ" \) dρ} } {} (17)
Figure 1
Fig. 2: Ground state energy of exciton as a function of dot radius and the dependence of binding energy of exciton type II on the rate between the two radii of the layers of the coated quantum dot
  1. Conclusion
The numerical results showing the dependence of the correction energy on the rate between the two radii are plotted in Fig. 2 for (16) and (15). It is easily seen that when the rate increases, the energy correction decreases, and when it tends to zero, the two results are the same as (8). It is surprising that the two plotted lines virtually coincide with each other. This astonishing result can be physically explain, that is because of the fact that the wave function of the hole is spherically symmetric and the Coulomb interaction is small. It is, therefore, the Akimov ansatz is reasonably applied for the coated quantum dot. For further research, the method can also be generalized to compute binding energy for multi-layer quantum dot having more than one exciton inside
  1. Acknowledgment
This work is supported by the National Basics Research Program 4.1.1601: “Theory of low dimensional systems and nano-structures”.

References

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