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Application of the photon's intrinsic flux to the 1s-2p and 2p-3d excitonic transitions in nanostructures.

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Ann. Phys. (Berlin) 18, No. 12, 910 – 912 (2009) / DOI 10.1002/andp.200910398
Application of the photon’s intrinsic flux to the 1s-2p and 2p-3d
excitonic transitions in nanostructures
Z. Saglam1,∗ and M. Saglam2,∗∗
1
2
Aksaray University Department of Physics, Science Faculty, Department of Physics, 68100-Aksaray,
Turkey
Ankara University, Department of Physics, 06100-Tandogan, Ankara, Turkey
Received 1 September 2009, accepted 13 September 2009
Published online 11 December 2009
Key words Quantized magnetic flux, Dirac hydrogen atom, optical transitions, excitons.
PACS 03.65.-w, 03.65.Ca, 05.30.-d
We have applied our two recent results [depending on its helicity photon carries a quantum flux of ±Φ0 =
±hc/e and the quantized magnetic fluxes through the electronic orbits of the Dirac hydrogen atom are given
by: Φ(n, l, mj ) = (n − l − mj )Φ0 )] to the 1s-2p and 2p-3d excitonic transitions in nanostructures. It is
shown that the flux changes for the non-zero matrix elements in the 1s-2p and 2p-3d excitonic transitions is
either ±Φ0 or zero. The present result supports the previous results stated above. It is also shown that spin
flip is possible in the 1s-2p and 2p-3d excitonic transitions.
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
In an earlier study [1] we calculated the quantized magnetic fluxes through the electronic orbits |n, l, mj of hydrogen atom in the absence of an external magnetic field. The source of the magnetic field was taken
to be the proton’s magnetic moment. The resultant quantum flux through the quantum orbit corresponding
to the |n, l, mj state was found to be: Φ(n, l, mj ) = (n − l − mj )Φ0 . Very recently Saglam and Sahin
[2, 3] proved that photon itself carries an intrinsic flux quantum of ±Φ0 = ±hc/e here the (+) and (−)
signs stand for the right hand and left circular helicity respectively] and the circumference of the flux tube
is equal to the wavelength of the photon. The aim of the present study is to apply the results of [1, 2] to
the 1s-2p and 2p-3d excitonic transitions in nanostructures. Excitons in semiconductors and nanostructures
constitute a well defined model system to study many body interactions between large number of quasiparticles, phonons and photons [4–8]. Especially their relations with semiconductor quantum dots hold
great promise, due to the recent advances in nanostructure. Because in these systems the long ranged
Coulomb interaction between electrons and holes can stabilize the formation of excitons as hydrogen-like
bound states with new physical properties. We show that the flux changes for the non-zero matrix elements
in the 1s-2p and 2p-3d excitonic transitions is either ±Φ0 or zero. This is consistent with our recent result
that photon carries a quantum flux of ±Φ0 = ±hc/e (here the (+) and (−) signs stand for the right hand
and left circular helicity respectively]. In passing we note that the superposition of the right and left hand
circular helicity gives us linear polarization which corresponds zero flux change. We have also shown that
during the excitonic transitions the spin flip-floppings are allowed as was also obserwed experimentally
[4–6].
∗
∗∗
Corresponding author E-mail: zsaglam@aksaray.edu.tr
E-mail: saglam@science.ankara.edu.tr
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Ann. Phys. (Berlin) 18, No. 12 (2009)
911
2 Formalism
To find the eigenvalues corresponding to the excitonic levels we first write the Dirac Hamiltonian for
hydrogen atom [1] :
HD = α · pc + βmc2 + V (r)
(1)
where V (r) = −e2 /r is the Coulomb potential, m is the mass of an electron, c is the velocity of light and
α and β are the standard Dirac matrices in the Dirac representation. The spin dependent eigenstates of the
Hamiltonian are given in [1]. To write the Dirac Hamiltonian for an exciton V (r) = −e2 /r is replaced
by V (r) = −e2 /εr (ε is the dielectric constant of the related nanostructure) and m is replaced by the
effective mass, m∗ . As a result of these replacements we get the effective Bohr radius a∗ = ε2 /m∗ e2
which is quite large compared to Bohr radius a0 of hydrogen atom. Then the spin dependent eigenvalues
for an exciton are written as:
l + mj + 12
mj − 12
Rnl (r)exciton Yl=j−
|n, l, ml , ↑exciton ≡ Ψn,j=l+ 12 ,mj =
1 χ+
2
2l + 1
l − mj + 12
mj + 12
Rnl (r)exciton Yl=j−
+
1 χ−
2
2l + 1
m − 12
m +1
= F1 Yj−j1
|n, l, ml , ↓exciton
χ+ + F2 Yj−j1 2 χ−
2
2
l − mj + 12
mj − 12
Rnl (r)exciton Yl=j+
≡ Ψn,j=l− 12 ,mj = −
1 χ+
2
2l + 1
l + mj + 12
mj + 12
Rnl (r)exciton Yl=j+
+
1 χ−
2
2l + 1
m − 12
= −F2 Yj+j1
2
m + 12
χ+ + F1 Yj+j1
2
χ− .
(2a)
(2b)
When an exciton is subject to an external electromagnetic field, it will have interaction through the Hamiltonian (H = −d · E). According to the Golden rule, The transition probability will be proportional to the
square of the matrix element of H between the initial and the final states:
f |H |i = n , l , mj |H |n, l, mj = n , l , mj | − d · E|n, l, mj (3)
which can be put in the form [9]:
f |H |i ≈ n , l , mj |r sin θe±iφ |n, l, mj + n , l , mj |r cos θ|n, l, mj .
(4)
For (1s→2p) we will have 6 matrix elements while for (2p→3d) we will have 60 matrix elements. The list
of non-zero matrix elements and the related flux changes for (1s→2p) transitions are:
2, 1, 0, −1/2| − d · E|1, 0, 0, 1/2 → ΔΦ = Φ0
2, 1, 1, −1/2| − d · E|1, 0, 0, 1/2 → ΔΦ = 0
(5)
2, 1, −1, 1/2| − d · E|1, 0, 0, 1/2 → ΔΦ = Φ0
2, 1, 0, 1/2| − d · E|1, 0, 0, 1/2
→ ΔΦ = 0
Similarly the list of non-zero matrix elements and the related flux changes for (2p→3d) are:
www.ann-phys.org
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
912
Z. Saglam and M. Saglam: Photon’s intrinsic flux to excitonic transitions in nanostructures
3, 2, 2, −1/2| − d · E|2, 1, 1, 1/2
→ ΔΦ = 0
3, 2, 1, −1/2| − d · E|2, 1, 1, 1/2
→ ΔΦ = Φ0
3, 2, 1, −1/2| − d · E|2, 1, 0, 1/2
→ ΔΦ = 0
3, 2, 0, −1/2| − d · E|2, 1, 0, 1/2
→ ΔΦ = Φ0
3, 2, 0, −1/2| − d · E|2, 1, −1, 1/2
→ ΔΦ = 0
3, 2, −1, −1/2| − d · E|2, 1, −1, 1/2 → ΔΦ = Φ0
3, 2, 1, 1/2| − d · E|2, 1, 1, 1/2
→ ΔΦ = 0
3, 2, 0, 1/2| − d · E|2, 1, 1, 1/2
→ ΔΦ = Φ0
3, 2, 0, 1/2| − d · E|2, 1, 0, 1/2
→ ΔΦ = 0
3, 2, −1, 1/2| − d · E|2, 1, 0, 1/2
→ ΔΦ = Φ0
3, 2, −1, 1/2| − d · E|2, 1, −1, 1/2
→ ΔΦ = 0
3, 2, −2, 1/2| − d · E|2, 1, −1, 1/2
→ ΔΦ = Φ0
3, 2, 1, −1/2| − d · E|2, 1, 1, −1/2
→ ΔΦ = 0
3, 2, 0, −1/2| − d · E|2, 1, 1, −1/2
→ ΔΦ = Φ0
3, 2, 0, −1/2| − d · E|2, 1, 0, −1/2
→ ΔΦ = 0
(6)
3, 2, −1, −1/2| − d · E|2, 1, 0, −1/2 → ΔΦ = Φ0
3 Conclusions
We have determined the non-zero matrix elements and the flux changes for the 1s→2p and 2p→3d excitonic transitions by using the solutions of Dirac hydrogen-like atoms. We have shown that the flux changes
for the non-zero matrix elements in the 1s-2p and 2p-3d excitonic transitions is either ±Φ0 or zero. This is
consistent with our recent result [1] which states that photon carries a quantum flux of ±Φ0 = ±hc/e (here
the (+) and (−) signs stand for the right hand and left circular helicity respectively). In passing we note
that the superposition of the right and left hand circular helicity gives us linear polarization which corresponds to zero flux change. We have also shown that during the excitonic transitions the spin flip-floppings
are allowed as was also obserwed experimentally [4–6].
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
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M. Saglam and G. Sahin, Int. J. Modern Phys. B 23, 23 (2009).
M. Saglam and G. Sahin, Proceedings of DSM 2008 4–6 December, MRS fall Meeting Boston, USA 2008.
M. Saglam, B. Boyacioglu, and Z. Saglam, J. Laser Optic 28(4), 377 (2007).
A. Vinattieri, J. Shah, T. C. Damen, D. S. Kim, L. N. Maialle, and L. J. Sham, Phys. Rev. B 50, 10868 (1994).
D. W. Snoke, W. W. Rühle, K. Köhler, and K. Ploog Phys. Rev. B 55, 13789-13794 (1997).
W. Heller and U. Bockelmann, Phys. Rev. B 55, R4871–R4874 (1997).
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96 227402 (2006).
[10] M. Saglam, B. Boyacioglu, and Z. Saglam, J. Laser Optic 28(4), 361 (2007).
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.ann-phys.org
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