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Volumn 103, Issue 11, 2008, Pages

Dichotomy of the exciton wave function in semiconductors under intense laser fields

Author keywords

[No Author keywords available]

Indexed keywords

DIFFERENCE EQUATIONS; ELECTRIC CONDUCTIVITY; EQUATIONS OF MOTION; EXCITONS; FOURIER TRANSFORMS; FUNCTION EVALUATION; LASER OPTICS; LASERS; MATHEMATICAL TRANSFORMATIONS; NUMERICAL METHODS; PULSED LASER DEPOSITION; SEMICONDUCTOR MATERIALS; WAVE FUNCTIONS;

EID: 45149085515     PISSN: 00218979     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2937087     Document Type: Article
Times cited : (32)

References (73)
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    • For simplicity, we shall restrict our study to nondegenerate, parabolic, isotropic bands. This is a good approximation for carriers with small wave numbers in most III-V semiconductors. This is particularly true for electrons in GaAs, in which the minimum of the conduction band (at K=0) has a 6 symmetry.
    • For simplicity, we shall restrict our study to nondegenerate, parabolic, isotropic bands. This is a good approximation for carriers with small wave numbers in most III-V semiconductors. This is particularly true for electrons in GaAs, in which the minimum of the conduction band (at K=0) has a 6 symmetry.
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    • The zero energy reference level was taken at the top of the valence band.
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    • It should be emphasized that, similarly to what is usually done by atomic physicists, we adopted in our computational programs a system of units that is more natural for working with excitons in semiconductors, namely the "excitonic" effective system of units. In this system, one has =e=μ= k/ r =1, which simplifies Eq. to [-1 / 2 ∇2 - 1/r] (r) = ε̃ (r), where ε̃ is the energy in e.u. The unit of energy is 1 Ht (=2 Ry) and the unit of length is 1 aB. This makes all mathematical expressions related to the SCD equation assume a more simple form, with the additional advantage of reducing the rounding-off errors in numerical computations since numbers whose scientific notation presents either very small or very large exponents are rare when e.u.'s are adopted.
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    • The operator U, the Hermitian adjoint of U, can be easily found by exchanging i by -i in Eq.
    • The operator U, the Hermitian adjoint of U, can be easily found by exchanging i by -i in Eq..
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    • At this point, the translation operator property ex(±i / αp) f (r) =f (r±α) is used thoroughly.
    • At this point, the translation operator property exp (±i / αp) f (r) =f (r±α) is used thoroughly.
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    • In the laboratory frame, the solution of the SCD equation in the momentum gauge is (r,t) =U ̃ (r,t) = ̃ (r-α (t),t) exp [-i e2 / (2μ) A2 dt] ≈ ̃ 0 (r-α (t),t) exp (-it), where (EKH + EFK) /. As customary, we have neglected the rapidly oscillating phase factor exp [-i e2 A02 / (8ωμ) sin (2ωt)] in this approximation. It is then apparent that the energy spectrum for the SCD equation in the laboratory frame is shifted with respect to that given by Eq. just by EFK.
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    • The shift from the laboratory to the Kramers accelerated frame of reference is only an interpretation of the K-H unitary transformation and no assumptions concerning the validity of Eq. are needed
    • The shift from the laboratory to the Kramers accelerated frame of reference is only an interpretation of the K-H unitary transformation and no assumptions concerning the validity of Eq. are needed.
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    • For a given laser source, whose frequency is and output power is I, the following practical formulas are useful: F0 (in kV/cm) ≈0.868 I / r 4 and α0 (in e.u.) ≈7.31 r -5 /4 I / 2, for in THz and I in kW / cm2. The inclusion of r in these formulas agrees with the distinction made by Liu and Zhu between the intensity of the laser source (our I) and the laser intensity inside the semiconductor material (see Ref.).
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    • For those interested in developing Fourier-Floquet computational calculations of higher orders, it will be advantageous to adopt the following integral expression for these coefficients: m (r; α0) = im / π -1 +1 V (r+ α0 uk) Tm (u) / 1- u2 du, where Tm (u) are Chebyshev polynomials.
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    • In GaAs, e.g., this opens an ample range of valid frequencies, as given by 0.4 THz 1727 THz, where a refractive index of 3.3 was assumed. The upper limit can be smaller, however, due to the nonresonance condition
    • In GaAs, e.g., this opens an ample range of valid frequencies, as given by 0.4 THz 1727 THz, where a refractive index of 3.3 was assumed. The upper limit can be smaller, however, due to the nonresonance condition.
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    • Note that the origin of the coordinates for the relative electron-hole position (i.e., the point at which r=0) is kept at the hole position. This is in agreement with our definition of r
    • Note that the origin of the coordinates for the relative electron-hole position (i.e., the point at which r=0) is kept at the hole position. This is in agreement with our definition of r.
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    • As pointed out by Gavrila and Kaminski (see Ref.), it is indeed possible to write 0 (r; α0) in terms of K (x), the complete elliptic integral of the first kind, as given by 0 (r; α0) =- (2/π) (k e2 / r) (1/ r+ r-) K [(1/ 2) 1- r + r -], where r± =r± α0 k. Since K (0) = π /2, then 0 (r; α0) →V (r), the field-free potential energy, for α0 r. Note that 0 (r; α0) has 1/ r singularities at the endpoints of the charge distribution (at ± α0 k), as well as a logarithmic singularity along the line segment between them, due to the behavior of K (x) for x→1.
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    • In fact, we found computationally advantageous to make use of the elliptic integral expression for 0 (r; α0) given in Ref.
    • In fact, we found computationally advantageous to make use of the elliptic integral expression for 0 (r; α0) given in Ref..
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    • Within the Born-Oppenheimer approximation, this potential energy reads -k e2 ([1/ r+ (R̃ /2) ] + [1/ r- (R̃ /2) ]), where R̃ is the internuclear distance.
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    • 45149114950 scopus 로고    scopus 로고
    • + molecule are simply φ (r,R) = B± (Β=1) [φ± (r,R) ± φ- (r,R)], where the + (-) sign is for the gerade (ungerade) state and φ± (r,R) are 1s hydrogen AOs centered at r± (R/2)
    • + molecule are simply φ (r,R) = B± (Β=1) [φ± (r,R) ± φ- (r,R)], where the + (-) sign is for the gerade (ungerade) state and φ± (r,R) are 1s hydrogen AOs centered at r± (R/2).
  • 67
    • 0003460779 scopus 로고
    • For more details on this coordinates system see, e.g., (McGraw-Hill, New York), Vol. 1, Appendix 1. See also, 2nd ed. (Prentice Hall, New York, 2003), A practical manner to evaluate two-center integrals is found in Appendix 10.
    • For more details on this coordinates system see, e.g., J. C. Slater, Quantum Theory of Molecules and Solids (McGraw-Hill, New York, 1963), Vol. 1, Appendix 1. See also B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, 2nd ed. (Prentice Hall, New York, 2003), pp. 490-495. A practical manner to evaluate two-center integrals is found in Appendix 10.
    • (1963) Quantum Theory of Molecules and Solids, Physics of Atoms and Molecules , pp. 490-495
    • Slater, J.C.1    Bransden, B.H.2    Joachain, C.J.3
  • 68
    • 45149119013 scopus 로고    scopus 로고
    • note
    • + molecule in the limit R→0 is -0.8 Ry, somewhat above the exact value of -1 Ry (the energy of the 2p AO for the He+ ion). Thus, the exact value projected for the exciton binding energy in the absence of a laser field is -0.25 Ry, instead of -0.20 Ry. Though this small difference can be important in some circumstances (e.g., when exact numerical results for small values of α0 are sought), it is not relevant here since we are more interested in the correct qualitative behavior of the energy eigenvalues and their corresponding eigenfunctions for large values of α0. Anyway, exact numerical solutions for Eq. valid for all α0 are being worked out by the authors.
  • 70
    • 45149083781 scopus 로고    scopus 로고
    • As pointed out by Pont and Gavrila (see Ref.), a hydrogenlike system does not suddenly appear in a region under ILF, but instead it is exposed to subcritical fields during the ramuof the laser pulse, and these fields can in effect ionize a large number of atoms
    • As pointed out by Pont and Gavrila (see Ref.), a hydrogenlike system does not suddenly appear in a region under ILF, but instead it is exposed to subcritical fields during the ramp up of the laser pulse, and these fields can in effect ionize a large number of atoms.
  • 71
    • 0020722092 scopus 로고
    • SSCOA4 0038-1098 10.1016/0038-1098(83)90799-8.
    • L. C. M. Miranda, Solid State Commun. SSCOA4 0038-1098 10.1016/0038-1098(83)90799-8 45, 783 (1983).
    • (1983) Solid State Commun. , vol.45 , pp. 783
    • Miranda, L.C.M.1
  • 73
    • 45149117553 scopus 로고    scopus 로고
    • The lowerature condition is crucial, because thermal ionization can mask the laser ionization process. Thermal ionization is expected to be efficient at temperatures above Tmax, given by 3/ 2 kB Tmax = 1/4 Ry. In GaAs one has Tmax =12.8 K, which is easily attainable in the laboratory
    • The lowerature condition is crucial, because thermal ionization can mask the laser ionization process. Thermal ionization is expected to be efficient at temperatures above Tmax, given by 3/ 2 kB Tmax = 1/4 Ry. In GaAs one has Tmax =12.8 K, which is easily attainable in the laboratory.


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