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Although the atomic units of power (P0) and area-i.e., a02, where a0 2 / (m0 k e2) is the atomic Bohr radius-can be taken to form an a.u. of intensity I0 = P0 / a02 (≈6.4× 1015 W/ cm2), we found it useful to adopt here the "electromagnetic" a.u., which corresponds to the intensity of a linearly polarized electromagnetic plane wave with an amplitude equal to an a.u. of electric field (E0). This results in Iγ = 0 c E0 / a02 = I0 / (8π ), where k e2 / (c) ≈ 1/ 137 is the (dimensionless) fine-structure constant. Note that the interaction of an electromagnetic field of intensity Iγ with a hydrogen atom in the ground state produces a force in the electron equal to the Coulomb attraction exerted on it by the nucleus.
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31
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45149093354
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note
-
Nowadays, excitons in semiconductors are routinely observed in photoluminescence experiments, in which a series of narrow lines at frequencies below the band gap resonance in the band-to-band absorption spectrum can be resolved. These lines are particularly pronounced at low temperatures.
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32
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45149133879
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Wannier-Mott excitons are weakly bound electron-hole pairs, with an average relative distance large in comparison with a lattice constant
-
Wannier-Mott excitons are weakly bound electron-hole pairs, with an average relative distance large in comparison with a lattice constant.
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-
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33
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45149124976
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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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34
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45149086489
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The zero energy reference level was taken at the toof the valence band
-
The zero energy reference level was taken at the top of the valence band.
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-
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36
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45149120668
-
-
note
-
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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37
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45149093903
-
-
note
-
In bulk GaAs at low temperatures, e.g., the electron and hole effective masses are 0.0665 m0 and 0.45 m0, respectively, where m0 is the electron rest mass. The relative dielectric constant r decreases from 13.18 to 10.90 with the increase of the ac-field frequency. We adopted the high-frequency value in this work.
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42
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45149086782
-
-
note
-
This approximation works only if the relative coordinate is limited to the size of the two-body system, which makes exp (±iKr) only slightly different from units. This condition can be written as 〈 Kr 〉 1. Since 〈 Kr 〉 ≈ μ/ m K aB and K= ω /c, where is the refractive index of the hosting material, one has ω m /μ × c/ ( aB). Note that the dipole approximation is actually an idealization for the pulsed beams used in most experiments since in practice they have a certain inhomogeneity in both space and time.
-
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44
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3643096294
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PRLTAO 0031-9007 10.1103/PhysRevLett.21.838.
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Henneberger, W.C.1
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45
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1542470699
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The physical irrelevance of the ponderomotive shift in a region of almost uniform field is consistent with the fact that the A2 term may be eliminated by a contact transformation, as discussed in detail in, PLRAAN 1050-2947 10.1103/PhysRevA.39.1139.
-
The physical irrelevance of the ponderomotive shift in a region of almost uniform field is consistent with the fact that the A2 term may be eliminated by a contact transformation, as discussed in detail in P. W. Milonni and J. R. Ackerhalt, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.39.1139 39, 1139 (1989).
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Milonni, P.W.1
Ackerhalt, J.R.2
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46
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45149084626
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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..
-
-
-
-
47
-
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45149083200
-
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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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48
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45149093613
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note
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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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-
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49
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45149083199
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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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50
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45149121302
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-
note
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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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-
-
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52
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45149098380
-
-
note
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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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56
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45149122692
-
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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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57
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45149084877
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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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58
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45149124725
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note
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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.
-
-
-
-
59
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45149126300
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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..
-
-
-
-
60
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45149131293
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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
-
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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61
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0000922514
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0008-4204;, Phys. Lett. A PYLAAG 0375-9601 10.1016/0375-9601(88)90051-5 126, 524 (1988).
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F. Ehlotzky, Can. J. Phys. 63, 907 (1985) 0008-4204; F. Ehlotzky, Phys. Lett. A PYLAAG 0375-9601 10.1016/0375-9601(88)90051-5 126, 524 (1988).
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Ehlotzky, F.1
Ehlotzky, F.2
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63
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0003462207
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+ eigenvalue problem is separable in confocal elliptic coordinates, a feature related to the conservation of a quantity that is a generalization of the one-center Runge-Lenz vector, as shown in, (Academic Press, New York), Cha. Once the separation is carried out, the problem becomes exactly solvable, but we preferred to follow a variational scheme instead of a direct numerical solution. The advantage is that our variational treatment allows a better understanding of the wave function dichotomy with only a small loss in accuracy.
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+ eigenvalue problem is separable in confocal elliptic coordinates, a feature related to the conservation of a quantity that is a generalization of the one-center Runge-Lenz vector, as shown in B. R. Judd, Angular Momentum Theory for Diatomic Molecules (Academic Press, New York, 1975), Chap.. Once the separation is carried out, the problem becomes exactly solvable, but we preferred to follow a variational scheme instead of a direct numerical solution. The advantage is that our variational treatment allows a better understanding of the wave function dichotomy with only a small loss in accuracy.
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(1975)
Angular Momentum Theory for Diatomic Molecules
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Judd, B.R.1
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64
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45149118166
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The reader should remember that Z= 1/2 for each "nucleus."
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The reader should remember that Z= 1/2 for each "nucleus."
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65
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0004086648
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2nd ed. (McGraw-Hill, New York)
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J. C. Slater, Quantum Theory of Matter, 2nd ed. (McGraw-Hill, New York, 1968), pp. 368-400.
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(1968)
Quantum Theory of Matter
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Slater, J.C.1
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66
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45149114950
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+ 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)
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+ 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).
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67
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0003460779
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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.
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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.
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(1963)
Quantum Theory of Molecules and Solids, Physics of Atoms and Molecules
, pp. 490-495
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Slater, J.C.1
Bransden, B.H.2
Joachain, C.J.3
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68
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45149119013
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note
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+ 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.
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69
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0037424149
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PRLTAO 0031-9007 10.1103/PhysRevLett.90.094801.
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M. Abo-Bakr, J. Feikes, K. Holldack, P. Kuske, W. B. Peatman, U. Schade, and G. Wustefeld, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.90. 094801 90, 094801 (2003).
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, pp. 094801
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Abo-Bakr, M.1
Feikes, J.2
Holldack, K.3
Kuske, P.4
Peatman, W.B.5
Schade, U.6
Wustefeld, G.7
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70
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45149083781
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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
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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.
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71
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0020722092
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SSCOA4 0038-1098 10.1016/0038-1098(83)90799-8.
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L. C. M. Miranda, Solid State Commun. SSCOA4 0038-1098 10.1016/0038-1098(83)90799-8 45, 783 (1983).
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(1983)
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, pp. 783
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Miranda, L.C.M.1
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72
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33746563396
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PRBMDO 0163-1829 10.1103/PhysRevB.74.045216.
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Y. Mizumoto, Y. Kayanuma, A. Srivastava, J. Kono, and A. H. Chin, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.74.045216 74, 045216 (2006).
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(2006)
Phys. Rev. B
, vol.74
, pp. 045216
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Mizumoto, Y.1
Kayanuma, Y.2
Srivastava, A.3
Kono, J.4
Chin, A.H.5
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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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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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