SUMMARY AND OUTLOOK
Summary and outlook
In this work two problems have been addressed. We first compared two different approaches for the calculation of optical properties of disordered semiconductors. It is shown that the equation of motion approach ink-space, formulated for the config-urationally averaged linear optical polarization and including the second-Born and Markov approximations, leads to results that partly disagree with those obtained for the same model from a tight-binding real-space approach. It should be noted in passing that R. Zimmermann pointed out [29] that instead of the second Born approximation one should rather use the Coherent Potential Approximation (CPA) which does not lead to unphysical spectra. However, for the time being it is not clear to us how to implement this approximation in calculations of nonlinear optical properties. For the rest of this work we therefore apply the tight-binding real-space approach with subsequent configurational averaging, although it is much more de-manding numerically if compared to thek-space approach.
The second problem focused on disorder-induced features of linear and nonlin-ear optical spectra. The influence of the length scale of disorder on shift and width of linear spectra of excitons was studied for the case of weak disorder. The results are interpreted in terms of the averaging of the disorder potential due to the relative motion of electrons and holes. This averaging mechanism is well known, however, its action for different length scales has not been studied in sufficient detail so far.
We find that for length scale smaller than the Bohr radius there is a universal be-havior, while for larger length scale the Coulomb interaction does no longer play a direct role.
We then focused on nonlinear 2DFT spectra. It was argued that in a disordered semiconductor various couplings (in the sense of Fano-couplings) become possible which are absent in the ordered counterpart. Modeling these couplings by a simple level model we showed in a tutorial manner how one can identify these couplings in 2DFT spectra. The calculation for the semiconductor model indeed showed fea-tures that closely resemble those found for the level system. It is therefore shown that in a disordered semiconductor Fano-like couplings are existing. They lead to disorder-induced dephasing and an interpretation of the excitonic line in terms of an energy dependent dephasing rate. It is also shown that two-exciton states ob-scure the spectra. It is therefore important to use such excitation conditions that do not yield large bi- or two-exciton contributions to the nonlinear response if disorder effects on the exciton are to be studied. We suggest to apply co-circular excitation for studies on disordered semiconductor structures.
Further work should be done in this field both experimentally and theoretically.
On the experimental side we would like to see 2DFT spectra for quantum wells that are characterized by disorder with different length scale. Preparation tech-niques exist that produce such samples. It would also be interesting to study the effect of disorder on bi- and two-exciton transitions in the 2DFTS scheme. There is some older work on Four-Wave-Mixing [40] which showed disorder effects of biex-citons. However, with 2DFTS we would expect much more information, in particu-lar, for weak disorder. From previous work [15] we learned that in this case FWM in the time domain is more problematic to interpret.
In parallel, on the theoretical side, one should study in detail disorder broaden-ing for situations where there are both heavy-hole and light-hole excitons which, depending on polarization of the excitation pulses, do or do not couple, at least within the Hartree-Fock limit. This coupling, however, is present in the ordered case due to correlations [20, 37], even if it is absent in the HF-limit. Disorder adds to the coupling scenario. Work is under way [39] that will certainly give interesting insights into these more complex structures. Experimental data exist that seem to show that both homogeneous and inhomogeneous broadenings of the two different excitons differ appreciably. An explanation for this finding does not yet exist.
Disorder effects on biexcitons would be another interesting problem to study.
In particular, we then have yet another length scale given by the biexcitonc spatial extension. For these studies one should consider linear cross-polarized pulses.
ZUSAMMENFASSUNG
Zusammenfassung
Diese Dissertation behandelt zwei Themenkomplexe. Zunächst werden zwei un-terschiedliche theoretische Methoden zur Berechnung der optischen linearen Polar-isation verglichen. Die erste Methode basiert auf Bewegungsgleichungen der kon-figurationsgemittelten optischen Polarisation. Hier entsteht neben der bekannten Vielteilchen-Hierarchie eine zweite Hierarchie von Gleichungen für unordnungsas-sistierte Polarisationen immer höheren Grades. Während die Vielteilchenhierar-chie bei der Beschreibung kohärent getriebener Halbleiter exakt behandelt werden kann, ist dies für die unordnungsassistierten Polarisationen nicht möglich. Eine zweite Born’sche Näherung führt bekanntlich zu unphysikalischen Spektren. Erst eine zusätzliche Markov-Näherung bringt hier Abhilfe. Diese unbefriedigende Sit-uation kann nur geklärt werden, wenn man eine zweite Methode auf das gleiche Modell anwendet, die frei von Näherungen ist. Dies ist das Tight-Binding Modell im Ortsraum mit nachfolgender Konfigurationsmittelung, das in der Vergangenheit vielfach mit Erfolg angewendet wurde. Allerdings stellt es unvergleichlich höhere Anforderungen an die Rechnerkapazität als die zuvor genannte Methode. Es zeigt sich, dass die Resultate beider Methoden zwar grob übereinstimmen, in Details aber Abweichungen existieren. Weiterhin wird daher mit dem Tight-Binding Modell gearbeitet.
Der zweite Themenkomplex befasst sich mit der Interpretation unordnungsin-duzierter spektraler Eigenschaften von exzitonischen Resonanzen im sowohl lin-earen als auch nichtlinlin-earen Regime. Im Detail wird die Verschiebung und Breite linearer exzitonischer Spektren als Funktion der Längenskala schwacher Unord-nung studiert. Die Interpretation macht von dem schon seit längerem bekannten Mechanismus der durch die Relativbewegung verursachten Mittelung Gebrauch.
Es zeigt sich, dass die Wirksamkeit dieser Mittelung entscheidend von der Län-genskala der Unordnung abhängt, wobei der Bohr’sche Radius eine wichtige Rolle spielt. Für kleinere Längenskala hat man ein universelles Verhalten, nicht jedoch für größere.
Schließlich werden nichtlineare Spektren betrachtet, wie sie im Rahmen der Meth-ode der "Two-Dimensional Fourier-Transform Spectroscopy" (2DFTS) erhalten wer-den. Da diese Methode eine Variante des Vierwellenmischens darstellt, ist sie in der Lage, Information über homogene und inhomogene Breite getrennt zu liefern, im Gegensatz zur linearen Antwort. Da die Unordnung die Impulserhaltung der Schw-erpunktsbewegung des Exzitons verletzt, kommt es zu zusätzlichen Kopplungen.
Gerade die 2DFTS-Methode zeigt Kopplungen in sehr klarer Weise, wenn man erst einmal gelernt hat, wie man die Spektren zu lesen hat. Um dies zu erreichen, wird in einer Serie von einfachen Niveau-Systemen zunehmender Komplexität das 2DFT
Spektrum jeweils berechnet und dargestellt, bis man schließlich die Fano-Situation erreicht hat. Es zeigt sich, dass die aus dem Halbleitermodell berechneten Spektren solche spektralen Merkmale aufweisen, wie sie für Fano-Systeme gefunden wur-den. Die Auswertung dieser Spektren liefert dann für den ungeordneten Fall eine homogene Breite, die um ca. 20 % größer ist als die für den geordneten Fall. Dies zeigt den Effekt des "unordnungs-induzierten Dephasierens", der schon früher für ein nichtwechselwirkendes System mit großer Unordnung vorhergesagt wurde. Es zeigt sich auch, dass man die Spektren des ungeordneten Halbleiters als inhomo-gene Linie interpretieren kann, deren einzelne homoinhomo-gene Linien durch eine mit der Energie zunehmende Dephasierungsrate gekennzeichnet sind. Auch dieser Effekt basiert auf einem Fano-Mechanismus.
Die Erkenntnisse dieser Arbeit sollen in naher Zukunft auf experimentelle Re-sultate angewendet werden, die mit der Methodes der 2DFTS erhalten wurden.
APPENDIX
Appendix
Appendix A
Abbreviation
com center of mass
FWHM full width at half maximum FWM four-wave mixing
HF Hartree-Fock hh heavy hole PB Pauli blocking
2DFTS two-dimensional Fourier-transform spectroscopy
Appendix B
Coulomb matrix element
The Coulomb matrix elementVq in Eq. (13), which is used to calculate the lin-ear absorption in the k-space approach, is given by the Fourier-transform of the Coulomb-interaction matrix elementV(z)which for our one-dimensional model is given by [25]
V(z) = V0
q
z2 + 4Rπ 2 (73)
as
Vq = 2 LK0
|q|4R π
(74) with the normalizing length L and regularization length R=8.25 nm. K0(x) is the modified Bessel function.
In the real space tight-binding model the Coulomb interaction matrix elementVij is accordingly given as
Vij = V0
q
(i−j)2a2+ 4Rπ 2. (75) The regularized Coulomb matrix element, which is used in the 2DFT calcula-tions, is given by [41]
Vij = V0
|i−j|a−a0
, (76)
a is the site separation and a0 = 0.5a is the regularization length. V0 is a positive constant, which is chosen such that the binding energy has a value close to a real semiconductor system.
APPENDIX
Appendix C
Cluster expansion
Many-body correlations of order N consist of N fermionic creation operators and N annihilation operators. To truncate the hierarchy problem, one has to find a consistent way of approximating hONi, which describes an N-particle operator [24, 27]:
hO2i = hO2iS+ ∆hO2i (77)
hO3i = hO3iS+hO1i∆hO2i+ ∆hO3i (78) ...
hONi = hONiS+hON−2iS∆hO2i+hON−4i∆hO2i∆hO2i+. . .+ ∆hONi. (79) Each term denotes a sum over all possibilities to reorganize the N coordinates among singlets, doublets, etc. The lowest-level truncation scheme resulting from the cluster expansion is the singlet approximation which is identical to the Hartree-Fock approximation. The Hartree-Hartree-Fock approximation, in which the full density matrix is represented by singlets, represents only the first step of the general cluster expansion. The next step to improve the level of approximation is to include also doublets. In this case, the full density matrix can describe uncorrelated carriers and an arbitrary amount of correlated pairs. The quantities∆hONicontain only the purely correlated part of theN-particle cluster. The advantage of the factorization is its direct physical interpretation.
Appendix D
Markov approximation
Here we follow the work of Hoyer [25]. We consider a general equation, i¯h∂
∂tX(t) = (EX −iγ)X(t) +A(t). (80) Its formal solution is given by
X(t) = −i
¯ h
Z t
∞
Aj(t′)e−¯hi(EX−iγ)(t−t′)dt′. (81) In order to separate the strongly and weakly time dependent terms under the inte-gral, we assume thatA(t)itself is mainly oscillating with a frequencyEA/¯hand can be written as
A(t) =A0(t)e−¯hiEAt, (82) whereA0(t)is slowly varying in time. Then we can take the slowly varying part out of the integral and integrate, obtaining:
X(t)≈ A(t)
EA−EX +iγ. (83)
More generally:
i¯h∂
∂tX(t) = (EX −iγ)X(t) +X
j
Aj(t) ⇒ X(t)≈X
j
Aj(t)
EAj−EX +iγ. (84)
APPENDIX
Appendix E
Equations
Here we present the derivation of various equations of motion in the linear re-sponse applying the rotating-wave approximation and the low-density limit.
k-space equations
With second-Born, without Markov approximation, correlated disorder:
i¯h∂
∂thpkkiE = (εck−εvk+X
q6=k
Vk−q− i¯h
T2)hpkkiE −E(t)d∗vc−X
q6=k
Vk−qhpqqiE +X
q
(hUk−qc pkqiE − hUq−kv pqkiE)
(85)
i¯h∂
∂thUk−lc pkliE = (εcl −εvk+X
q6=k
Vk−q− i¯h T2
)hUk−lc pkliE −X
q6=k
Vk−qhUk−lc pq,l−k+qiE
+hUk−lc Ul−kc iEhpkkiE−hUck−lUvl−kiEhplliE
(86)
i¯h∂
∂thUk−lv pkliE = (εcl −εvk+X
q6=k
Vk−q− i¯h T2
)hUk−lv pkliE −X
q6=k
Vk−qhUk−lv pq,l−k+qiE +hUvk−lUcl−kiEhpkkiE− hUk−lv Ul−kv iEhplliE
(87)
With Markov approximation, correlated disorder:
i¯h∂
∂thpkkiE = εck−εvk+X
q6=k
Vk−q− i¯h T2
!
hpkkiE −E(t)d∗vc−X
q6=k
Vk−qhpqqiE
+X
q,q′
hUk−qc Uq−qc ′iE hpkq′iE
εcq′ −εcq+Ti¯h2−hUck−qUvq′−kiE hpq′qiE εvk−εvq′ + Ti¯h
2
−hUvq−kUck−q′iE hpqq′iE εcq′ −εck+Ti¯h
2
+hUq−kv Uqv′−qiE hpq′kiE εvq −εvq′ +Ti¯h2
!
(88) For anticorrelated disorder bold-faced terms have opposite signs. For uncorrelated disorder they vanish.
exciton basis, all excitonic states
By transforming the equations into an exciton basis we obtain the following equations.
With second-Born, without Markov approximation:
i¯h∂
∂thXν0iE =EνhXν0iE −E(t)d∗vcX
˜k
φ∗ν(˜k) +X
ν′
X
q1
hXν′,−q1Uqc1iEX
k˜
φ∗ν(˜k)φν′(˜k−q1h)
− hXν′,−q1Uqv1iEX
˜k
φ∗ν(˜k)φν′(˜k+q1e)
(89)
i¯h∂
∂thXνqU−qc iE =
Eν+ ¯h2q2 2M
hXνqU−qc iE +X
ν′
X
q1
hXν′,q−q1iE
hU−qc Uqc1iE
X
˜k
φ∗ν(˜k)φν′(˜k−q1h)
−hUc−qUvq
1iEX
˜k
φ∗ν(k)φ˜ ν′(˜k+qe1)
(90)
i¯h∂
∂thXνqU−qv iE =
Eν +¯h2q2 2M
hXνqU−qv iE
+X
ν′
X
q1
hXν′,q−q1iE
hUv−qUcq
1iEX
˜k
φ∗ν(˜k)φν′(k˜−qh1)
− hU−qv Uqv1iE
X
˜k
φ∗ν(˜k)φν′(˜k+q1e)
(91)
With Markov approximation:
i¯h∂
∂thXν0iE =EνhXν0iE−E(t)d∗vcX
˜k
φ∗ν(˜k)
+X
ν′
X
q1
X
ν′′
X
q2
hXν′′,−q1−q2iE
Eν′′−Eν′ +¯h2M2q22 +¯h22M2q1q2 +iγ·
hUqc1Uqc2iE
X
k˜
φ∗ν′(˜k)φν′′(˜k−q2h)X
˜k
φ∗ν(˜k)φν′(˜k−q1h)
−hUcq
1Uvq
2iEX
˜k
φ∗ν′(˜k)φν′′(˜k+qe2)X
k˜
φ∗ν(k)φ˜ ν′(˜k−qh1)
−hUvq1Ucq2iEX
˜k
φ∗ν′(˜k)φν′′(˜k−qh2)X
k˜
φ∗ν(k)φ˜ ν′(˜k+qe1)
+hUqv1Uqv2iEX
˜k
φ∗ν′(˜k)φν′′(˜k+q2e)X
˜k
φ∗ν(˜k)φν′(˜k+qe1) !
(92)
APPENDIX
If we introduce the notation:
Uex(ν, ν′, q1) = Uqc1
X
˜k
φ∗ν(˜k)φν′(˜k−q1h)−Uqv1
X
˜k
φ∗ν(˜k)φν′(˜k+q1e), (93) we cannot choose the anticorrelated and uncorrelated cases.
In second-Born approximation:
i¯h∂
∂thXν0iE =EνhXν0iE−E(t)d∗vcX
˜k
φ∗ν(˜k) +X
ν′
X
q1
hXν′,−q1Uex(ν, ν′, q1)iE (94)
i¯h∂
∂thXν′,−q1Uex(ν, ν′, q1)iE =
Eν′ +¯h2q12 2M
hXν′,−q1Uex(ν, ν′, q1)iE +X
ν′′
X
q2
hXν′′,−q1−q2iEhUex(ν, ν′, q1)Uex(ν′, ν′′, q2)iE (95)
In second-Born - Markov approximation:
i¯h∂
∂thXν0iE =EνhXν0iE −E(t)d∗vcX
k˜
φ∗ν(˜k) +X
ν′
X
q1
X
ν′′
X
q2
hXν′′,−q1−q2iEhUex(ν, ν′, q1)Uex(ν′, ν′′, q2)iE Eν′′−Eν′ +¯h2M2q22 + ¯h22M2q1q2 +iγ
(96)
1s exciton equations
With second-Born, without Markov approximation:
i¯h ∂
∂thX0iE =E0hX0iE −E(t)d∗vcX
˜k
φ∗(˜k) +X
q1
hX−q1Uqc1iE
X
k˜
φ∗(˜k)φ(k˜−q1h)− hX−q1Uqv1iE
X
k˜
φ∗(˜k)φ(˜k+q1e)
(97)
i¯h∂
∂thXqU−qc iE =
E0+¯h2q2 2M
hXqU−qc iE +X
q1
hXq−q1iE
hU−qc Uqc1iEX
˜k
φ∗(˜k)φ(˜k−q1h)
−hUc−qUvq
1iEX
˜k
φ∗(k)φ(˜ ˜k+qe1)
(98)
i¯h∂
∂thXqU−qv iE =
E0 +¯h2q2 2M
hXqU−qv iE +X
q1
hXq−q1iE
hUv−qUcq1iEX
˜k
φ∗(k)φ˜ (˜k−qh1)
− hU−qv Uqv1iEX
˜k
φ∗(˜k)φ(˜k+qe1)
(99)
With Markov approximation:
i¯h∂
∂thX0iE =E0hX0iE−E(t)d∗vcX
˜k
φ∗(˜k)
+X
q1
X
q2
hX−q1−q2iE
¯ h2q22
2M +¯h22M2q1q2 +iγ·
hUqc1Uqc2iEX
˜k
φ∗(˜k)φ(˜k−q2h)X
˜k
φ∗(˜k)φ(˜k−q1h)
−hUcq1Uvq2iEX
k˜
φ∗(˜k)φ(k˜+qe2)X
˜k
φ∗(k)φ(˜ ˜k−qh1)
−hUvq1Ucq2iEX
k˜
φ∗(˜k)φ(k˜−qh2)X
˜k
φ∗(k)φ(˜ k˜+qe1)
+hUqv1Uqv2iEX
k˜
φ∗(˜k)φ(˜k+q2e)X
k˜
φ∗(˜k)φ(˜k+q1e) !
(100)
With the notation:
Uex(q1) =Uqc1
X
k
φ∗(k)φ(k−q1h)−Uqv1
X
k
φ∗(k)φ(k+q1e) (101)
Only second-Born approximation:
i¯h∂
∂thX0iE =
E0− i¯h T2
hX0iE +X
q1
hX−q1Uex(q1)iE−E(t)d∗vcX
˜k
φ∗(˜k) (102)
i¯h∂
∂thX−q1Uex(q1)iE =
E0+¯h2q12 2M − i¯h
T2
hX−q1Uex(q1)iE (103) +X
q2
hX−q1−q2iEhUex(q1)Uex(q2)iE (104)
With Markov approximation:
i¯h∂
∂thX0iE =
E0− i¯h T2
hX0iE+X
q1
X
q2
hX0iEhUex(q1)Uex(−q2)ie
¯ h2q21
2M + ¯h22Mq1q2 +Ti¯h
2
−E(t)d∗vcX
˜k
φ∗(˜k) (105)
APPENDIX
If we use thathUex(q1)Uex(q2)iE ∝δq1,−q2: With second-Born approximation:
i¯h∂
∂thX0iE =
E0− i¯h T2
hX0iE+X
q1
hX−q1Uex(q1)iE −E(t)d∗vcX
k˜
φ∗(˜k) (106)
i¯h∂
∂thX−q1Uex(q1)iE =
E0 +¯h2q21 2M − i¯h
T2
hX−q1Uex(q1)iE+hX0iEhUex(q1)Uex(−q1)iE (107) Second-Born - Markov approximation:
i¯h∂
∂thX0iE =
E0 − i¯h T2
+X
q1
hUex(q1)Uex(−q1)iE
−¯h
2q21
2M +Ti¯h2
hX0iE −E(t)d∗vcX
˜(k)
φ∗(˜(k)) (108)
Appendix F
Mathematical explanation of the double-peaked spectrum with sec-ond Born approximation
Here we follow Zimmermann [29]. The appearance of the two-peak structure of the spectrum, which is certainly unphysical, can most easily be understood by considering the lowest-energy exciton.
i¯h∂
∂thX0iE = (E0−iγ)hX0iE +X
q
hX−qUex(q)iE −E(t)d∗vcX
˜k
φ∗(˜k) (109)
i¯h∂
∂thX−qUex(q)iE =
E0+¯h2q2 2M − i¯h
T2
hX−qUex(q)iE +hX0iEhUex(q)Uex(−q)iE (110) The formal solution for the exciton amplitude can be obtained by Fourier trans-formation as
¯
hωhX0(ω)iE = (E0−iγ)hX0(ω)iE+X
q
hUexc(q)X−q(ω)iE−E(ω)d∗vcX
˜k
φ∗(˜k) (111) where the disorder-assisted amplitude follows from
¯
hωhUexc(q)X−q(ω)iE = (E0+¯h2q2
2M −iγ)hUexc(q)X−q(ω)iE+hUexc(q)Uexc(−q)iEhX0(ω)iE (112) thus
hUexc(q)X−q(ω)iE = hUexc(q)Uexc(−q)iEhX0(ω)iE
¯
hω−E0 −¯h2M2q2 +iγ . (113) Finally
¯
hω−E0− ¯h2q2
2M +iγ+σ
hX0(ω)iE =E(ω)d∗vcX
˜k
φ∗(˜k), (114) whereσis the self energy
σ=−X
q
hUexc(q)Uexc(−q)iE
¯
hω−E0− ¯h2M2q2 +iγ =−X
q
P V 1
¯
hω−E0 −¯h2M2q2
!
hUexc(q)Uexc(−q)iE
+iπX
q
δ
¯
hω−E0−¯h2q2 2M
hUexc(q)Uexc(−q)iE (115)
APPENDIX
where PV stands for principal value determining the real partσ′.
This expression for the self energy shows that it has an imaginary partσ′′only for photon energies¯hωthat are above the lowest excitonic resonance atE0in the ordered system. Thus, for energies above E0 we expect approximately a half-Lorentzian.
For energies belowE0 a contribution to the spectrum solely arises from the zero of
¯
hω−Eq−σ′, which leads to a broadenedδ-like peak there.
−5 −4 −3 −2 −1
Energy [meV]
Absorption [arb.u.]
Figure 32: Calculated spectra ink-space with second-Born approximation (shaded area) and the theoretical result from the analytical solution
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Acknowledgement
I would like to thank everyone who has contributed to the completion of this thesis in one way or another.
I am deeply indebted to my supervisor Prof. Peter Thomas for his support in helping me accomplish this work. His enduring patience and infinite understanding permitted to work in my own manner. His continuous encouragement helped me to swing over the low points during my work. Thank you for the anecdotes and the non-scientific activities which made the days colorful.
I thank Prof. Peter Thomas and Prof. Florian Gebhard for the opportunity to work and study in the European Graduate College "Electron-electron interactions in solids". In these years the different scientific activities (workshops, seminars) improved my skills which are required in the academic life.
I acknowledge the financial support by the European Graduate Collage and Prof.
Stephan Koch.
I want to thank my colleagues for the pleasant working atmosphere and their useful advices. A special thank to Ada, Christina and Hanno for helping me when-ever I had problems of any kind.
I would like to say thanks to some former colleagues, Peti, Irina and Walter for the many explanations, instructive discussions and efficient joint work even after they left our group.
I also thank Prof. Torsten Meier for his valuable comments and the opportunity to use the supercomputer facilities in Jülich.
I am glad to say thanks to "my Marburg family": Andi, Ági, Igor and Kornél.
You always made sure that my life wasn’t boring for a minute. I think my stay in Marburg wouldn’t have been as eventful without all of you.
There is a big thanks towards my family (Mom, Dad, Norbi, Lili, Zsófi and Anett) for their constant support, and passionately believing in my capabilities. Thank you for accepting my decision to study in abroad. Although you live far away, you have been always there for me.
To everyone else that has been part of my wonderful Marburg-life and who I have failed to mention, thank you for the memories.
I wouldn’t have done it without you.
Thank you. Danke. Köszönöm.