Abstract

We study high-order harmonic generation (HHG) in armchair-type single-wall carbon nanotubes (SWNTs) driven by ultrashort, mid-infrared laser pulses. For a SWNT with chiral indices (n, n), we demonstrate that HHG is dominated by bands |m| = n − 1 and that the cut-off frequency saturates with intensity, as it occurs in the case of single layer graphene. As a consequence, HHG in SWNTs can be described effectively as a one-dimensional periodic system, whose high-frequency emission can be modified through the proper control of the structural parameters. Additionally, we show that the HHG mechanism in nanotubes has some similarities to that previously reported in single layer graphene. However, as a main difference, the electron-hole pair excitation in SWNTs is connected to the non-adiabatic crossing through the first van Hove singularity of the |m| = n − 1 bands, instead of the crossing through the Dirac point that takes place in graphene.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]

2020 (1)

S. de Vega, J. D. Cox, F. Sols, and F. J. García de Abajo, “Strong-field-driven dynamics and high-harmonic generation in interacting one dimensional systems,” Phys. Rev. Res. 2(1), 013313 (2020).
[Crossref]

2019 (2)

O. Zurrón-Cifuentes, R. Boyero-García, C. Hernández-García, A. Picón, and L. Plaja, “Optical anisotropy of non-perturbative high-order harmonic generation in gapless graphene,” Opt. Express 27(5), 7776–7786 (2019).
[Crossref]

A. Nayak, M. Dumergue, S. Kühn, S. Mondal, T. Csizmadia, N. G. Harshitha, M. Füle, M. U. Kahaly, B. Farkas, B. Major, V. Szaszkó-Bogár, P. Földi, S. Majorosi, N. Tsatrafyllis, E. Skantzakis, L. Neoričić, M. Shirozhan, G. Vampa, K. Varjú, P. Tzallas, G. Sansone, D. Charalambidis, and S. Kahaly, “Saddle point approaches in strong field physics and generation of attosecond pulses,” Phys. Rep. 833, 1–52 (2019).
[Crossref]

2018 (1)

O. Zurrón, A. Picón, and L. Plaja, “Theory of high-order harmonic generation for gapless graphene,” New J. Phys. 20(5), 053033 (2018).
[Crossref]

2017 (2)

G. Vampa and T. Brabec, “Merge of high harmonic generation from gases and solids and its implications for attosecond science,” J. Phys. B: At., Mol. Opt. Phys. 50(8), 083001 (2017).
[Crossref]

N. Yoshikawa, T. Tamaya, and K. Tanaka, “High-harmonic generation in graphene enhanced by elliptically polarized light excitation,” Science 356(6339), 736–738 (2017).
[Crossref]

2015 (1)

J. M. Iglesias, M. J. Martín, E. Pascual, and R. Rengel, “Carrier-carrier and carrier-phonon interactions in the dynamics of photoexcited electrons in graphene,” J. Phys.: Conf. Ser. 647, 012003 (2015).
[Crossref]

2014 (2)

H. K. Kelardeh, V. Apalkov, and M. I. Stockman, “Wannier-Stark states of graphene in strong electric field,” Phys. Rev. B 90(8), 085313 (2014).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical Analysis of High-Harmonic Generation in Solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref]

2013 (1)

D. Brida, A. Tomadin, C. Manzoni, Y. J. Kim, A. Lombardo, S. Milana, R. R. Nair, K. S. Novoselov, A. C. Ferrari, G. Cerullo, and M. Polini, “Ultrafast collinear scattering and carrier multiplication in graphene,” Nat. Commun. 4(1), 1987 (2013).
[Crossref]

2012 (1)

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref]

2011 (2)

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

A. Roberts, D. Cormode, C. Reynolds, T. Newhouse-Illige, B. J. LeRoy, and A. S. Sandhu, “Response of graphene to femtosecond high-intensity laser irradiation,” Appl. Phys. Lett. 99(5), 051912 (2011).
[Crossref]

2010 (1)

S. V. Goupalov, A. Zarifi, and T. G. Pedersen, “Calculation of optical matrix elements in carbon nanotubes,” Phys. Rev. B 81(15), 153402 (2010).
[Crossref]

2009 (1)

M. Breusing, C. Ropers, and T. Elsaesser, “Ultrafast Carrier Dynamics in Graphite,” Phys. Rev. Lett. 102(8), 086809 (2009).
[Crossref]

2008 (1)

D. Golde, T. Meier, and S. W. Koch, “High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations,” Phys. Rev. B 77(7), 075330 (2008).
[Crossref]

2005 (1)

S. V. Goupalov, “Optical transitions in carbon nanotubes,” Phys. Rev. B 72(19), 195403 (2005).
[Crossref]

1997 (1)

F. H. M. Faisal and J. Z. Kamiński, “Floquet theory of high-harmonic generation in periodic structures,” Phys. Rev. A 56(1), 748–762 (1997).
[Crossref]

1994 (2)

K. A. Pronin, A. D. Bandrauk, and A. A. Ovchinnikov, “Harmonic generation by a one-dimensional conductor: Exact results,” Phys. Rev. B 50(5), 3473–3476 (1994).
[Crossref]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref]

1993 (4)

S. Iijima and T. Ichihashi, “Single-shell carbon nanotubes of 1 nm diameter,” Nature 363(6430), 603–605 (1993).
[Crossref]

D. S. Bethune, C. H. Klang, M. S. de Vries, G. Gorman, R. Savoy, J. Vazquez, and R. Beyers, “Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls,” Nature 363(6430), 605–607 (1993).
[Crossref]

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[Crossref]

K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett. 70(11), 1599–1602 (1993).
[Crossref]

1992 (4)

J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68(24), 3535–3538 (1992).
[Crossref]

L. Plaja and L. Roso, “High-order harmonic generation in a crystalline solid,” Phys. Rev. B 45(15), 8334–8341 (1992).
[Crossref]

N. Hamada, S. I. Sawada, and A. Oshiyama, “New one-dimensional conductors: Graphitic microtubules,” Phys. Rev. Lett. 68(10), 1579–1581 (1992).
[Crossref]

R. Saito, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, “Electronic structure of chiral graphene tubules,” Appl. Phys. Lett. 60(18), 2204–2206 (1992).
[Crossref]

1991 (1)

S. Iijima, “Helical microtubules of graphitic carbon,” Nature 354(6348), 56–58 (1991).
[Crossref]

1985 (1)

N. Bozović, I. Bozović, and M. Damnjanović, “Selection rules for polymers and quasi-one-dimensional crystals: IV. Kronecker products for the line groups isogonal to Dnh,” J. Phys. A: Math. Gen. 18(6), 923–937 (1985).
[Crossref]

Agostini, P.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Alisauskas, S.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref]

Andriukaitis, G.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref]

Apalkov, V.

H. K. Kelardeh, V. Apalkov, and M. I. Stockman, “Wannier-Stark states of graphene in strong electric field,” Phys. Rev. B 90(8), 085313 (2014).
[Crossref]

Arpin, P.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref]

Balciunas, T.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref]

Balcou, P.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref]

Baltuska, A.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref]

Bandrauk, A. D.

K. A. Pronin, A. D. Bandrauk, and A. A. Ovchinnikov, “Harmonic generation by a one-dimensional conductor: Exact results,” Phys. Rev. B 50(5), 3473–3476 (1994).
[Crossref]

Becker, A.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref]

Bethune, D. S.

D. S. Bethune, C. H. Klang, M. S. de Vries, G. Gorman, R. Savoy, J. Vazquez, and R. Beyers, “Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls,” Nature 363(6430), 605–607 (1993).
[Crossref]

Beyers, R.

D. S. Bethune, C. H. Klang, M. S. de Vries, G. Gorman, R. Savoy, J. Vazquez, and R. Beyers, “Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls,” Nature 363(6430), 605–607 (1993).
[Crossref]

Boyero-García, R.

Bozovic, I.

N. Bozović, I. Bozović, and M. Damnjanović, “Selection rules for polymers and quasi-one-dimensional crystals: IV. Kronecker products for the line groups isogonal to Dnh,” J. Phys. A: Math. Gen. 18(6), 923–937 (1985).
[Crossref]

Bozovic, N.

N. Bozović, I. Bozović, and M. Damnjanović, “Selection rules for polymers and quasi-one-dimensional crystals: IV. Kronecker products for the line groups isogonal to Dnh,” J. Phys. A: Math. Gen. 18(6), 923–937 (1985).
[Crossref]

Brabec, T.

G. Vampa and T. Brabec, “Merge of high harmonic generation from gases and solids and its implications for attosecond science,” J. Phys. B: At., Mol. Opt. Phys. 50(8), 083001 (2017).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical Analysis of High-Harmonic Generation in Solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref]

Breusing, M.

M. Breusing, C. Ropers, and T. Elsaesser, “Ultrafast Carrier Dynamics in Graphite,” Phys. Rev. Lett. 102(8), 086809 (2009).
[Crossref]

Brida, D.

D. Brida, A. Tomadin, C. Manzoni, Y. J. Kim, A. Lombardo, S. Milana, R. R. Nair, K. S. Novoselov, A. C. Ferrari, G. Cerullo, and M. Polini, “Ultrafast collinear scattering and carrier multiplication in graphene,” Nat. Commun. 4(1), 1987 (2013).
[Crossref]

Brown, S.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref]

Cerullo, G.

D. Brida, A. Tomadin, C. Manzoni, Y. J. Kim, A. Lombardo, S. Milana, R. R. Nair, K. S. Novoselov, A. C. Ferrari, G. Cerullo, and M. Polini, “Ultrafast collinear scattering and carrier multiplication in graphene,” Nat. Commun. 4(1), 1987 (2013).
[Crossref]

Charalambidis, D.

A. Nayak, M. Dumergue, S. Kühn, S. Mondal, T. Csizmadia, N. G. Harshitha, M. Füle, M. U. Kahaly, B. Farkas, B. Major, V. Szaszkó-Bogár, P. Földi, S. Majorosi, N. Tsatrafyllis, E. Skantzakis, L. Neoričić, M. Shirozhan, G. Vampa, K. Varjú, P. Tzallas, G. Sansone, D. Charalambidis, and S. Kahaly, “Saddle point approaches in strong field physics and generation of attosecond pulses,” Phys. Rep. 833, 1–52 (2019).
[Crossref]

Charlier, J. C.

L. E. F. Foa-Torres, S. Roche, and J. C. Charlier, Introduction to graphene-based nanomaterials: from electronic structure to quantum transport (Cambridge University, 2014).

Chen, M. C.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref]

Corkum, P. B.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical Analysis of High-Harmonic Generation in Solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref]

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[Crossref]

Cormode, D.

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A. Nayak, M. Dumergue, S. Kühn, S. Mondal, T. Csizmadia, N. G. Harshitha, M. Füle, M. U. Kahaly, B. Farkas, B. Major, V. Szaszkó-Bogár, P. Földi, S. Majorosi, N. Tsatrafyllis, E. Skantzakis, L. Neoričić, M. Shirozhan, G. Vampa, K. Varjú, P. Tzallas, G. Sansone, D. Charalambidis, and S. Kahaly, “Saddle point approaches in strong field physics and generation of attosecond pulses,” Phys. Rep. 833, 1–52 (2019).
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S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
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Phys. Rep. (1)

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Phys. Rev. Res. (1)

S. de Vega, J. D. Cox, F. Sols, and F. J. García de Abajo, “Strong-field-driven dynamics and high-harmonic generation in interacting one dimensional systems,” Phys. Rev. Res. 2(1), 013313 (2020).
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Science (2)

N. Yoshikawa, T. Tamaya, and K. Tanaka, “High-harmonic generation in graphene enhanced by elliptically polarized light excitation,” Science 356(6339), 736–738 (2017).
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T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref]

Other (4)

H. Nishidome, K. Nagai, K. Uchida, Y. Ichinose, Y. Yomogida, K. Tanaka, and K. Yanagi, “Control of high-harmonic generation by tuning the electronic structure and carrier injection,” arXiv:2004.11000v1 [physics.optics] (2020).

L. Plaja, R. Torres, and A. Zaïr, Attosecond Physics. Attosecond Measurements and Control of Physical Systems (Springer-Verlag, 2013).

S. Reich, C. Thomsen, and J. Maultzsch, Carbon nanotubes: basic concepts and physical properties (WILEY-CVH Verlag GmbH & Co. KGaA, 2004).

L. E. F. Foa-Torres, S. Roche, and J. C. Charlier, Introduction to graphene-based nanomaterials: from electronic structure to quantum transport (Cambridge University, 2014).

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Figures (8)

Fig. 1.
Fig. 1. (a) View of the structure and symmetries of a (9,9) armchair nanotutube. Magenta and green circles represent the atoms in the two sublattices of the unrolled graphene sheet. (b) Allowed k vectors in the first Brillouin zone of (9,9) armchair tube (green lines). This nanotube has 36 C-atoms in its unit cell, $q=18$ , $m=0,\pm 1,\pm 2,\ldots ,\pm 8, 9$ and $d_t=12.2$ Å. The red hexagon shows the boundary of the graphene’s BZ along with the high symmetry points K and $\Gamma$ .
Fig. 2.
Fig. 2. Band structure of (9,9) armchair. The energy is given in units of the frequency of a 3 $\mu$ m driving field ( $\hbar \omega _0=0.41$ eV). The horizontal axis represents the wave vector times the translational period $a_0$ . The solid (dashed) black lines indicate the $m=0$ ( $m=9$ ) singlets. The rest of the bands represent double degenerated states $E_{|m|}^\pm$ , with $E=A/B$ . Parities under $\sigma _h$ reflections at $a_0k=0$ and $\pi$ are also indicated. The pink-filled profile in the background shows the density of states, dominated by the van Hove singularities.
Fig. 3.
Fig. 3. The solid line represents the total harmonic yield from (9,9) $\mathcal {A}$ -tube driven by a $3\mu$ m wavelength, 28 fs (2.9 cycles) FWHM pulse at $5\times 10^{10}$ W/cm $^2$ peak intensity. The dotted line indicates the intraband spectral component.
Fig. 4.
Fig. 4. (a) Matrix element $|D_m(k)|$ corresponding to the (9,9) armchair tube for different $m$ values. The dotted line at $a_0k=2\pi /3$ indicates the band’s crossing point for $m=9$ . (b) Contribution to the harmonic yield from the different values of $m$ . The driver’s parameters are the same as in Fig.  3.
Fig. 5.
Fig. 5. Harmonic yield from (9,9), (10,10), (11,11) and (12,12) armchair tubes. The diameters of these tubes are $d_t=1.22, 1.36 ,1.50$ and 1.63 nm, respectively. The electric field parameters are those in Figs.  3 and 4(b).
Fig. 6.
Fig. 6. (a) Harmonic yield from (9,9) $\mathcal {A}$ -tube driven by a $3\mu$ m wavelength, 28 fs (2.9 cycles) FWHM pulse at $5\times 10^{12}$ W/cm $^2$ peak intensity. (b) Cut-off scaling with intensity from (9,9) armchair. The blue diamonds are the result of the numerical integration of the TDSE. The red circles are predicted by the semiclassical SPAM considering electron-hole creation at the first van Hove singularity: $a_0k=2\pi /3$ , $m=8$ . The filled area in the background corresponds to intensities above the damage threshold.
Fig. 7.
Fig. 7. (a) Map of the energy of the emitted photon for different classical trajectories computed with the SPAM, being $t_H$ the time of creation of the electron-hole pair and $t$ the potential time of the photon emission. The points where the electron-hole trajectories intersect in direct space at time $t$ are represented by the red area. (b) Electron and hole trajectories corresponding to point A illustrated in panel (a). The trajectories of the electron (black solid line) and the hole (dashed line) are represented as a function of time. The red curve represents the energy gap $E_g(t)$ between the electron and the hole during their oscillation in the bands. The figure corresponds to a driving laser pulse of 3 $\mu$ m wavelength with constant intensity of $5\times 10^{11}$ W/cm $^2$ targeting a (9,9) armchair tube.
Fig. 8.
Fig. 8. Energy map for the emitted harmonic photon from a (9,9) armchair tube according to SPAM for a driver peak intensity of $10^{13}$ W/cm $^2$ . The points corresponding to intersecting electron-hole trajectories are highlighted in red.

Equations (12)

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f ( k ) = e i a 0 k / 3 ( 1 + 2 e i 3 a 0 k / 2 cos a 0 k 2 ) ,
Ψ ( r , t ) = m Ψ m ( k ; r , t ) d k = m [ C m + ( k , t ) Φ m + ( k ; r ) + C m ( k , t ) Φ m ( k ; r ) ] d k .
C m M ( κ t , t ) = C m + ( κ t , t ) C m ( κ t , t )
C ~ m P ( κ t , t ) = e i ϕ m ( κ t ) [ C m + ( κ t , t ) + C m ( κ t , t ) ] ,
i d d t C m M ( κ t , t ) = E m + ( κ t ) + E m ( κ t ) 2 C m M ( κ t , t ) + E m + ( κ t ) E m ( κ t ) 2 e i ϕ m ( κ t ) C ~ m P ( κ t , t )
i d d t C ~ m P ( κ t , t ) = E m + ( κ t ) + E m ( κ t ) 2 C ~ m P ( κ t , t ) + E m + ( κ t ) E m ( κ t ) 2 e i ϕ m ( κ t ) C m M ( κ t , t ) ,
d ( t ) = i q e 2 m [ C m M C m M κ t + C ~ m P C ~ m P κ t ] d k .
a m intra ( t ) = q 2 2 F ( t ) [ | C m + | 2 2 E m + κ t 2 + | C m | 2 2 E m κ t 2 ] d k .
D m ( k ) = a 0 q e 2 sin ( a 0 k 2 ) sin ( m π n ) 1 + 4 cos ( a 0 k 2 ) [ cos ( m π n ) + cos ( a 0 k 2 ) ] .
d ~ m ( q ω 0 ) = i D 0 m q e k e i [ S m ( k , t , t H ) + q ω 0 t ] D m ( κ t ) d k d t ,
t H t v m + ( κ τ ) d τ = t H t v m ( κ τ ) d τ ,
E m + ( κ t ) E m ( κ t ) = q ω 0 ,

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