22C08
Theoretical Study on Population Inversion of Hydrogenlike Nitrogen in
Recombining Plasma Ⅱ
水素様窒素再結合プラズマの反転分布形成に関する理論的研究ⅡTakuya Ozawa, Hideaki Kumai, Eiki Hotta, Kazuhiko Horioka and Tohru Kawamura
小澤拓也, 熊井英章，堀田栄喜，堀岡一彦，河村 徹
Tokyo Institute of Technology, Department of Energy Sciences,Nagatsuta 4259, Midori, Yokohama, Kanagawa, Japan
東工大総理工 〒226-8503 横浜市緑区長津田町4259 メールボックスG3-40
Discharge produced plasma (DPP) is a promising tool to generate effective extreme ultraviolet (EUV) light.
A hydrogen-like nitrogen ion emits light at 13.4 nm with an optical allowed transition from n = 3 to 2,
where n stands for a principal quantum number. We focus on the properties of population inversion
between n = 3 and 2 in recombining phase. Population inversion is calculated in the framework of
population kinetics, and a gain coefficient G is estimated. In this study, dielectronic-capture ladderlike(DL) processes are included. At high density and low temperature, the DL deexcitations make a large
contribution to the population inversion.

1. Introduction
A short-wavelength light source has been usedfor such industrial applications as nanostructurescience and technology. Extreme ultraviolet (EUV)
light is one of the most promising candidates for the
next generation lithography. The wavelength of
EUV light is typically 5 ~ 40 nm, and can beobtained by creation of hot dense plasma. The EUV
light generation has been experimentally done byintense laser light and high current discharge. Ingeneral, hot dense plasma creation by discharge
seems to be more efficient due to its compact andsimple scheme compared with that by laser
irradiation.
In this study, hydrogen-like nitrogen is focused
as one of EUV light emitters of which wavelengthis 13.4 nm. The wavelength is near that for the next
generation semiconductor lithography, and may be
expected to have potential for the industrial use
described above. The 13.4 nm light comes from an
optical allowed transition from n = 3 to 2(Balmer-α). To get enough intensity of the radiation,
nitrogen must be ionized up to the ionization degreeof six. In a recombining scheme, hot dense fully
stripped nitrogen plasma is created and thehydrogen-like nitrogen is populated by three bodyrecombination with rapid plasma expansion andcooling, resulting in the creation of population
inversion and the lasing of Balmer-α. Although arelated experiment has been performed and a pulse
waveform shaping is designed for the rapid cooling,
there is no lasing [1]. A related numericalsimulation with a magnetohydrodynamic (MHD)
and an atomic population kinetics codes for Z-pinch
recombination pumping was done [2]. In theresearch, a recommended pressure is estimated andthe maximum gain of 0.11 cm − 1 is obtained.
However, because of plasma motion, understandingof the lasing characteristics in the viewpoint ofatomic population kinetics is not clear. We
calculate population kinetics with a stepwise
cooling to gain an insight into the lasing of nitrogenat 13.4 nm.
In the calculation, doubly excited states whose
outermost electron is bound in a higher orbital areconsidered. Those states are implicitly included in
an atomic model proposed by Fujimoto et al. [3].
The processes in the model are called dielectroniccapture ladderlike (DL) processes. Such atomic
processes are not considered in the study done by
Vrba et al. [2]. In a recombining plasma scheme,
the DL deexcitation may play an important role tocreate the population inversion. Finally, thecontribution of the DL processes to the population
inversion is quantitatively estimated.
2. Estimation of lasing performance
The usual goal for lasing has been to achieve again-length product of GL = 5. With the crosssection of stimulated emission σstim, the gain
coefficient of Balmer-α is
G = Nn = 3σstim ×[1 − (Nn = 2 /gn = 2 )/(Nn = 3 /gn = 3)],(2)

Nn = 2 and Nn = 3 are the population density of n = 2and 3, respectively. gn = 3 and gn = 2 are thecorresponding statistical weights. Because of atime-dependent property of population inversion, atime-averaged gain Gave is estimated, namely, Gave =τ∫ Gdt τ , where τ is the time at G = Gmax /e after
0reaching G = Gmax.

3.5

THTL
0timeFig. 1. Time evolution of electron temperature.

We focus the recombining phase after themaximum compression relevant to discharge
produced plasma (DPP). To demonstrate therecombining phase, atomic processes are calculatedwith rapid plasma cooling shown in Fig. 1. This
cooling condition is expected to give us strong
nonequilibrium properties on atomic processes.
Since the plasma cooling is assumed to be astepwise to exclude the effect of plasma motion, wecan see the property of the population inversion in
the viewpoint of only population kinetics.
4. Simulation results and discussion
4TH = 300 eV
TL = 10 eV

3n i = 10 18 cm –3
n=2n=32
100.00.20.4

TL = 10 eVn i = 1018 cm ‒33.00.6time [ns]

0.8

1.0

Fig. 2. Population densities of hydrogen-like nitrogenions per statistical weights at ni = 1018 cm−3(n stands for a principal quantum number).

Figure 2 shows the population densities ofhydrogen-like nitrogen ions per statistical weights(Nn/gn) without the DL processes. The initial
electron temperature TH is assumed to be 300 eV,
and the final one TL is 10 eV. The ion density ni is1018 cm−3. The population inversion is created, andG ave [cm ‒3 ]electrontemperature

3. Calculation conditions

N n /g n ×10 14 [cm –3 ]€we can see that Nn = 3 /gn = 3 is higher than Nn = 2 /gn = 2.
The time-averarged gain Gave is ~ 2.8 cm−1. The
plasma length L can be approximately estimated by
the product of τ and the speed of light c, and L is ~8.4 cm at ni = 1018 cm−3.

w/o DLwith DL

2.5
2.0
1.5
1.0
0.5
0.0

0100

200

TH [eV]

300

400

Fig. 3. Time-averaged gain with and without DL
processes at ni = 1018 cm−3.

At ni = 1018 cm−3, DL deexcitations from n = 4 to2 and from n = 3 to 2 are dominant. These
processses make a large population in n = 2 andreduce an order of magnitude of the time-averaged
gain shown in Fig. 3.
5. Conclusions
To get an optimum condition for the generationof 13.4 nm laser light relevant to discharge
produced nitrogen plasma, we focus on thepouplation inversion and the gain coefficient ofBalmer-α in recombining phase. In recombining
phase, the DL deexcitation is found to be dominant
to determine the population inversion between n = 2and 3. With TL ~ 10 eV, the DL deexcitation make alarge contribution to the population inversion,
resulting in the reduction of the gain. The processes
are indispensable for accurate estimation of thegain.
References[1] Y. Sakai, S. Takahashi, T. Hosokai, M. Watanabe,
G-H. Kim and E. Hotta: J. Appl. Phys. 107 (2010)
083303.
[2] P. Vrba, N. A. Bobrova, P. V. Sasorov, M. Vrbova
and J. Hubner: Phys. Plasmas 16, (2009) 073105.
[3] T. Fujimoto and T. Kato: Phys. Rev. A 32 (1985)
1663.