9.7 Band–Band Transitions
285
Fig. 9.30 Absorption of
GaAs (low temperature,
T = 10 K) as a function of
the electron–hole density n
(theory). Adapted
from [889]
GaAs
n=0
n=5 10 cm
n=3 10 cm
n=8 10 cm
15
-3
16
-3
-3
16
1.500
1.495
1.490
5
- 1
0.0
0.2
0.4
0.6
0.8
1.0
-0.2
1.2
(a)
7
6
5
4
3
2
1
0 0
0.5
1.0 1.5
2.0
(10 cm )
17
-3
2.5
T (K)
T c
exp
EHL
EHL
EHG
c
exp
sp
Ge
(b)
screw
EHL
Fig. 9.31 a Temperature–density phase diagram of electrons and holes in Ge. The regions of electron–hole gas (EHG)
and liquid (EHL) and the droplet phase are labeled. Solid line is theoretical calculation, symbols are experimental data
from [892]. The dash-dotted line denoted ρ sp is the experimentally obtained temperature dependence of the liquid density
due to single-particle excitations. ρ
exp
c and T
exp
c
denote the experimental critical density and temperature, respectively.
Adapted from [893]. b Photographic image of radiative recombination (at 1.75 µm wavelength) from a 300-µm diameter
droplet of electron–hole liquid (EHL) in a stressed (001) Ge disk (diameter 4 mm, thickness 1.8 mm) at T = 2 K. The
stress is applied from the top by a nylon screw along a 110 direction. Adapted from [894], reprinted with permission,
©1977 APS
9.7.13 Electron–Hole Droplets
At low temperature and high density, electron–hole pairs in Ge and Si can undergo a phase transition
into a liquid state. This electron–hole liquid (EHL) was suggested in [890] and is a Fermi liquid
exhibiting the high conductivity of a metal and the surface and density of a liquid. The condensation is
due to exchange interaction and correlation. The formation is fostered by the band structure of Ge [891]
and the long lifetime of carriers in the indirect band structure. In unstressed Ge typically a cloud of
electron–hole droplets with diameter in the µm range exists. The phase diagram is shown in Fig. 9.31a.
In suitably stressed Ge electron–hole droplets with several hundred µm diameter form around the point
of maximum shear strain in inhomogeneously strained crystals, as shown in Fig. 9.31b. The pair density
in such a liquid is of the order of 10
17 cm
−3 .
We note that the metallic EHL state hinders observation of the Bose–Einstein condensation (BEC)
of (bosonic) excitons. The light-exciton mass offers a high condensation temperature in the 1 K range
285
Fig. 9.30 Absorption of
GaAs (low temperature,
T = 10 K) as a function of
the electron–hole density n
(theory). Adapted
from [889]
GaAs
n=0
n=5 10 cm
n=3 10 cm
n=8 10 cm
15
-3
16
-3
-3
16
1.500
1.495
1.490
5
- 1
0.0
0.2
0.4
0.6
0.8
1.0
-0.2
1.2
(a)
7
6
5
4
3
2
1
0 0
0.5
1.0 1.5
2.0
(10 cm )
17
-3
2.5
T (K)
T c
exp
EHL
EHL
EHG
c
exp
sp
Ge
(b)
screw
EHL
Fig. 9.31 a Temperature–density phase diagram of electrons and holes in Ge. The regions of electron–hole gas (EHG)
and liquid (EHL) and the droplet phase are labeled. Solid line is theoretical calculation, symbols are experimental data
from [892]. The dash-dotted line denoted ρ sp is the experimentally obtained temperature dependence of the liquid density
due to single-particle excitations. ρ
exp
c and T
exp
c
denote the experimental critical density and temperature, respectively.
Adapted from [893]. b Photographic image of radiative recombination (at 1.75 µm wavelength) from a 300-µm diameter
droplet of electron–hole liquid (EHL) in a stressed (001) Ge disk (diameter 4 mm, thickness 1.8 mm) at T = 2 K. The
stress is applied from the top by a nylon screw along a 110 direction. Adapted from [894], reprinted with permission,
©1977 APS
9.7.13 Electron–Hole Droplets
At low temperature and high density, electron–hole pairs in Ge and Si can undergo a phase transition
into a liquid state. This electron–hole liquid (EHL) was suggested in [890] and is a Fermi liquid
exhibiting the high conductivity of a metal and the surface and density of a liquid. The condensation is
due to exchange interaction and correlation. The formation is fostered by the band structure of Ge [891]
and the long lifetime of carriers in the indirect band structure. In unstressed Ge typically a cloud of
electron–hole droplets with diameter in the µm range exists. The phase diagram is shown in Fig. 9.31a.
In suitably stressed Ge electron–hole droplets with several hundred µm diameter form around the point
of maximum shear strain in inhomogeneously strained crystals, as shown in Fig. 9.31b. The pair density
in such a liquid is of the order of 10
17 cm
−3 .
We note that the metallic EHL state hinders observation of the Bose–Einstein condensation (BEC)
of (bosonic) excitons. The light-exciton mass offers a high condensation temperature in the 1 K range