CHAPTER III
Feasibility Analysis of Sound Lasers (SASER)
3.1 Introduction
In Chapter II we have discussed the feasibility in achieving stimulated LO decay by
externally injected LA phonons.
Now it is natural to ponder the idea of realizing
coherent LA emission from the same mechanism, just as coherently amplified EM
radiations can be obtained from the stimulated transition between atomic states (Siegman,
1986). In analogue to LASER, this type of devices, which can export high-frequency
coherent sound waves, could be called SASER (Sound Amplification by Stimulated
Emission of Radiation). In this Chapter, we will make such an analysis. First, let us give
a review of the former attempts in building sasers.
The obvious similarities between the properties of quanta of light (photons) and sound
(acoustic phonons) have inexorably led a number of researchers to make attempts to
transfer the concepts and techniques from one field of knowledge to the other. Some of
the concepts, such as, the geometrical optical techniques, are relatively easy to transfer
from optics to acoustics and back, while other optical techniques have been proven to be
far harder to implement in acoustics. The concept of coherent acoustic phonon source, or
47
saser, looms large among the devices whose benefits and basic features are easy to grasp,
yet extremely difficult to attain in practice.
In 1961, Tucker demonstrated the amplification of 9.3GHz ultrasonic waves in a ruby
rod. The acoustic waves were generated by the spin-phonon interaction in a 3700 gauss
magnetic field, and the populations of the spin levels were inverted. Tucker obtained a
gain of 0.12 per centimeter (Tucker, 1961). Thirty years later, Prieur and Devaud (1996)
did experiments in Tetrasil glass with the rapid-adiabatic-passage method to invert the
populations of the two level systems. The maximum amplification coefficient was 2dB
in their experiments.
During the same year when Tucker amplified the ultrasonic waves in ruby, Huston and
Mcfee (1961) reported their experimental results on the amplification of ultrasonic waves
in photoconductive CdS by applying a strong dc electric field along the direction of
propagation. The measured gains range from 18db for 15 Mc/s (Mc/s=MHz) to 38db for
45 Mc/s in a 7mm sample. Their experiments demonstrated the possibility of generating
acoustic waves in piezoelectric materials. Today, transducers based on piezoelectric
effect are widely used to generate GHz surface acoustic waves (Auld, 1990).
In 1997, Fokker and Dijkhuis (1997) studied the stimulated phonon emission in an
acoustic cavity. The population inversion of the meta-stable Zeeman-split doublet in
Ruby (Al2O3:Cr3+) was prepared by optical pumping, and they observed the generations
of 10~100 GHz acoustic phonons with strain up to 0.4%.
In 1998, Bartels and Dekorsy (1998) generated 0.45THz acoustic waves in GaAs/AlAs
superlattices by femto-second laser pulse excitation. Their experiments were performed
48
at room temperature, and the acoustic modes were resonantly excited by the E1HH1 (E:
Electron Level, HH: Heaven Hole Level) transition in the superlattices.
In 2001, Özgür and Lee (2001) showed that it was possible to coherently control the
acoustic phonons in semiconductor quantum wells (QW). The zone-folded LA phonons
were generated by sub-picosecond optical pump, similar to the impulsive stimulated
Raman Scattering (ISRS) technique discussed by Merlin (1997). The oscillation period P
of the acoustic waves is proportional to the multi-QW period d, so the frequency of the
LA phonons is adjustable.
Besides these experimental efforts, theoretic work has also been done. For example,
in 1995, Zavtrak suggested to use liquid dielectric in a resonator with dispersed particles
to generate acoustic waves up to 2KHz. His proposal is similar to a free-electron laser.
Zavtrak (1995) argued that an oscillating electrical field could deform the particle
volume, and the dispersed particles would oscillate and acoustic waves would be emitted.
In 2000, Komirenko and Kim (2000) discussed the Cerenkov emission of high frequency
(~0.1THz) acoustic phonons by the drifting electrons in quantum wells.
And their
calculation showed that we could obtain a gain of hundreds per cm in a Si/SiGe/Si
structure.
Different ways of generating optical phonons have also been discussed (Komirenko
and Kim etc al, 2001; Merlin, 1997). But since optical phonons have close-to-zero group
velocities, their potential applications are doubtful and we do not discuss them here.
In the below sections we pinpoint the main difference between the acoustic phonons
and photons, which make the threshold in sasers orders of magnitude higher – the fact
49
that the acoustic phonons are much slower than photons. Then we quickly survey various
phonon-emitting mechanisms that might be suitable for the saser operation, and identify
the main property of the promising candidates: that the final products should be two
highly dissimilar phonon states.
A brief estimate is then provided to ascertain the
practicality of one of the schemes.
3.2. Obstacles on the Way to Sasers
Basic physics being similar, why are the photon lasers seen everywhere while their
phonon analogs, sasers, do not exist? The answer to this question is easy to obtain.
|2 >
q
|1 >
Figure 3.1: General photon or phonon producing process
Consider the generalized process of transition between two states, upper |2,nq> and
lower |1,nq+1>, as shown in Fig 3.1, where both |2> and |1> can be either electronic states
or phonon states. If nq is the quanta occupation number of photons or phonons of
interest, whose wave vector is q, we can write the following equation for the rate of
change of quanta |q> in the transition from state |2> to state |1>, as
50
dnq M 2
=
[n2 ( nq + 1) − n1nq ]δ ( E2 − E1 − ωq )δ k2 ,k1 +q
dt
V
(3.1a)
when the states |2> and |1> are electronic states and
dnq
dt
=
M2
[n2 ( nq + n1 + 1) − n1nq ]δ ( E2 − E1 − ωq )δ k2 ,k1 +q
V
(3.1b)
when the states |2> and |1> are phonon states. The additional term on the right-hand side
of Eq. (3.1b) is due to the bosonic nature of phonons. Here V is the volume, k1 and k2 are
the wave vectors of the lower and upper states, respectively,
quanta |q>, and M 2 ~ 1 H 2
2
q
is the energy of the
is proportional to the square of the transition matrix
element, including all the information about the physical process. A salient feature of the
"best-case" approach developed below is the fact that it is not necessary to know the
absolute value of M 2 in order to ascertain the parameters of the sasers to within an order
of magnitude.
Spontaneous decay of state |2> involves the emission of the quanta with all different
wave vectors – when the summation over these wave vectors is performed, the presence
of the delta-function in Eq. (3.1) yields the density of final states (Okubo and Tamura,
1983; Chen and Khurgin etc al, 2002)
g (ωq ) = V −1
q ,k1
δ (ω2 − ω1 − ωq ) δ k -k ,q
2
1
(3.2)
When it comes to the stimulated decay, the photon (phonon) wave vector q is
determined by the resonance of the cavity, or by the wave vector of externally injected
photon (phonon). But now the resonance condition in Eq. (3.1) is relaxed by broadening,
the reason of which is twofold: first, the emitted photons (phonons) are not truly
monochromatic – scattering processes limit their lifetimes.
51
Second, and the most
important, the states |2> and |1> themselves are broadened due to their finite lifetimes.
Assuming that the broadening is homogeneous in nature, its shape can be described by a
Lorentzian curve with full-width half-maximum (FWHM) Γ. Thus we obtain the balance
equations for the population volume densities of all three quanta, Ni
N +c N
N
dN 2
= R2 − M 2 N 2 [ q e, p 1 + g (ωq )] + M 2 N1 q
dt
πΓ
πΓ
(3.3a)
N +c N
N
dN1
N
= M 2 N 2 [ q e , p 1 + γ 1 g (ωq )] − M 2 N1 q − 1
dt
πΓ
πΓ τ 1
(3.3b)
dN q
N +c N
N
N
= M 2 N 2 [ q e , p 1 + γ q g (ωq )] − M 2 N1 q − q
πΓ
πΓ τ q
dt
(3.3c)
where ce,p=0(1) for electron (phonon) states |1> and |2>, R2 is the rate of some pumping
processes that supply upper laser level, γ 1 and γ q denote the fraction of the spontaneous
decay of the state |2> that ends up in the states |1> and |q>, and, finally τ 1 and τ q are the
lifetimes of these states determined by a variety of scattering processes. τ q also includes
the loss of photons (phonons) leaving the lasing cavity.
Note that since we are
considering the "best-case" scenario we assume that the upper state |2> lifetime is
determined exclusively by the spontaneous decay to level |1> with the emission of
photons (phonons) – thus no additional decay term appears on the right-hand side of Eq.
(3.3a). Also note that in deriving Eq. (3.3) we have neglected the thermal population of
photons (phonons), which is correct for low temperatures T<< ωq / k B .
Higher
temperature cases will be discussed later. Disregarding for now the parametric term
proportional to ce, p N1 N 2 (this term exists only for the bosonic upper and lower states,
52
and, as shown later, has to be small in a good saser) we obtain the expression for the net
gain of the photons (phonons) flux J q = vq N q
d
d
J q = N q = Gq I q
dz
dt
(3.4)
where vq is the group velocity of quanta, Lq = vqτ q is the mean free path, and
Gq =
M 2 ( N 2 − N1 ) 1
−
πΓvq
Lq
(3.5)
In order to ascertain the best-case scenario we shall now assume, for a time being, that
the lower state population N1 is negligibly small. Most of the four-level schemes would
naturally fall into this category (Saleh and Teich, 1991). Of course, as discussed in the
following sections, the depopulating of the lower state is extremely important, and, if the
lower state lifetime is very long, it can prevent the continuous-wave (CW) lasing. But
now we are looking for the main factor that differentiates the thresholds of lasers and
sasers, we shall first consider the best-case scenario. Set the net gain Gq and population
Nq in Eq. (3.3) and Eq. (3.4) to zero, we obtain the expression for the threshold pump rate
Rth = π g (ωq )
Γ
τq
(3.6)
Note that the actual strength of the transition, M 2 , is absent in this expression obtained
under the best-case scenario outlined above.
Assuming that photons (phonons) |q> have linear dispersion and the states |1> and |2>
are dispersionless, we obtain
Rth =
2
1 ωq Γ
Γ
= 2π 2
2
2π vq Lq
λq Lq
53
(3.7)
Herein lies the main reason for the high threshold of the sasers – the small wavelength
of the acoustic wave (some nm) caused by the low velocity of sound. In addition, the
mean free path of acoustic phonons is typically much shorter than that of photons, which,
once again can be traced to the small velocity of sound.
Of course, the only decay channel considered in this simple exercise is the
spontaneous emission; needless to say, other processes can be equally important.
However, our simple expression states unequivocally: in the best-case scenario, the
threshold for phonon lasing is orders of magnitude higher than that for photon lasing at a
comparable frequency. It is worthwhile to repeat that in our best-case scenario the
strength of the emitting process, M 2 , does not enter the final expression.
Now we can make our first recommendation on how to achieve the phonon gain –
reduce the density of final states for the phonons. The most straightforward way is,
obviously, following Eq. (3.7), to reduce the phonon frequency, i.e., to operate close to
the center of the Brillouin Zone. Another direction is to look for the places close to the
edges of the Brillouin Zone, where the density of states can also be quite low.
Unfortunately, the lifetime of phonons there can turn out to be quite short (Tamura,
1984&1985). For the acoustic phonons the scattering and anharmonic decay rates are
proportional to the fourth and fifth powers of frequency (Tamura, 1984; Klemens, 1957;
Okubo and Tamura, 1983), thus going to the zone edge may have the effect of increasing
the loss and would thus raise the threshold. A more interesting alternative approach to
lower the density of phonon states is to consider low-dimensional structures with gaps in
the phonon spectrum, which will be discussed later.
54
Before considering impact of other factors, we re-work the expression of the threshold
pump rate by introducing the quality factors of the transition Qtr and the cavity Qcav
Qtr =
ω21
Γ
(3.8a)
ωq
∆ωcav
(3.8b)
ωq 1
∝ ωq4
λq3 QtrQcav
(3.9)
Qcav = 2π
Lq
λq
=
The threshold Rth is then
Rth = 4π 2
Indeed, it is a very elegant expression indicating that the stimulated emission involves
two coupled resonant harmonic oscillators – the atomic transition and the phonon itself,
and increasing their Q’s values will reduce the threshold. But the common factor ωq4
clearly demonstrates the paramount importance of lowering the angular frequency ωq to
reduce the threshold.
3.3 Choice of Mechanism
Up until now, we have disregarded the lower state |1>. It is quite important to
consider what happens with that state, since, in the absence of efficient relaxation process
from that state, the population buildup will ensue and the efficiency of emission will be
reduced. Here we should consider two different cases: in the first case, the phonon
emission takes place between two electronic states, and in the second case, between two
phonon states.
55
3.3.1 Electronic-to-Electronic States Transition
Since the gain and threshold critically depend on the broadening Γ, the most suitable
arrangement would probably involve an intersubband transitions in quantum wells, as
shown in Fig. 3.2.
q
|2>
|1>
|0>
Figure 3.2: Illustration of phonon-assisted transition between electronic
states in a quantum well
Let us now see how Eq. (3.3) can be simplified for this case. First of all, for fermionic
states ce , p = 0 and the parametric term ~N1N2 is absent. Secondly, every action of
phonon emission will result in populating the lower state |1>, therefore γ 1 ≈ 1 . We can
introduce the lifetime of the upper laser state as
τ 2−1 = M 2 g (ω ) + R '
(3.10)
where R ' is the decay rate due to processes other than the emission of phonons of
interest. It may involve other types of phonons, plasmons, etc.
In addition, since the phonons move we shall introduce the spatial dependence for
them. We then obtain
56
N ( N − N1 ) N 2
dN 2
= R2 − β q 2
−
dt
N thτ 2
τ2
(3.11a)
N ( N − N1 ) N 2 N1
dN1
=β q 2
+
−
dt
N thτ 2
τ 2 τ1
(3.11b)
dN q
dN q
N ( N − N1 )
N
N
+ vq
=β q 2
+γq 2 − q
dt
dx
N thτ 2
τ2 τq
(3.11c)
where we have introduced the quantum efficiency of the transition
β = Mg (ωq )τ 2
(3.12)
N th = π g (ωq ) Γ
(3.13)
and the threshold phonon density
Nth is the phonon density at which the stimulated decay rate of the upper state equals its
spontaneous rate. Usually γ q << 1 because the solid angle of the stimulated emission is a
very small fraction of that of the spontaneous decay ( 4π ), and for the stimulated
emission of phonons to our interest, ignoring this term will introduce little error.
Solving Eq. (3.11) for the steady state
( ∂N i / ∂t = 0) ,
we obtain the following
expression for the unsaturated gain when N th >> N q
Gq =
β R2
N th vq
1−
τ1
1
−
τ2
Lq
(3.14)
Therefore, it is necessary to have the lifetime of the lower state |1> shorter than that of
the upper state |2>. As in the case of quantum cascade lasers the lower level lifetime can
be reduced by means of phonon engineering (Faist and Hofstetter etc al, 2002; Julien and
Sa’ar etc al, 1995), but, in general the requirement is quite strict. If we are interested in
generating acoustic phonons, the energy splitting is very small, in the range of a few
57
millielectronvolts; thus the tunneling resonance width should be substantially less than
can be accomplished with today's technology. Alternatively, we may try to reduce the
lifetime of the lower level by introducing an additional level |0> below it at a sufficiently
low energy, as shown in Fig. 3.2. Unfortunately, that will also cause strong relaxation
from level |2> to level |0> and will not lower the threshold. Of course, we can most
probably make an efficient LO phonon emitter, but, since the group velocity of LO
phonons is close to zero, this would be of dubious practical value.
However, the fact that it is easy to obtain nonequilibrium density of LO phonons
suggests an exciting opportunity: using them as the upper state and using the anharmonic
process LO–>LTA+LTA (LTO) as the operating principle behind the sasers.
3.3.2 Phonon-to-Phonon States Transition
The opportunity of using the anharmonic phonon decay as a source of coherent
phonons was first discussed by Makler and Tuyarot (1996) and Makler and Vasilevskiy
(1998). They considered an intersubband monochromatic LO emitter with subsequent
decay of the zone-center LO phonons into the edge LO phonons and TA phonons, as
shown in Fig. 3.3 and Fig. 3.4b. More recently Romanov and Glavin (1999) have shown
that the LO phonons do not have to be monochromatic for the stimulated emission to
become comparable with the spontaneous emission. However, the authors (Romanov
and Glavin etc al, 1999) have considered only the situation when all the stimulated LA
phonons have the same frequencies (Klemens Channel, Fig. 3.4a), but have their wave
vectors directed in a full 4π angle, which is clearly different from the lasing. We shall
58
show that the Klemens mechanism is far from being optimal for the coherent acoustic
phonon emitters (sasers).
LO
f
1
LO
2
TA
Figure 3.3: LO phonon-emitting process (Makler and Vasilevskiy etc al, 1998)
Consider the situation in which a LO phonon decays into two states |A> and |B> that
can be either acoustical or optical phonons. To modify Eq. (3.3) for this case, we notice
that ce , p = 1 thus a parametric term needs to be included. Furthermore, conservation of
momentum requires that the phonon states |A> and |B> should always appear together
and thus γ A = γ B = γ . As mentioned in Section 3.3.1, it is important to note that γ << 1
because the fraction of the spontaneous decay that goes into the mode of interest is very
smaller due to the high density of phonon states. The balance equations, Eq. (3.3), can
thus be written as
dN LO
N N + N LO N B − N A N B N LO
= RLO − β LO A
−
dt
N thτ LO
τ LO
(3.15a)
dN A
dN
N N + N LO N B − N A N B
N
N
+ vA A = β LO A
+ γ LO − A
dt
dx
N thτ LO
τ LO τ A
(3.15b)
59
dN B
dN B
N N + N LO N B − N A N B
N
N
− vB
= β LO A
+ γ LO − B
τ LO τ B
dt
dx
N thτ LO
(3.15c)
where β, as before, is the quantum efficiency, i.e the relative strength of the chosen LO
phonons decay channel, and vA and vB are the group velocities of the phonon states |A>
and |B>.
f
LO (TO)
f
LO
LA (TA)
TA
k
-q
(a)
k
q
-q
(b)
q
Figure 3.4: Energy diagram of (a) Klemens Channel and (b) the
Channel used by Makler and Vasilevskiy etc al (1998)
Now, the equations are symmetric relative to the states |A> and |B>. In fact, one of
them can be considered to be an equivalent of laser light and the other one an equivalent
to the lower laser state. Let us re-write the numerator in the stimulated decay term as
N LO ( N A + N B ) − N A N B , which consists of the gain term proportional to the sum of
phonon densities in the states |A> and |B> (logical “or”) and the loss term proportional to
their product (logical “and”). To maximize the net gain, it is desirable to have the
densities of phonons NA and NB as different from each other as possible. Therefore, it is
desirable to have the lifetimes of two states as different from each other as possible – the
long lived state will play the role of “light” |q> and the short lived state will play the role
60
of the lower laser state |1>. It is why we disregarded the term ~ N 2 N1 in Section 3.2,
since N 2 N1 should be much smaller than N 2 N q . Clearly, the Klemens process is the
least advantageous since it maximizes the loss term ~ N A N B .
Mirror
Coherent 1.26THz
LA output
Pumping
Hot
LO
TO
Other Low Energy Phonons
GaInAs:0.586nm
InP:1.172nm
Figure 3.5: InP sasers based on the anharmonic process LO TO+LA
3.4 Proposed Terahertz Sasers
Based on all the conclusions and recommendations reached in the previous section, we
propose a different anharmonic decay process as the candidate for the phonon lasing, first
mentioned by Chen and Khurgin (2002) in the context of reducing the lifetime of LO
phonons by stimulated phonon emission. As shown in Fig. 3.5, this process, involving
the decay of the zone-center LO phonons into TO phonons and LA phonons, is a
dominant LO decay channel in InP (Debernardi, 1998). TO plays the role of the lower
laser level in the proposed sasers. Indeed, it has a lifetime of the order of only 10ps at 4K
61
(Irmer and Wenzel etc al, 1996), while that of the LO phonons is 200ps (Vallée, 1994).
The LA phonon clearly plays the role of a photon: it has a rather long lifetime (~6ns)
(Chen and Khurgin etc al, 2002) and reasonably high group velocity (4520m/s) (Auld,
1990) resulting in mean free path of about 30 m. Since the frequency of the LA phonon
is only 1.26THz, the corresponding density of states is not outrageously high.
We can now write the balance equations in a way more familiar to the laser physicists
dN LO
( N − N TO ) N LA + N LO N TO N LO
= RLO − β LO
−
'
dt
N thτ LO
τ LO
(3.16a)
dN TO
( N LO − N TO ) N LA + N LO N TO
N
N
=β
+ γ LO − TO
'
dt
N thτ LO
τ LO τ TO
(3.16b)
dN LA
( N LO − N TO ) N LA + N LO N TO
N
N
=β
+ γ LO − LA
'
dt
N thτ LO
τ LO τ LA
(3.16c)
where we have used the fact that the group velocity of TO phonons is close to zero and
introduced the delayed time
t' = t −
x
vLA
(3.17)
Now, in order to achieve the phonon lasing, we need to introduce the feedback for the
LA phonons. It can be achieved by using an acoustic Distributed Bragg Reflector (DBR)
(see Chapter 5.2), which is a multilayer reflector consisting of half-wave layer pairs of
materials with different acoustic impedance Z = ρ v (Auld, 1990). We can consider using
the lattice-matched Ga0.47In0.53As and InP (Bhattacharya, 1997), whose relevant
parameters are listed in Table 3.1.
Since we can only grow epitaxial film with thickness equal to integer number of
monolayers, the exact quarter-wave thickness cannot be achieved for the 1.26THz LA
62
phonons. However, if we choose the Ga0.47In0.53As thickness to be one lattice constant
and two lattice constants for InP, the combined thickness of a pair is very close to half
wavelength of the 1.26THz LA phonons and a good acoustic reflector can be achieved.
Using the propagation matrix method (see Chapter 5.2), we find that the required
reflectivity R~0.9 is achievable with 24 pairs of layers, and 40 pairs of layers can provide
reflectivity as high as 0.99.
Table 3.1: Relevant parameters for InP and Ga0.47In0.53As
(Thobel and Baudry etc al, 1990; Madelung, 1996)
Density (kg/m3)
LA Velocity (m/s)
Lattice Constant (nm)
InP
4810
4520
0.586
Ga0.47In0.53As
5550
4280
0.586
The other side of the cavity is the air/InP interface, which provides a reflectivity of
~99.99%. So the loss through this interface can be neglected.
We can then introduce the effective lifetime of LA phonon in the cavity of length d as
1
τ
'
LA
=
1
τ LA
−
vLA
ln R
2d
(3.18)
Next, we shall make the assumption that the TO phonon density is much less than that
of the LA phonons, which is easy to justify since the lifetime of TO phonons is 3 orders
of magnitude smaller than the lifetime of LA phonons. We shall also ignore γ , which is
crucial for initiation of stimulated emission, but plays no role in the steady state region.
With these assumptions we obtain
63
dN LO
( N LO − N TO ) N LA N LO
= RLO − β
−
'
dt
N thτ LO
τ LO
(3.19a)
dN TO
( N LO − N TO ) N LA N TO
=β
−
'
dt
N thτ LO
τ TO
(3. 19b)
dN LA
( N − N TO ) N LA N LA
= β LO
− '
'
dt
N thτ LO
τ LA
(3. 19c)
We can now obtain the expression of the threshold inversion density δ N th
δ N th = ( N LO − N TO )th = N th
τ LO
'
βτ LA
(3.20)
and the threshold pump rate Rth
Rth =
N th
'
βτ LA
(3.21)
Introducing the new normalized variables, as
nLO =
pLO
N LO
N
N
; nTO = TO ; nLA = β LA
δ N th
δ N th
N th
R
= LO ;
Rth
τ
α = LO
τ TO
(3.22)
we obtain the saser balance equations similar to the standard set of optical laser equations
as
dnLO
1
=
[ p − n − (nLO − nTO )nLA ]
'
dt
τ LO LO LO
(3.23a)
dnTO
1
=
[−α nTO + (nLO − nTO )nLA ]
'
dt
τ LO
(3.23b)
dnLA nLA
= ' ( nLO − nTO − 1)
dt '
τ LA
The steady-state solution of these equations is
64
(3.23c)
nLO = 1 +
nTO =
pLO − 1
1+α
(3.24a)
pLO − 1
1+α
nLA = ( pLO − 1)
(3.24b)
α
(3.24c)
1+α
The latter refers to the LA phonon density inside the cavity, and for R=90% mirror
reflectivity, it is almost a constant inside the cavity. If we want to find the LA phonon
flux density JLA outside the cavity, it is not difficult to obtain
J LA = N LAvLA
1− R
N
α (1 − R )
'
= ( RLOτ LA
− 0 )vLA
2
1+α 2
β
(3.25)
Then the output power of the saser is
Wout =
ω LA
α
(Win − Wth )
ηout
ω el
1+α
(3.26)
where Win is the total power input and Wth is the threshold pump power
Win = ωelVcav RLO ;
Wth = ωelVcav
N th
'
βτ LA
(3.27)
Vcav = Ad is the cavity volume, A is the output surface area, d is the cavity length, ωel is
the mean energy loss of scattered carriers, and ηout is the output coupling efficiency
ηout =
ln(1/ R)
T
~
−1 −1
2dv τ + ln(1/ R) 2dvLAτ LA + T
−1 −1
LA LA
(3.28)
where T is the mirror transmission ratio.
Here we make a rather conservative assumption that ω el ≈ 1.5ω LO . This is based on
the fact that the energy of an LO phonon in InP corresponds to the kinetic energy of an
electron with drifting velocity ~107cm/s, i.e. almost a saturation value. Therefore, we
65
should not expect the electrons to reach much higher kinetic energies than LO phonons,
and most of the kinetic energy of electrons is transferred to LO phonons.
L~0.5micro
Mirror
R~0.99
InP
Cavity
VG
Source
Carriers
VD=2V
Gate
Drain
+ ++++ + +++++++
+ + + ++ ++++
+ +++
Mirror
R~0.9
LO
d~0.5micro
Substrate
Figure 3.6: Proposed saser structure based on MESFET
Let us now evaluate the values of the threshold phonon density and pump density.
The expression for the threshold phonon density is, of course, Eq. (3.13)
N th = πΓg (ωLA ) = Γ
2
ωLA
3
2π vLA
(3.29)
The broadening Γ has both homogeneous and inhomogeneous components,
Γ = Γ h + Γinh . The homogenous component Γ h is associated with the finite lifetimes of
the optical phonons
Γh =
1
τ LO
+
1
τ TO
=
1
τ TO
(1 +
1
α
)
(3.30)
The inhomogeneous broadening Γinh comes from the fact that the zone-center LO
phonons generated by electrons are distributed in a small region q ≤ δ q ~ 2ω LO mc
66
−1
,
where mc is the effective mass of electrons. As a result, there has inhomogeneous
broadening associated with the dispersion of the TO branch,
Γinh ~ vTOδ q ~ vTO
2ωLO mc
(3.31)
Since the group velocity of TO phonons is very small, we can neglect the inhomogeneous
broadening. Note that this would not be the case if the Klemens channel was considered.
With that in mind and α = τ LO / τ TO ~20, we can evaluate the threshold LA phonon
density in InP as
N th = 2π
f LA2
~ 1 × 1019 cm −3
3
τ LO vLA
(3.32)
Looking at the expression of the output coupling efficiency Eq. (3.28) and noticing
that the cavity length d is determined by the depth of the channel of the devices, i.e.,
d~0.5µm, which is far smaller than the mean free path of the 1.26THz LA phonons
~30µm, we can achieve almost 75% output coupling efficiency with the phonon mirror
transmission ratio T be 0.1.
So the LA phonons have an effective lifetime of
'
τ LA
~ τ LA / 4 ~ 1.5ns . Therefore, with β ~0.9 and the parameters shown in Table 4.3 for
InP, we obtain the following result for the threshold power pump density Pth:
Pth = hf LO Rth ≈
12π h f LO
β
f LA2
τ TO v τ
3
LA LA
≈ 8 × 107W / cm 3
(3.33)
Let us compare this value with the power density in the fully opened channel of an InP
Metal-Semiconductor Field-Effect Transistor (MESFET), operating in the saturation
velocity region (Chang and Kai, 1994), as shown in Fig. 3.6
PMESFET = evsat N D
67
VDS
V
= I DS DS
L
Vcav
(3.34)
where ND is the doping level of the channel, vsat is the saturation velocity, VDS is the
drain-to-source voltage, IDS is the drain-to-source current, L is the channel length, w is the
gate width, and Vcav = dLw is the cavity volume.
If we assume N D = 1× 1017 cm −3 , VDS=2V, L=0.5µm and vsat=107cm/s, we obtain
PMESFET =
Win
~ 6.4 × 109 W / cm 3
Vcav
(3.35)
The total power of the MESFET with 50µm gate width is then around 80mW, while the
threshold power is less than 1mW.
Let us now estimate the density of LA phonons inside the cavity when the saser
operates, e.g., at ten times of the threshold value. According to Eq. (3.24) the phonon
density inside the cavity is then roughly N LA ≈ 10 N 0 ≈ 1 × 1020 cm −3 . The density of LA
phonons in a cubic crystal is related to the strain S as
N LA ωLA =
2
c11S 2 ρ vLA
S2
=
2
2
(3.36)
where c11 is the elastic constant and ρ is the mass density, so we can estimate the strain
S to be about 0.1%, i.e., well within the limit that can be supported by InP.
With this order-of-magnitude estimate in place, we can now suggest a particular saser
structure based on MESFET, as shown in Fig. 3.6. Grown on the InP substrate is a
reflector comprised of alternating lattice-matched InP and Ga0.47In0.53As layers with
overall thickness of 42nm and 90% reflectivity. On top of the mirror is the n-doped InP
active layer of half-micron thickness followed by the standard MESFET metallization
forming the source, drain and gate electrodes (Chang and Kai, 1994). The gate voltage
68
controls the chain of events: “Gate Voltage
Generation
Drain–to–Source Current
LO Phonon
Phonon Lasing”.
3500
T=0.01
T=0.05
T=0.10
T=0.20
3000
Wout (µW)
2500
2000
1500
1000
500
0
0
5
10
15
IDS (mA)
20
25
Figure 3.7: Output power Wout as a function of drain-current IDS for
different mirror transmission ratios T
In Fig. 3.7 we show the dependence of the output LA phonon’s power Wout on the
drain-to-source current IDS (assuming a 2V drain-to-source voltage) for different mirror
transmission ratio T. As we can see, with modest input current the device is capable of
producing a few milli-watts of coherent phonons, and the overall efficiency can be as
high as 5%, which is about one half of the theoretical maximum ω LA / ω el .
In Fig. 3.8 we show the dependence of the output power Wout on the transmission T for
different drain-to-source current IDS. Clearly, optimization of mirror transmission can be
performed by differentiating Eq. (3.26). The optimal mirror transmission T is then
Topt =
Win
2d
−1
Wth ,0
vLAτ LA
where
69
(3.37)
Wth ,0 = ωel
N 0Vcav
(3.38)
βτ LA
is the threshold power for T=0. It should be noted, however, that at high density of LA
phonons the efficiency of phonon lasing will eventually decrease due to four-phonon and
more-phonon stimulated Umklapp processes, which can open additional decay channels
for LO phonons.
Wout (µW)
600
IDS=0.75mA
500
IDS=1.50mA
IDS=2.50mA
400
IDS=5.00mA
300
200
100
0
0.0
0.2
0.4
T
0.6
0.8
1.0
Figure 3.8: Output power Wout as a function of mirror
transmission ratios T for different IDS
3.5. DISCUSSION
It is important to mention here that our analysis was performed for the case of low
temperature (4K). When the temperature increases, the lifetimes of the three phonons
involved in the process will decrease, which will cast doubt on the operation possibility
of the proposed structure.
70
1)
Liquid N2 Temperature 77K: The lifetime of LA phonon will experience the
sharpest decline, since the energies of the final states of LA phonon decay are
commensurate with the thermal energy kBT. In our estimate that reduction
should be of the order of 10, meaning that the mean free path (~ 3µ m ) is still
substantially longer than the cavity length d. According to Irmer and Wenzel
(1996) and Vallée (1994), at 77K the LO and TO lifetimes should also be
reduced to 90ps and 8ps, respectively. As a result, both the spontaneous LO
−1
decay and the linewidth Γ ~ τ TO
are expected to increase leading to about a
three-fold increase in the threshold phonon density N th and threshold power Pth.
So, it is reasonable to expect that the proposed saser will operate at liquid N2
temperature.
2)
Room Temperature 300K: All bets are off. The LO and TO lifetimes are
around 30ps and 2ps (Irmer and Wenzel etc al, 1996; Vallée, 1994). With other
broadening factors accounted for, N th is increased by a factor of at least 50,
while the mean free path of the LA phonons becomes comparable or even
smaller than the cavity length. These make the continue-wave (CW) operation
unattainable, but perhaps low duty cycle operation is feasible.
As to the potential improvement on the device performance, we can make some
considerations on how to decrease the threshold density N th . It seems that we can
decrease the angular frequency ωLA to get a reduced threshold, following the 4th order
law, but since this angular frequency is determined by the requirement of energy
conservation, generally speaking, this method is impracticable. Some practical ways are
71
1) To reduce the density of final states g (ωLA )
2) To increase the lifetime of LO and LA phonons
Figure 3.9: (a) Dispersions of folded LA phonons in a GaAs/AlAs superlattice
(b) Zoom in of the square shown in (a), dots are experimental results
(c) Patterns of folded modes, dotted region is AlAs
These two ways can be taken simultaneously by utilizing the confinement of phonon
modes in semiconductor nanostructures.
The confinement of acoustic phonons in
quantum wells can be modeled by an elastic continuum model (Colvard and Gant etc al,
1985), which was demonstrated experimentally, as shown in Fig 3.10 for a GaAs/AlAs
superlattice (Colvard and Merlin etc al, 1980). The folded dispersion curves in the
reduced Brillouin Zone (BZ) have open gaps at the BZ edge and center, the widths of
which are determined by the elastic properties of the two materials forming the quantum
wells. Around these gaps, the phonon density of states is modified from that in bulk
materials, as shown in Figs. 3.10. So, it is possible for us to adjust the period of the
72
superlattice so that the LO phonons live a longer (shorter) life because of the reduced
(increased) density of final states, if the daughter LA phonons are located at the point B
(A).
A
B
A
ρ : density of states (1D)
Figure 3.10: Illustration of gap and density of states for folded LA
branch in a GaAs/AlAs superlattice
The above sample is for phonon confinement among 1 direction. Confinements in 2
directions (quantum lines) and in 3 directions (quantum dots) are also possible. And
analogous to photonic crystals (Yablonovitch, 1987), phononic crystals (Kushwaha and
Halevi etc al, 1993) have also been employed to inhibit the spontaneous decay of LO
phonons.
Although, in our opinion, generation of coherent terahertz phonons would be a great
feat just for pure science sake, we cannot leave the issue of practical applications
untouched. A number of authors (for example, Prieur and Devaud etc al, 1996) have
already speculated on this subject.
73
The main advantage that acoustic phonons have over the photons is the fact that for
the same frequency the phonons have much shorter wavelength and thus can provide far
better resolution when used in imaging or micro-fabrication applications. This is rather
ironic, in view of the fact that as shown in Chapter 3.2, the short wavelength of the
phonons is precisely the factor impeding the development of sasers. Nevertheless, we
have shown that the beams of coherent phonons can be generated. The mean free path of
terahertz sound is measured in tens or hundreds of micrometers, thus, in our opinion, we
could use coherent beams of terahertz phonons for imaging, processing and diagnosing of
nanostructures. Indeed, for the 1.26THz sound wave the wavelength is 3.6nm, which is a
typical dimension for many nanostructures, to achieve wavelength that short in optics we
would use soft X-rays of 0.34keV energy.
We can also consider the effects of the interactions between the terahertz phonons and
other particles. The most obvious one is the acousto-optic effect (Saleh and Teich, 1991)
– the terahertz phonon represents nothing but a moving grating for the soft X-rays of
comparable wavelengths – and thus can be used to manipulate them. Furthermore, the
THz phonons can interact with the THz electromagnetic waves via the piezoelectric
effect. More interesting, however, is the interaction between the terahertz phonons and
electrons: the deformation potential pattern produced by the terahertz phonons is nothing
but a moving superlattice for the electrons. Thus turning the acoustic waves on and off
can change the properties of electrons, and an electronic equivalent of the acousto-optic
device can be developed.
74
3.6 Conclusion
In this chapter, we have identified the main reason impeding the coherent generation
of acoustic phonons in solid state – the inherently high density of phonon states. Based
on the results of our analysis we formulated a set of conditions that may make saser
practical and pointed out the most promising mechanism of acoustic phonon lasing – the
LO LA+TO decay in InP. We then developed a complete set of saser equations and
designed a saser structure based on InP MESFET. We showed that we could obtain a
high power terahertz coherent sound emission with a threshold pump power as small as
0.7mW and a up to 5% slope efficiency. We also discussed the potential applications and
improvements of this saser.
75