Theoretical investigation of metal–metal waveguides for terahertz quantum-cascade lasers
Micha Szyma nski
Anna Szerling
Kamil Kosiel
We report our theoretical investigations of metal-metal waveguides for terahertz quantum-cascade lasers. The device is considered as a planar, multilayer structure. The optical properties of constituent materials are calculated according to Drude-Lorenz model. The Helmholtz equation is solved numerically using transfer matrix method. We concentrate on selecting the proper metallic material for claddings to minimize the waveguide losses. In addition, we analyze the consequences of inserting Ti separation layers between claddings and semiconductor core for blocking the destructive diffusion of metals into the active layer and improving the adhesion. The terahertz region of the electromagnetic spectrum proves its usability in number of applications including security screening, (bio)chemical detection, remote sensing, nondestructive materials evaluation, communications, astronomy, biology and medicine (Williams 2007; Belkin 2009). Therefore the development of compact, cheap, high-power and convenient continuous-wave radiation sources in this range is strongly desired. The terahertzemitting quantum-cascade lasers (THz QCLs) seem to be an excellent response for these needs. The first THz QCL was demonstrated in 2002 as a cryogenic device. Obviously, for most practical applications, such feature is not acceptable. A lot of efforts are made to fabricate
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Table 1 List of symbols used in the paper
= n2
Thickness (subscript in the text denote layer numbers or materials)
Free space wavevector
Confinement factor
Background dielectric constant
Dielectric constant (subscripts in the text denote materials)
Free space wavelength
devices able to lase in higher, at least TEC-controlled, temperatures. Here, it is useful to
recall the formula for threshold gain, valid for any semiconductor laser:
gth =
List of symbols can be found in Table 1. All the parameters on the right side of Eq. (1)
are determined by the passive waveguide structure, while the material gain depends on the
carrier transport and radiative properties of the quantum-well gain medium. Minimizing the
threshold gain generally results in reduced threshold current densities and increased operating
temperatures (Kohen 2005). Therefore two ways leading to improvements of THz QCL can
be distinguished: (i) designing new lasing schemes with higher gain and (ii) lowering the
waveguide loss. For example, present temperature performance record of 200 K has been
achieved mainly due to approach (i), namely by optimizing the lasing transition oscillator
strength of the resonant phonon based three-well design (Fathololoumi 2012).
In this theoretical work, we explore the path (ii). Our investigations deal with metal
metal waveguides commonly used for THz QCLs. Our goal is to select the proper metallic
material for claddings and thus to minimize the waveguide losses wg . In addition, we
analyze the consequences of inserting separation layers between claddings and semiconductor
core for blocking the destructive diffusion of metals into the active layer and improving the
adhesion.
Fig. 1 Geometry of the planar waveguide. Coordinates x, y, z indicate the directions: across the epitaxial
layers, lateral and of wave propagation, respectively. To the right one can see the field components
2 The model
THz QCLs can be considered as planar waveguides. Schematic view of such a structure is
shown in Fig. 1. It is well known that QCLs support TM modes only (Sirtori 2002). The
further analysis will be based on these two facts.
2.1 Modes of the laser waveguide
The total electromagnetic field of TM mode can be determined by Hy , which satisfies the
Helmholtz equation (Marcuse 1974):
Hy (x ) = 0,
According to the transfer matrix method and boundary conditions imposed on the layer
interfaces we get the following dispersion equation (Chilwell and Hodgkinson 1984):
where m pq are elements of the transfer matrix defined as
() = M+1m11 + M+10m12 + m21 + 0m22 = 0,
M =
Mm =
In Eqs. (3) and (4) m = n2mmk0 0/ 0, m = (k02n 2m 2) and m = m dm . Numerical
solution of Eq. (3) provides the propagation constant = n e f f k0. The waveguide loss can
be calculated as:
2.2 Material parameters
The optical properties of metals as well as highly doped semiconductors are determined by
the behaviour of free carriers, which exhibit well known collective charge oscillations, i.e.
plasmons. Introducing the volume plasma frequency
and using the classical DrudeLorenz model, it is possible to calculate the refractive index
as (Lu 2009):
where = 1 + 1/( )2 and = me f f e/e is the relaxation time.
For calculating n Ga As with different doping levels, we created our own software. We
assumed me f f = 0.063 me (http://www.ioffe.ru/SVA/NSM/), e = 0.85 m2/(Vs) for low
doping (http://www.ioffe.ru/SVA/NSM/), e = 0.2 m2/(Vs) for highly doped contact
lay2
ers (Zivanov and Zivanov 1995) and b = n Ga As , where n Ga As = 2.99 is the value measured
for 0 = 100 m by trans (...truncated)