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Eur. Phys. J. B (2019) 92: 43
https://doi.org/10.1140/epjb/e2019-90629-5 THE EUROPEAN
PHYSICAL JOURNAL B
Regular Article
Coupled elastic membranes model for quantum heat transport
in semiconductor nanowires
Julian A. Lawnaand Daniel S. Kosov
College of Science and Engineering, James Cook University, Townsville, QLD 4811, Australia
Received 28 October 2018 / Received in final form 10 January 2019
Published online 18 February 2019
c
EDP Sciences / Societ`a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature,
2019
Abstract. Presented here is a nanowire model, consisting of coupled elastic membranes with the purpose of
investigating thermal transport in quasi-one-dimensional quantum systems. The vibrations of each elastic
membrane are quantized and the flow of the vibrational energy between adjacent membranes is allowed.
The ends of the nanowire are attached to thermal baths held at different temperatures. We derived quan-
tum master equation for energy flow across the nanowire and obtained thermal currents and other key
observables. We study the effects of a disordered boundary on the thermal current by randomizing the
membrane radii. We evaluate the model as a nanowire analogue as well as study the effects of a disordered
boundary on thermal conductivity. The calculations show that the membrane lattice model demonstrates
diameter phonon confinement and a severe reduction in thermal conductivity due to surface roughness
which is characteristic of semiconductor nanowires. The surface roughness also produces a length depen-
dence of the thermal conductivity of the form κ=αLβ, with βdependent on disorder characteristics, in
the otherwise ballistic regime. Finally, the parameters of the model are fitted to available experimental
data for silicon nanowires and the results of the calculations are assessed against the experimental data.
1 Introduction
Thermal transport in bulk materials is generally well
described by Fourier’s law J=−κOT, where the ther-
mal current is driven by the local temperature gradient
proportional to the thermal conductivity, κ. The thermal
conductivity was considered to be an intrinsic property
of material, independent of the geometry the material.
However it was shown that for a one dimensional lat-
tice of coupled harmonic oscillators that thermal energy
was transported ballistically [1], like a wave, which is
sometimes referred to as “second sound” [2,3]. Since then
it has been shown for low dimensional systems that
the thermal conductivity diverged with system length L
such that κ∼Lβ, such as disordered harmonic[4,5] and
anharmonic chains [5,6], truncated Toda lattices [7] and
Fermi-Ulam-Pasta chains [8,9].
Semiconductor nanowires have been of interest due to
the reduced thermal conductivity that they exhibit in
comparison to bulk materials [10]. This presented a the-
oretical challenge and an opportunity for technological
application, particularly for thermo-electric devices [11].
It was shown that the small diameter of silicon nanowires
compared to phonon wavelengths reduced the thermal
conductivity [10]. Due to large surface to volume ratio
of nanowires, thermal conductivity is also significantly
ae-mail: julian.lawn@my.jcu.edu.au
affected by the geometry of the surface. In experimental
samples where the surfaces is disordered due to etch-
ing or corrugation, the thermal conductivity is reduced
towards the amorphous limit [12]. The combination of the
reduced dimensions and roughness effects led to numer-
ous theoretical works describing the phenomena [13–20].
The reduction of the thermal conductivity due to sur-
face roughness can reduce the thermal conductivity below
Casimir’s classical limit where boundary effects dominate
thermal characteristics [20–22]. It was also found, both
theoretically [22] and experimentally [23], that the corre-
lation length of the surface roughness plays a significant
role, with shorter correlation lengths being key to reducing
the thermal conductivity. Additionally, computer simula-
tions using classical molecular dynamics suggested that
nanowire lengths below the phonon mean free path limit
phonon-phonon interactions, leading to super-diffusive
behaviour and length dependence on thermal conductivity
[24] where the thermal conductivity written as a function
of the nanowire length κ=αLβwas linearly depen-
dent (β= 1) on the system length, up to around 60nm;
beyond which the exponent βreduces towards Fourier
like behaviour. Recently semi-ballistic phonon transport
was confirmed experimentally for silicon nanowires at
temperatures around 4K [25].
Figure 1illustrates the model developed here which
constructs a nanowire analogue out of a series of coupled
Page 2 of 12 Eur. Phys. J. B (2019) 92: 43
Fig. 1. Sketch of the coupled elastic membranes model of
quantum wire.
elastic membranes which each have their own (local)
vibrational spectrum which depends on the membrane
radius. By choosing the size of each individual membrane
the surface roughness directly influences the local vibra-
tional spectra of the nanowire. The system is coupled
weakly at either end to thermal reservoirs held at dif-
ferent temperatures to drive a thermal flux resulting in a
combined system-environment Hamiltonian. In a quantum
master equation approach the nanowire is treated as an
open quantum system. The application of the sequence of
approximations, the Born approximation (keeping terms
up to second order for nanowire-environment coupling in
the Liouville equation for the reduced density matrix),
the Markov approximation (assumption that the corre-
lation functions of the electrodes decay on a time scale
much faster than tunneling events) and the rotating wave
approximation, leads to a Lindblad master equation. The
steady state observables, such as thermal conductivity, are
derived analytically using this master equation.
The outline of the paper is as follows. Section 2describes
the general theory, including the model Hamiltonian,
derivation of the quantum master equation and expres-
sions for key physical observables. In Section 3we applied
the theory to study heat transport through model semi-
conductor nanowires with different types of static surface
disorder. Section 4summarizes the main results of the
paper.
2 Theory
2.1 Physical model and Hamiltonian
The undoped semiconductor nanowire conducts heat by
phonons and contribution from electron and electron-hole
degrees of freedom is negligible. To develop a physical
model for heat transport, we “slice” the nanowire into
thin discs. The width of the disk is assumed to be much
smaller than radius of the nanowire and we assume that
the disc vibration spectrum can be approximated by the
eigen-frequencies of a vibrating thin elastic membrane.
The vibration of a thin membrane is, for small dis-
placements, modelled well by the two-dimensional wave
equation
∂2u
∂t2=c2∂2u
∂x2+∂2u
∂y2.(1)
Here u(x, y, t) is the displacement of the membrane from
its equilibrium position, which assumed to be zero on the
surface of nanowire. Here cis a property of the particular
nanowire material and its value will be determined later.
We use cylindrical coordinates and also assume that the
membrane vibrations have radial symmetry, that means
u(r, φ, t) = u(r, t). The wave equation becomes
∂2u
∂t2=c2∂2u
∂r2+1
r
∂u
∂r .(2)
This differential equation is solved by the separation
of variables and the solution has the following time
dependence
u(r, t)∼eiλctf(r).(3)
Therefore, the period of vibration is 2π/λc, where cis a
parameter which has dimension of velocity. Parameter λ
is given through zeros of a Bessel function of the first kind
[26]
J0(λkR)=0, k = 1,2,3, . . . (4)
where Ris the radius of the membrane. The corresponding
quantized energy spectrum of elastic membrane is Ek=
hλkc/2π=~λkc.
Therefore, the quantized Hamiltonian for the individual
elastic membrane is
X
k
Ekb†
kbk+1
2,(5)
where b†
kand bkare bosonic creation and annihilation
operators satisfying the standard commutation relations
[bk, b†
k0] = δkk0,[bk, bk0]=[b†
k, b†
k0] = 0.(6)
We introduce coupling between membranes to enable
vibrational energy transfer between them. The Hamilto-
nian of the quantum wire becomes
HW=
N
X
αX
k
Ekb†
αkbαk
+
N−1
X
αX
kk0
vkk0hb†
αkbα+1k0+b†
α+1kbαk0i,(7)
where αis the membrane index and Nis the number of
membranes that comprise the wire. We omitted the zero
point energy part of this Hamiltonian, since it does not
contribute to any quantities we compute. The amplitude
for inter-membrane coupling is taken in the form
vkk0=v0
|Eαk −Eα+1k0|/q + 1.(8)
The physical reasoning behind this choice of the interac-
tion is the following. It describes the transfer of energy
between the vibrations of the nearest-neighbour mem-
branes, it is maximal when the membrane vibrational
Eur. Phys. J. B (2019) 92: 43 Page 3 of 12
energy levels are in resonance and decays as the energy gap
between the vibrational states of interest grows. Value v0
gives the strength of inter-membrane coupling and param-
eter qsuppresses (for large q) or amplifies (for small q) the
resonance-dominated energy transfer. The wire is attached
to two thermal baths – macroscopically large ideal gases
of phonons with spectra land rheld at different tem-
peratures TLand TR, respectively. The Hamiltonians for
the thermal baths are
HL=X
l
lb†
lbl, HR=X
r
rb†
rbr.(9)
Phonon–exchange interaction between the wire and baths
is taken in the energy transfer form
V=tLX
lk b†
lb1k+b†
1kbl+tRX
rk b†
rbNk +b†
Nk br.
(10)
Here the first term describes the vibration quanta trans-
fer between the left bath and the first membrane with
amplitude tLand the second term describes the vibration
quanta transfer between the last membrane and the right
bath with amplitude tR.
2.2 Canonical transformation from local membrane
vibration to wire normal modes
The wire is described by the Hamiltonian which is in
bosonic quadratic form:
HW=X
αk X
α0k0
hαk,α0k0b†
αkbα0k0,(11)
where the hαk,α0k0are the matrix elements of the block-
tridiagonal Hamiltonian matrix
h=
hαk vkk00
vkk0
...vkk0
vkk0hαk vkk0
vkk0
...vkk0
0vkk0hαk
(12)
where
hαk =
Eα1
...
EαKα
(13)
is the primary block diagonal consisting of the diago-
nal matricies of local vibrational energies Eαk , and the
upper and lower block diagonals containing the inter-
membrane coupling elements vkk0. This Hamiltonian is
diagonalized by the canonical transformation from ini-
tial localized membrane vibrations b†
αk to global nanowire
normal modes B†
s
B†
s=X
αk
Uαk,sb†
αk Bs=X
αk
Uαk,sbαk .(14)
The transformation is unitary in order to preserve the
bosonic commutation relations
X
αk
Uαk,sUαk,s0=δss0,X
s
Uαk,sUα0k0,s =δkk0δαα0.
(15)
If we require that, the transformation matrix satisfies
the eigenvalue problem
X
α0k0
hαk,α0k0Uα0k0,s =sUαk,s,(16)
then the Hamiltonian becomes diagonal in normal mode
creation and annihilation operators
HW=X
s
sB†
sBs.(17)
The coupling of the wire to the thermal bath is also
transformed to normal mode operators
V=X
ls
νLs b†
lBs+B†
sbl+X
rks
νRs b†
rBs+B†
sbr,
(18)
where we introduced the amplitudes for energy trans-
fer between phonons in the thermal bath and nanowire
normal modes
νLs =tLX
k
U1k,s νRs =tRX
k
UNk ,s.(19)
2.3 Quantum master equation for nanowire density
matrix and observables
Using the Born-Markov and rotating wave approximations
(details of the derivation are given in Appendix A) we get
a Lindblad type master equation for the nanowire density
matrix ρ(t)
i~˙ρ(t) = "X
s
˜sB†
sBs, ρ(t)#
−iX
s,µ=L,R
Γµ(s)1
2(1 + 2nµ(s)){B†
sBs, ρ(t)}
−(1 + nµ(s))BsρB†
s−nµ(s)B†
sρBs+nµ(s)ρ.
(20)
The energies of the nanowire normal modes are shifted
by the coupling to the left and right baths
˜s=s+ ∆L(s)+∆R(s) (21)
by the Lamb shifts (real parts of bath self-energies)
∆L(s)=[νLs]2X
l
s−l
(l−s)2+v2
∆R(s)=[νRs]2X
r
s−r
(r−s)2+v2.(22)
The dissipative part of the density matrix time evo-
lution is governed by the normal mode energy level
Page 4 of 12 Eur. Phys. J. B (2019) 92: 43
broadening function (imaginary parts of the baths self-
energies)
ΓL(s)=2π[νLs]2ρL(s) ΓR(s) = 2π[νRs ]2ρR(s).(23)
Let us now demonstrate how to use this master equation
to compute observables of interest. The average value of
an arbitrary operator Oat time tis
hOit= Tr[Oρ(t)].(24)
Differentiating with respect to tgives the equation of
motion for a time-dependent expectation value of operator
O:
i~d
dthOit=X
s
˜sh[O, B†
sBs]it
−iX
s,µ=L,R
Γµ(s)h1
2(1 + 2nµ(s))h[B†
sBs, O]it
−nµ(s)h[Bs, O]B†
sit−(1 + nµ(s))h[B†
s, O]Bsiti.
(25)
Using (25) we compute various nonequilibrium quanti-
ties to characterize properties of the nanowire. We begin
with the number of vibrational quanta populating the nor-
mal mode s. Substituting the corresponding operator into
(25) we get ns=B†
sBs
i~d
dthnsit=iX
µ=L,R
Γµ(s)hnµ(s)h[Bs, ns]B†
sit
+ (1 + nµ(s))h[B†
s, ns]Bsiti,(26)
which gives
d
dthnsit=X
µ=L,R
Γµ(s)
~[nµ(s)− hnsit].(27)
This differential equation has the following solution:
hnsit=ΓL(s)nL(s)+ΓR(s)nR(s)
ΓL(s)+ΓR(s)
+ exp −ΓL(s)+ΓR(s)
~t
×hhnsit=0 −ΓL(s)nL(s)+ΓR(s)nR(s)
ΓL(s)+ΓR(s)i.(28)
If we tend tto infinity, then the nanowire reaches a
nonequilibrium steady state regime and the populations
of the nanowire normal modes become stationary
hnsi=ΓL(s)nL(s)+ΓR(s)nR(s)
ΓL(s)+ΓR(s).(29)
Having computed the nonequilibrium population of
nanowire normal modes (29) we can easily compute the
amount of vibrational energy stored in the nanowire
hHWit=X
s
s
ΓL(s)nL(s)+ΓR(s)nR(s)
ΓL(s)+ΓR(s).(30)
We define the heat current using the continuity equation
for the vibrational energy flow. The energy conservation
gives
d
dthHWit=JL(t) + JR(t),(31)
where JL(JR) is the energy flowing into the system from
the left (right) heat baths. The rate of change of the
vibrational energy stored in nanowire is
d
dthHWit=X
s
sΓL(s)
~[nL(s)− hnsit]
+ΓR(s)
~[nR(s)− hnsit].(32)
Comparing (32) with the right hand side of the continuity
equation (31), we identify the heat currents for the energy
flowing into the wire from the left bath
JL(t) = X
s
s
ΓL(s)
~[nL(s)− hnsit],(33)
and from the right bath
JR(t) = X
s
s
ΓR(s)
~[nR(s)− hnsit].(34)
In the steady state regime, the current becomes time-
independent and it is given by
JL=1
~X
s
s
ΓL(s)ΓR(s)
ΓL(s)+ΓR(s)[nL(s)−nR(s)],(35)
and
JR=1
~X
s
s
ΓL(s)ΓR(s)
ΓL(s)+ΓR(s)[nR(s)−nL(s)].(36)
Therefore, the total heat current is
J=JL=−JR.(37)
The expression for heat conductivity is obtained con-
sidering the linear response regime. Suppose that the
TL=T+ ∆T/2 and TR=T−∆T /2, where ∆Tis the
temperature difference between left and right ends of the
nanowire. Performing Taylor expansion in ∆T
J=K∆T+· · · (38)
Eur. Phys. J. B (2019) 92: 43 Page 5 of 12
we identify the expression for the proportionality
coefficient
K=1
~kBT2X
s
2
s
ΓL(s)ΓR(s)
ΓL(s)+ΓR(s)
es/kBT
(es/kBT−1)2.(39)
Comparing to Fourier’s Law in one dimension
J
A=κ∆T
L,(40)
where Ais the cross-section area of the nanowire and
Lis its length, we can infer the expression for thermal
conductivity is
κ=KL
A.
3 Results
3.1 Model parameters
For the results found throughout this paper the physically
relevant quantities were calculated using molecular units
(m.u.), which has energies of kJ/mol, for the numerical
component of calculations (B.4). Additionally for com-
parison to the experimental data for silicon nanowires
there are a number of model parameters to consider. For
the individual membranes the mechanical constant cwas
taken to be the bulk counterpart, the speed of sound in
silicon (c= 8433 m/s).
The cut-off for local vibrations Kαand the reservoir
couplings (B.5) were approximated by comparing the
model to the experimental data from Li et al. [10]. The
comparison was made by calculating the conductivity-
temperature profile for smooth nanowires and varying
the cut-off until it provided a good fit across the dif-
ferent diameters, giving preference to better describing
low temperatures (less than 150 K). Additionally a ref-
erence energy of 1.1 kJ/mol is used to ensure that the
phonon energies remain positive after diagonalization.
The reservoir couplings were found by minimising the
mean square residuals between the model and experi-
ment. The resulting couplings are roughly proportional
to the cross-sectional area of the nanowires. The choice of
preferencing the low temperature fit is due to the lack
of temperature dependent effects, particularly thermal
expansion. As a result when the phonon transport is sat-
urated the thermal conductivity plateaus whereas in the
experimental results temperature dependent effects, such
as thermal expansion, begin to dominate changes in the
conductivity leading to a falloff in the in the conductivity
at high temperatures. As a result of the decisions above we
use the following set of fitted parameters in the majority
of our calculations (for nanowire with diameter of 37 nm):
Kα= 3.3 kJ/mol and ΓL= ΓR= 23.1 kJ/mol.
3.2 Role of disorder (surface roughness)
The surface roughness is introduced as a randomisation of
the diameter of a subset of the membranes which make up
the nanowire. This randomisation of the diameters change
the local vibrational spectrum for each membrane (5). The
radius of each of the membranes is randomized such that
R0
α=Rα−∆Rα,(41)
where the ∆Rαrepresents the magnitude of the disor-
der and Rαis the unmodified radius of the nanowire
with R0
αrepresenting the radius of disordered or “etched”
nanowire. The disorder ∆Rαis achieved by randomly
sampling the uniform distribution between 0 and a rough-
ness depth σ. To introduce a tuneable corrugation length
into the realisations, each of these membranes are placed
half the desired corrugation length apart and the mem-
branes that lie between them are interpolated to produce
smoothed corrugations. Finally the membranes which are
coupled directly to the driving reservoirs remain unmodi-
fied to ensure symmetry in the reservoir couplings (B.5).
Figure 2shows examples of nanowires generated in this
fashion. These realisations illustrate how the corrugation
length affects the roughness profile.
Throughout this paper for each set of system param-
eters observables are averaged over 500 realisations of
disordered nanowires. The introduction of disorder results
in a reduction of the thermal current of up two orders
of magnitude with roughness less than half the mem-
brane radius. This reduction is significant for even small
amounts of disorder while further increases in disorder
have less of an effect. Figure 3illustrates this behaviour
in the decreasing magnitude of the current with increasing
disorder in the current-temperature profiles. The results
clearly demonstrates the diminishing effect with increas-
ing disorder depth by comparing the thermal current of
nanowires at 300K for various roughness depths.
Conversely as can be seen in Figure 4, short corruga-
tion lengths result in a significant reduction in thermal
current, with thermal current increasing with corruga-
tion length. This interplay between roughness depth and
roughness corrugation length is qualitatively similar to
what was observed in experimental results investigating
the effects of correlation length on the thermal con-
ductivity of vapour-liquid-solid (VLS) grown nanowires
[23].
3.2.1 Contributions from individual normal modes
As we discussed in the previous section, the nanowire sur-
face roughness results in an altered temperature profile
where the thermal conductivity does not increase with
temperature as much as in the smooth nanowires. This
behaviour can be explained by looking at the contribu-
tions to the thermal current from the individual nanowire
normal modes. In comparison to the smooth system, in
the disordered systems higher energy phonons contribute
less to thermal transport while lower energy levels are
saturated leading to a shallower thermal current vs tem-
perature profile. The shift towards lower energy phonons
is due to disorder introducing a mismatching of local
vibrational energies between neighbouring membranes.
Disorder through the randomisation of the radii alters
Page 6 of 12 Eur. Phys. J. B (2019) 92: 43
Fig. 2. Illustration of membranes making up two disordered nanowires (a) with a corrugation length of σL= 4 and (b) with a
corrugation length of σL= 10. Both are 50 membranes long and have maximum diameter of 37 nm and a roughness depth of
σ= 2 nm.
Fig. 3. Current-temperature profiles for various surface rough-
ness depths σwith corrugation length σL= 4 for a nanowire
of diameter D= 37 nm.
the dispersion of the local vibrational modes (4). Differ-
ences in the dispersion of the local energy spectrum lead to
smaller energy differences between the low energy vibra-
tions of neighbouring membranes while at higher energies
the mismatch is greater. This mismatch in neighbour-
ing vibrational energies results in a weaker neighbouring
coupling (8) and hence lowers transport.
Figure 5shows this effect for two different coupling
regimes. The figure shows the contribution of various
phonon energies to the thermal current. Figures 5a–5d
the nanowires have the same resonance parameter
(q= 0.01) and the same “initial” radius (D= 37 nm)
while Figures 5b and 5d have disordered radii. Figures 5a
and 5b demonstrate that a weaker nearest neighbour cou-
pling (v0= 0.01) results in separated and non-overlapping
vibrational subbands in the nanowire vibrational spectra.
These subbands are related to the one local vibrational
modes contributed by each membrane. In this weak cou-
pling regime local vibrational modes of one membrane
only couple to local vibrational modes closest in energy.
Thereby the lowest energy subband is the contribution of
Fig. 4. Thermal current as a function of temperature for
different roughness corrugation lengths.
the lowest energy local vibrational mode from each of the
membranes. When the coupling strength is increased local
vibrational modes can couple to energy levels other than
the closest energetically. Figures 5c and 5d correspond
to a stronger coupling (v0= 0.5). The coupling between
more local vibrational modes leads to an overlapping of
the subbands leaving no gaps.
Neither regime has an effect on the thermal current
for smooth nanowires (without disorder). However the
coupling parameters v0and qwhich govern the transi-
tion between the two regimes play an important role in
determining how strong the effect of disorder on thermal
transport in the system and consequently whether the sys-
tem transports thermal energy in a more ballistic or wave
like fashion or more diffusivity akin to Fourier’s law.
3.3 Length dependence of thermal current
In a system with no disorder in the membrane diame-
ter the system remains in a ballistic regime due to the
Hamiltonian (11) which does not have phonon-phonon
Eur. Phys. J. B (2019) 92: 43 Page 7 of 12
Fig. 5. The contribution to the thermal flux from phonons of various energy for smooth nanowires (a) and (c); and disordered
nanowires (b) and (d) σ= 0.75, σL= 4. (a) and (b) have coupling parameter v0= 0.01 while (c) and (d) have coupling
parameter v0= 0.5. Coupling parameter q= 0.01 for all sub-figures.
interactions in the longitudinal direction. The introduc-
tion of even a small amount of disorder into the membrane
radii reduces the thermal conductivity by an order of mag-
nitude and introduces a length dependence of the form
κ∝Lβor similarly in the thermal current J∝Lβ−1.
This power law behaviour is indicative of disordered 1D
lattices which conserve total phonon momentum [4–9] and
in silicon nanowires using molecular dynamic simulations
[24] for systems below the phonon mean free path. This
is expected as the Hamiltonian (11) lacks an-harmonic
phonon-phonon interaction terms and likewise conserves
total phonon momentum. However unlike what has been
reported previously for 1D momentum conserving systems
the magnitude of βis dependent on both the magnitude
of the disorder and the corrugation length of the disorder.
In other 1D momentum conserving lattices the power
law behaviour is dependent on the nature of the system
interactions and reservoir couplings such as in disordered
harmonic systems where a value of β= 3/2 was found
for free boundary conditions [27] and β= 1/2 for fixed
boundary conditions. This is independent of the magni-
tude or corrugation length of the disorder introduced. For
anharmonic systems the power law behaviour is depen-
dent on the nature of the anharmonicity. For example the
diatomic Toda lattice has β≈0.35 [7] and the FPU chains
exhibit β= 1/3 when anharmonicity is strong. When dis-
order is more dominant than the anharmonicity the length
Fig. 6. Log–log scale plot of the length dependence of ther-
mal current at T= 300 K for various roughness depths. All
curves have a corrugation length of σL= 1 nm and coupling
parameters v0= 0.5 and q= 0.01.
dependence of the system is similar to the harmonic sys-
tems due to disorder suppressing the anharmonicity. In
these other 1D systems the power law behaviour has
a single characteristic parameter βwith regards to dis-
order whereas in the thin membrane lattice model this
Page 8 of 12 Eur. Phys. J. B (2019) 92: 43
Table 1. Values for βcorresponding to a power-law length
dependence, J=Lβ−1of the thermal current for the vari-
ous roughness depths in Figure 6.phσ2iis the root mean
squared roughness depth.
Roughness 0 nm 0.25 nm 0.5 1 2 3
depth (nm)
phσ2i0 0.087 0.17 0.35 0.70 1.04
β1.000 0.868 0.811 0.755 0.594 0.503
Fig. 7. Length dependence of thermal current at T= 300 K
for various roughness corrugation lengths. All curves have a
roughness depth of σ= 1 nm and hopping parameters v0= 0.5
and q= 0.01.
behaviour is disorder dependant. From here the magni-
tude and corrugation length of the disorder as well as the
inter-membrane coupling (8) are altered to demonstrate
their effect on the power law dependence of the thermal
current. The value of βis found using a least squares
regression for lines of best fit on the numerical results. In
addition to summarising this value, the tables also char-
acterize the root mean squared roughness depth for the
realisations.
Figure 6shows the length dependence of the ther-
mal current for a series of D= 37 nm nanowires with
different roughness depths but the same roughness corru-
gation length and coupling parameters. It illustrates how
the power law behaviour is affected by the depth of the
disorder. With increasing disorder leading to a more pro-
nounced power law dependence of the thermal current.
Table 1summarizes the results in Figure 6, quantifying
the value of βfor lines of best fit. In addition Table as
well as the root mean square roughness depth for each set
of realisations.
3.3.1 Roughness corrugation length
The roughness corrugation length has a more signifi-
cant impact on the length dependence of the thermal
conductivity than the roughness depth.
Figure 7shows how increasing the corrugation length
of the surface roughness increases the dependence of the
Table 2. Values for βcorresponding to a power-law length
dependence, J=Lβ−1of the thermal current for the
various roughness corrugation lengths in Figure 7.
Roughness 4 6 8 10
corrugation length
phσ2i0.35 0.35 0.34 0.34
β0.742 0.801 0.850 0.860
Fig. 8. Length dependence of thermal current at T= 300 K
for various values for the coupling parameter v0. All curves
have a roughness depth of σ= 2 nm and corrugation length of
σL= 4 and coupling parameter q= 0.01.
Table 3. Values for βcorresponding to a power-law length
dependence, J=Lβ−1of the thermal current for the
various roughness corrugation lengths in Figure 8.
Coupling 0.01 0.03 0.05 0.1
parameter v0
phσ2i0.69 0.69 0.70 0.69
β0.778 0.836 0.706 0.782
thermal conductivity on length. All nanowires in the figure
have an unaltered diameter of D= 37 nm and have the
same roughness depth and coupling parameters. Table 2
summarizes this through quantification of βas well as
characterising the root mean squared roughness depth for
each of the different corrugation length series.
3.3.2 Inter-membrane coupling
The disorder induced length dependence is also influ-
enced by the strength of the inter-membrane coupling
(8) with a strong coupling increasing the length depen-
dence. Lowering the parameter qstrengthens the coupling
between non-equal local vibrational modes and increasing
v0strengthens inter-membrane coupling generally. Each
leading to a decrease in the value of β.
Figure 8illustrates the length dependence of the
thermal current on the coupling parameter v0for
D= 37 nm nanowires with the same surface roughness
Eur. Phys. J. B (2019) 92: 43 Page 9 of 12
Fig. 9. Length dependence of thermal current at T= 300 K for
various values of the coupling parameter q. All curves have a
roughness depth of σ= 2 nm and a corrugation length σL= 4
and a coupling parameter v0= 0.5.
(σ= 2), roughness corrugation length (σL= 4) and
resonance parameter (q= 0.01). Table 3outlines the cor-
responding characteristics of the roughness and power law
behaviour. An interesting feature in Figure 8is the inter-
section of the v0= 0.03 and v0= 0.05 series indicating
the transition between the weak coupling regime and the
strong coupling regime which corresponds to the tran-
sition from discrete bands in Figures 5a and 5b to the
overlapping bands in Figures 5c and 5d.
Similarly Figure 9and Table 4characterize the power
law length dependence of the thermal conductivity and
its relation to the coupling parameter qfor D= 37 nm
nanowires while keeping the same surface roughness
(σ= 2), roughness corrugation length (σL= 4) and
coupling strength (v0= 0.5). Value of the parameter q
(8) indicates the relative importance of resonance and
off-resonance energy transfer. Increasing the parameter
decreases the effect of energy mismatch between coupled
energy levels and in turn decreases the effect that disor-
der has on the thermal current. Conversely decreasing the
coupling parameter increases the energy mismatch and
increases the effect of disorder.
Recent advances in the use of graphical processing
units for computing have allowed molecular dynamics
simulations of silicon nanowires to reach sizes compa-
rable to the sizes in experiments [28]. The simulations
show that the thermal conductivity approaches a size
independent regime (β= 0) as they increase in size.
Our membrane lattice model could suggest that the rate
at which the nanowire approaches the size independent
regime would depend on the nanowire material, through
the inter-membrane coupling and the surface roughness
characteristics.
3.4 Comparison with the experiment
The thermal conductivity of nanowire treated in the pro-
posed model is reduced as the diameter of the nanowire
Table 4. Values for αand βcorresponding to a power-law
length dependence, J=Lβ−1of the thermal current for
the various roughness corrugation lengths in Figure 9.
Coupling 0.001 0.005 0.1 0.5
parameter q
phσ2i0.70 0.70 0.69 0.69
β0.386 0.508 0.594 0.882
Fig. 10. Shows thermal conductivity as a function of
nanowire temperature (the average of the two driving reser-
voirs (TL+TR)/2). Diameters illustrated are 22 nm, 37 nm,
56 nm, 115 nm to draw comparison with the experimental data
(dots) from Li et al. [10].
is decreased similar to the experimental results for sili-
con nanowires [10]. However the temperature profile of
the membrane lattice deviates from experimental profiles
of silicon nanowires [10]. Several factors contribute to
this discrepancy. The model relies on a simplified local
vibrational spectrum based on a classical elastic mem-
brane with an introduced vibrational cutoff in a similar
vein as a Debye cutoff. This local vibrational spectrum
is also not temperature dependent, not accounting for
thermal expansion. In the experimental data this leads
to a decrease in the thermal conductivity after the ini-
tial plateau which occurs between 100 K and 150 K which
is not present in our thin-membrane lattice model. This
discrepancy is apparent when comparing to the original
experiment by Li et al. [10] and less so when compared to
the VLS grown nanowires in [23] which is more subtle in
the as grown nanowires.
Figure 10 shows the fitted curves outlined in Section 3.1
on model parameters and compares them to the experi-
mental data from Li et al. [10] which they were fitted to.
It also illustrates the discrepancy between the model and
experiment at high temperatures. This was due to the sim-
plistic description for the local vibrational modes without
taking into account the thermal expansion of the mem-
brane radius. At these higher temperatures where phonon
transport is saturated these effects dominate. This dis-
crepancy is the motivating reason for preferencing the
Page 10 of 12 Eur. Phys. J. B (2019) 92: 43
Table 5. The values used to generate the fitted curves
in Figure 3.Arepresents the cross sectional area of the
nanowire.
Diameter (nm) 22 37 56 115
Cut off (kBT) 3.3 3.3 3.3 3.3
Reservoir coupling (γ) 3.8 23.1 64.0 218.1
γ/A ×1030.01 0.021 0.026 0.021
low temperature fit when developing the model param-
eters in Section 3.1. Table 5presents a summary of the
key parameters of the model which provides the fit to the
experimental data and used to generate the model curves
in Figure 10. The table also illustrates that the reservoir
couplings are proportional to the cross sectional areas of
the nanowires.
4 Conclusion
We have developed a physical model for heat transport in
semiconductor nanowires. The nanowire is “sliced” into
thin discs and the width of the disk is assumed to be
much smaller than radius of the nanowire, therefore the
disc vibration spectrum was approximated by the char-
acteristic frequencies of vibrating thin elastic membrane.
The left and right ends of the nanowire are attached to
two macroscopic thermal baths held at different temper-
atures. We treated the nanowire as an open quantum
system and derived a Lindblad master equation for the
nanowire density matrix.
Solving Linblad master equations for smooth and dis-
ordered nanowire we obtained key observables such as
thermal conductivity coefficient and thermal flux.
The main observations are as follows:
– Thermal current is significantly influenced not only
by the nanowire surface disorder but also by the
corrugation length of the disordered wire surface.
The magnitude and corrugation length both play
an important role in determining the magnitude
and length dependence of the thermal current. An
increase in the roughness depth from σ= 0.25 nm
to σ= 3nm resulted in a 100 m.u. reduction in the
thermal current for nanowires at a temperature of
T= 100K. Increasing the corrugation length of the
surface roughness from four membranes (σL= 4) to
ten (σL= 10) showed an increase in the thermal cur-
rent from J= 75 m.u. to J= 110 m.u. for nanowires
at T= 100 K.
– The introduction of disorder does not stop the
thermal conductivity from diverging producing the
power order length dependence of thermal conduc-
tivity κ∝Lβ. The value of βdepends on the mag-
nitude as well as corrugation length of the disorder.
For, example, introducing a roughness depth of σ=
3 nm changes the ballistic transport regime, β= 1.0,
to diffusive transport with significant length depen-
dent thermal conductivity, β= 0.5. For nanowires
with a roughness depth the model demonstrated
that doubling the corrugation length from four mem-
branes to eight showed an increase in the length
dependence parameter from β= 0.74 to β= 0.85.
– The inter-membrane coupling also effects the length
dependence of the thermal current. As the intro-
duction of disorder moves the energy levels of
neighbouring membranes away from resonance, the
inter-membrane coupling determines to what extent
off-resonance energy transfer contributes to the ther-
mal current. Changing the value of the hopping
parameter v0shows a small change in the length
dependence of the system; between β= 0.7 and β=
0.84 over a order of magnitude change in v0. Whereas
in comparison an order of magnitude change in the
inter-membrane coupling parameter qresults in a
change in the length dependence between β= 0.88
and β= 0.51.
– The model is checked against the available experi-
mental date. It is found that the model reproduces
some of the qualitative effects of disorder that are
present in semiconductor nanowires however the sim-
plifying assumptions of the model limited its ability
to reproduce the entire range of experimental results.
On one hand the model qualitatively agrees with
observations made previously about the interplay
between disorder and the lateral length scale of
the disorder. While on the other hand, the model
does not account for thermal radius expansion and
phonon-phonon interactions. This somehow limits its
practical use as a predictive model as well as its abil-
ity to investigate whether disorder effects the length
dependence of the system for nanowires beyond the
mean-free path of the phonons.
We would like to thank Peter Stokes and Samuel Rudge for
many valuable discussions.
Author contribution statement
JL and DK contributed equally throughout the process of
developing the model, performing calculations, analysing
results and writing the manuscript.
Appendix A: Derivation of the master
equation for quantum wire density matrix
The open quantum systems is generically described by the
following Hamiltonian
H=HS+HB+HSB ,(A.1)
where HSis the system Hamiltonian, HBis the bath
Hamiltonian, and HSB is the system-bath interaction. The
system-bath interaction can be written as a product of
operators in the system space Siand bath operators Bi:
V=X
µ
SµBµ=SµBµ.(A.2)
Eur. Phys. J. B (2019) 92: 43 Page 11 of 12
Using the Born–Markov approximation
i~˙σ(t) = [HS, σ(t)]
−i
~Z∞
0
dτ hGµα(τ)Sµe−i
~HSτSαei
~HSτσ(t)
−Gαµ(−τ)Sµσ(t)e−i
~HSτSαei
~HSτ+ h.c.i
(A.3)
where the bath correlation function
Gµα(τ) = TrBhBµe−i(HB+HS)τ/~Bαi.(A.4)
This master equation will be the starting point for our
calculations. This master equation (especially when it is
written in the basis of eigenstates of Hamiltonian HS) is
often called the Redfield master equation. We represent
the energy transfer interaction between the nanowire and
left and right thermal bath (18) in the form suitable for
the Redfield equation (A.2) by introducing the set of 4
non-Hermitian operators
S1=X
s
vLsBs, S2=X
s
vLsB†
s,(A.5)
S3=X
s
vRsBs, S4=X
s
vRsB†
s,(A.6)
B1=X
l
b†
l, B2=X
l
bl,(A.7)
B3=X
r
b†
r, B4=X
r
br.(A.8)
The bath correlation function has the following nonzero
matrix elements G12, G21, G34 , G43, which can be easily
computed: correlation functions for the left bath are
G12(τ) = X
l
ei
~lτnl,(A.9)
G21(τ) = X
l
e−i
~lτ(1 + nl) (A.10)
and likewise, the correlation functions for the right bath
are
G34(τ) = X
r
ei
~rτnr,(A.11)
G43(τ) = X
r
e−i
~rτ(1 + nr).(A.12)
Substituting these correlation functions and operators
into the general Redfield master equation (A.3) and using
rotating wave approximation we get
i~˙ρ= [(HW+X
s
∆L(s)B†
sBs+X
s
∆R(s)B†
sBs, ρ]
−iX
s,α=L,R
Γα(s)h1
2(1 + 2nα(s)){B†
sBs, ρ}
−(1 + nα(s))BsρB†
−nα(s)B†
sρBs+nα(s)ρi.(A.13)
Here the Lamb shifts due to left (α=L) and right
(α=R) baths are
∆α(s)=[vαs]2X
k∈α
s−k
(k−s)2+ν2(A.14)
and the level broadening functions are
Γα(s)=2π[vαs]2ρα(s).(A.15)
Appendix B: Practical calculations
Outlined here are the key steps for practical calculations
to obtain observables from the model.
Inputs: T– temperature , N– length (number of mem-
branes), Rii= 1, . . . , N radius, v0, q – coupling between
membranes, γL/R = 2πt2ρL/R – couplings between the
end membranes and reservoirs.
– Set up the Hamiltonian for the wire. Compute fre-
quencies of intrinsic vibrations of the membrane
(α= 1, . . . , N ,k= 1, . . . , Kα)
Eαk =~λαkc, (B.1)
where λαk is determined from zeroes of Bessel func-
tion J0(λαkRα) = 0. Here Kαis the natural cut-off
for internal vibrations of membrane α. This natural
cutoff is in analogue to a Debye cutoff introducing
the discrete atomic structure and its restriction on
higher energy modes.
HW=
N
X
α
Kα
X
k=1
Eαkb†
αkbαk
+v
N−1
X
αX
kk0hb†
αkbα+1k0+b†
α+1kbαk0i.(B.2)
– Form and diagonalize the Hamiltonian matrix Form
matrix hαk,α0k0– it has dimension (K1K2. . . KN)×
(K1K2. . . KN)
HW=X
αk X
α0k0
hαk,α0k0b†
αkbα0k0.(B.3)
Page 12 of 12 Eur. Phys. J. B (2019) 92: 43
Diagonalize the Hamiltonian by solving the eigen-
value problem
X
α0k0
hαk,α0k0Uα0k0,s =sUαk,s.(B.4)
– Evaluate the master equation normalisation and dis-
sipation terms in the broad band limit so that
∆L,R(s)→0 and the reservoir couplings are
ΓL(s) = γL
K1
X
k=1
U1k,s ΓR(s) = γR
KN
X
k=1
UNk ,s.
(B.5)
– Compute heat conductivity and other observables
using analytical formulae (nS=K1K2. . . KN)
κ=L
~kBT2A
nS
X
s=1
2
s
ΓL(s)ΓR(s)
ΓL(s)+ΓR(s)
es/kBT
(es/kBT−1)2.
(B.6)
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