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11institutetext: University of Lethbridge,
4401 University Drive, Lethbridge AB, Canada, T1K3M4

# Cooling by Thermodynamic Induction

S. N. Patitsas

(Received: date / Accepted: date)

###### Abstract

A method is described for cooling conductive channels to below ambient
temperature. The thermodynamic induction principle dictates that the
electrically biased channel will cool if the electrical conductance decreases
with temperature. The extent of this cooling is calculated in detail for both
case of ballistic and conventional transport with specific calculations for
carbon nanotubes and conventional metals, followed by discussions for
semiconductors, graphene, and metal-insulator transition systems. A theorem is
established for ballistic transport stating that net cooling is not possible.
For conventional transport net cooling is possible over a broad temperature
range, with the range being size-dependent. A temperature clamping scheme for
establishing a metastable nonequilibrium stationary state is detailed and
followed with discussion of possible applications to on-chip thermoelectric
cooling in integrated circuitry and quantum computer systems.

###### Keywords:

Electronic transport Thermoelectric effects Low temperature Thermodynamic
induction

††journal: Journal of Low Temperature Physics

##  1 Introduction

Many major advances in physics have come about after the development of new
cooling techniques. Examples include the discovery of superconductivity and
superfluidity after the development of techniques for liquifying gases such as
hydrogen, oxygen, and helium, and more recently the discovery of gaseous Bose-
Einstein condensation after the techniques of Doppler cooling and evaporative
cooling were developed Pobell ; Schawlow1975 ; Wineland1978 ; Wieman1995 ;
Ketterle1996 .

Here, I present a new way to achieve cooling which exploits the thermodynamic
induction (TI) effect. How this induction arises in a dissipative
nonequilibrium system has been described previously Patitsas2014 ;
Patitsas2015 . In such systems, certain subsystems which have a gating
property can be induced away from thermodynamic equilibrium in unexpected ways
Patitsas2014 . The rate of entropy generation plays an important role in the
theory and the analysis shows that there exists an important energy scale
given by the product of the rate of entropy production, the characteristic
time scale for fluctuations, and the temperature. This energy scale influences
the system dynamics and gives rise to TI. In fact for certain far-from-
equilibrium situations this energy scale exceeds all of the other energy
scales, the direct Hamiltonian approach for describing the system dynamics is
difficult to implement, and the dissipative dynamical approach based on
entropy production provides the best approach for describing the induction
effects that emerge in these systems. For example, entropy production
principles are believed to play an important role in the way individual atoms
are manipulated by the tip of a STM Patitsas2015 .

I show that in certain thermoelectric systems the gate can be induced to
achieve lower temperature than otherwise expected. This effect is not
described within linear nonequilibrium theory and may be thought of as a
nonlinear complement to traditional Peltier cooling AM . It is worth exploring
if TI can be exploited towards achieving a practical method for cooling.
Results presented here indicate that this may indeed be the case for certain
systems. In Section 2 the basic theory for TI is briefly reviewed, before
carefully describing the physical system in Section 3. This leads to the main
stationary state result presented in Section 4. The case of pure ballistic
transport is treated in Section 5 with numerical calculations on zigzag carbon
nanotubes, and then conventional transport with calculations for highly
purified silver and copper samples are presented in Section 6. Finally I
discuss my findings and conclude.

##  2 Thermodynamic Induction Theory

For general TI theory, one may refer to Patitsas2014  for the case of $n$
variables and for the continuum limit see Ref. Patitsas2015 . For the
theoretical development here, a discrete 2-variable approach will suffice for
the most part, so this will be briefly developed here. I emphasize that the
dynamical equations describing the approach to equilibrium for these two
variables are uncoupled to first order, i.e., the two Onsager coefficients
$L_{12}=L_{21}$ are zero. It is in the second order with nonlinear terms that
TI effects spring forth and create the interesting coupling that is at the
heart of this work.

These variables $x_{1}$ and $x_{2}$ are thermodynamic in nature with
equilibrium values $x_{1_{0}}$ and $x_{2_{0}}$. The difference variables
$a_{1}=x_{1}-x_{1_{0}}$ and $a_{2}=x_{2}-x_{2_{0}}$ are used to make an
expansion of the total entropy $S_{T}$ with small deviations from equilibrium
as

$\Delta S_{T}\equiv
S_{T}(a_{1},a_{2})-S_{T}(0,0)=-\frac{1}{2}\sum_{p=1}^{2}\sum_{q=1}^{2}c_{pq}^{-1}a_{p}a_{q}~{}.$
(1)

Generalized thermodynamic forces (affinities) are defined by
$X_{i}=\frac{\partial\Delta S_{T}}{\partial a_{i}}$ mazur . These conjugate
parameters are interrelated as:

$a_{i}=-\sum_{j=1}^{2}c_{ij}X_{j}~{},$ (2)

where $c_{12}=c_{21}$, i.e., the generalized capacitance matrix $c_{ij}$ is
symmetric and positive definite, thus ensuring thermodynamic stability. These
matrix elements also govern the extent of fluctuations mazur . For example, in
the decoupled case where $c_{12}=0$ then $k_{B}c_{22}$ is the root-mean-square
deviation of the probability distribution for $a_{2}$, i.e,

$P_{0}(a_{2})=c~{}\exp{\left(\frac{-a_{2}^{2}}{2k_{B}c_{22}}\right)}~{}\,,$
(3)

where $c$ is a normalization constant. If one or both of these variables is
pushed away from equilibrium then the dynamics is concerned with the approach
back to equilibrium. Such dynamics are often governed by transport processes
such as thermal conductivity, electrical conductivity, etc. These transport
processes are irreversible: the total system entropy increases with time as
the system approaches equilibrium. Note that

$\frac{dS_{T}}{dt}\equiv\dot{S_{T}}=\sum_{p=1}^{2}X_{p}\dot{a}_{p}~{}.$ (4)

In Eq. (4) the time derivative $\dot{a}_{p}$ must be handled with care. A
proper dynamical approach uses coarse-grained averages of the time
derivatives. For variable $x_{i}$ the time step $\Delta t_{i}$ chosen for the
dynamics must be much larger than the relevant correlation time $\tau_{i}^{*}$
for fluctuations. Fluctuations play an important role and are the ultimate
source of TI. They are also the fastest process in this analysis, with
$\tau_{i}^{*}$ being a measure of the width of the fluctuation force
correlation function, typically very short Reif . We closely follow the
notation in Ref. mazur , and use a bar to denote the coarse-grained time
derivative as $\bar{\dot{a}}_{i}$. The definition is

$\bar{\dot{a}}_{i}\equiv\frac{1}{\Delta t_{i}}\int_{t}^{t+\Delta
t_{i}}{\langle\dot{a}_{i}\rangle dt^{\prime}}=\frac{1}{\Delta t_{i}}\langle
a_{i}(t+\Delta t_{i})-a_{i}(t)\rangle~{}.$ (5)

Note that the symbols $\langle~{}\rangle$ denote ensemble averages over
accessible microstates of the system. Addition of the subscript, 0, refers to
equilibrium states in particular.

To evaluate this coarse-grained derivative I closely follow the approach in
Ref. Reif . In this approach, a nonequilibrium average value is determined by
evaluating an equilibrium average accompanied by the insertion of the
ubiquitous $\exp(\Delta S_{T}/k_{B})$ probability weighting factor mazur ;
LandauL . We emphasize that this must be the total system entropy change.
Pedagogically, this approach is known to establish the Onsager symmetry
relations Reif . Explicitly then:

$\langle\dot{a}_{i}(t^{\prime})\rangle=\left\langle\dot{a}_{i}(t^{\prime})e^{\frac{\Delta
S_{T}(t^{\prime}-t)}{k_{B}}}\right\rangle_{0}~{},$ (6)

where $\Delta S_{T}(t^{\prime}-t)\equiv S_{T}(t^{\prime})-S_{T}(t)$. The
physical interpretation is simple; For a given state at time $t$, every
possible outcome at time $t^{\prime}$ is considered with each outcome
statistically weighted by the number of accessible system microstates. Since
the change in entropy is small, a linear expansion is warranted, leaving:

$\langle\dot{a}_{i}(t^{\prime})\rangle=\frac{1}{k_{B}}{\langle\dot{a}_{i}(t^{\prime})\Delta
S_{T}(t^{\prime}-t)}\rangle_{0}~{},$ (7)

since $\langle\dot{a}_{i}\rangle_{0}=0$. Integrating both sides of Eq. (7)
over the time interval $\Delta t_{i}$ gives the coarse-grained time
derivative:

$\bar{\dot{a}}_{i}=\frac{1}{k_{B}\Delta t_{i}}\int_{t}^{t+\Delta
t_{i}}{dt^{\prime}\int_{t}^{t^{\prime}}{dt^{\prime\prime}\langle\dot{a}_{i}(t^{\prime})\dot{S_{T}}(t^{\prime\prime})\rangle_{0}}}\,.$
(8)

At this point a distinction is made between the two variables. Variable 1 will
be designated as the dynamical reservoir (DR) and variable 2 is the gate (GT)
variable. The DR variable $x_{DR}$ approaches equilibrium slowly and with a
virtually limitless supply. An example is the charge variable in an RC circuit
with a very large capacitance. The gate variable $x_{GT}$ controls the slower
DR variable by having the conductance of the DR depend on the state of the
gate. Following Ref. Patitsas2014  the dynamics for the DR is straightforward
and given by

$\dot{a}_{DR}=M_{11}X_{DR}\,.$ (9)

By Eqs. (4) and (9), entropy is created by the DR at a rate
$\dot{S}_{DR}=M_{11}X_{DR}^{2}$. If $M_{11}$ is constant then this is a
standard result Reif . Also, as mentioned above, a linear coupling term of the
form $L_{12}X_{GT}$ is not present. If it were present, then the nonlinear
analysis presented below would merely provide small, second order corrections
to the dynamics. It cannot be emphasized enough that the analysis presented
below will provide a potent nonlinear dynamical coupling between otherwise
uncoupled variables. Here, it is not the case that $M_{11}$ is constant, and
$M_{11}$ depends on the gate variable $x_{2}=x_{GT}$. The simplest way to
express this dependence is

$M_{11}=L_{11}+\gamma a_{GT}\,,$ (10)

where $L_{11}$ and $\gamma$ are constants. Using Eq. (4) the total rate of
entropy production has contribution given by
$L_{11}X_{DR}^{2}+L_{22}X_{GT}^{2}$ mazur , as well a term coming from the
nonlinear part of Eq. (10) which is given by:

$\dot{S}_{T_{nonlin}}=\gamma X_{DR}^{2}a_{GT}\,.$ (11)

When applying Eq. (8) to the gate variable, one recovers the expected linear
term $L_{22}X_{2}$, but by implementing Eq. (11) one also finds Patitsas2014
the interesting induction term, to give

${{\dot{a}}}_{GT}=L_{22}X_{GT}+\tau_{GT}^{*}\gamma L_{22}X_{DR}^{2}\,.$ (12)

The fluctuation time $\tau_{GT}^{*}$ plays an important role throughout this
work. This time comes out of the time integration for the linear term and gets
absorbed into the transport coefficient $L_{22}$. For the nonlinear term the
extra time integral gives the extra factor of $\tau_{GT}^{*}$, prominent in
the second term on the right hand side of Eq. (12), and for this reason
$\tau_{GT}^{*}$ finds itself in so many of the equations which follow.

A stationary state for the gate is defined by setting ${{\dot{a}}}_{GT}=0$
which gives

$X_{GT_{ss}}=-\tau_{GT}^{*}\gamma X_{DR}^{2}\,,$ (13)

and

$a_{GT_{ss}}=\tau_{GT}^{*}c_{22}\gamma X_{DR}^{2}\,.$ (14)

Stationary states have interesting properties. Compared to the situation with
$a_{GT}=0$, entropy production by the DR is increased from the rate
$L_{11}X_{DR}^{2}$ by a positive definite amount

$\dot{S}_{extra}=\tau_{GT}^{*}c_{22}\gamma^{2}X_{DR}^{4}~{}\,,$ (15)

so that

$\dot{S}_{DR}=M_{11}X_{DR}^{2}=L_{11}X_{DR}^{2}+\gamma
a_{GT}X_{DR}^{2}=L_{11}X_{DR}^{2}+\dot{S}_{extra}~{}\,.$ (16)

Also, the entropy of the gate changes from the equilibrium value by an amount:

$\Delta{S}_{GT}=-\tau_{GT}^{*}\dot{S}_{extra}~{}\,.$ (17)

This drop in entropy can mean many things; Here, under the right conditions,
the gate will cool, in the stationary state, to below-ambient temperature.

##  3 Thermoelectric System

Figure 1:  Schematic circuit diagrams depicting a conductive channel of length
$l$, thickness $t$, connecting a cathode to an anode. The channel has
rectangular cross section $w\times t$ ($w$ into page). Electron flow
associated with the dynamical reservoir is depicted by a thick solid arrow.
Dashed arrows depict the thermodynamically-induced heat flow away from the
channel for the case of negative $\gamma$. In case (a) the heat flow is
perpendicular to the electron flow, as heat is deposited into the electrically
insulating cladding layers. In case (b) heat flows from the channel and into
the cathode and anode.

Specifically, I analyze the systems depicted in Fig. 1, each of which
experiences two nonequilibrium processes, one concerned with electrical
transport and the other with thermal transport. Process 1, electrical
transport through a conductive channel, will play the role of the DR. More
specifically, the DR is not just the physical system of cathode, anode, emf
source, and conductive channel, but is also an associated process, here of
electronic transport.

The thermal transfer process will have a gating effect on the electrical
conductance of the reservoir, simply by having the conductive channel’s
electrical conductivity, $\sigma$, depend on temperature. The channel is the
most important component of these systems and must also conduct heat into or
out of an adjacent heat bath. The channel connecting anode and cathode across
a distance $l$ has rectangular cross section $w\times t$ and is much narrower
than the leads, thus forming an electrical bottleneck. The thickness dimension
$t$ is considered to be small compared to the size of the leads, but the depth
$w$ is not necessarily small. Similar to the DR, the gate is not just the
physical system of the channel shown in Fig. 1. The gate subsystem is
comprised of the channel, the anode and cathode, an associated process of heat
transfer, and a thermal bath to absorb or release heat. In the case of Fig.
1(a) the two electrically-insulating cladding layers compose the bath, while
in the case of Fig. 1(b) the channel is free-standing and the cathode and
anode together form the bath. Qualitatively, the effect of TI is that when
bias is applied, heat will flow either into or out of the channel, always in a
way as to increase the electrical conductance of the channel. As pointed out
in Refs. Patitsas2014 ; Patitsas2015  the effects of TI are imperceptible in
many real systems. Thus, the analysis presented here focuses on finding
conditions where the effects of TI are detectable.

In this discrete thermodynamic model, $x_{DR}$ is the number of electrons in
the anode Patitsas2014 . The thermodynamic force conjugate to this variable is
known to be simply related to the change in electron chemical potential $\mu$
relative to the cathode:

$X_{DR}\equiv\frac{\partial\Delta S_{T}}{\partial
x_{DR}}=-\frac{\Delta\mu_{DR}}{T}=+\frac{e}{T}W\,,$ (18)

where $W$ is the applied electric potential difference Patitsas2015 ; mazur .
Since the electrical current is $I=e\dot{x}_{DR}$ then Eq. (9) represents
Ohm’s law and the channel conductance $G$ is related to the Onsager
coefficient $M_{11}$ as Patitsas2015 :

$M_{11}=\frac{T}{e^{2}}G\,.$ (19)

Ignored at this level of analysis is the corresponding flow of thermal energy
caused by the electric field inside the channel AM ; mazur . Though this is a
valid concern, as long as appreciable thermal gradients in the channel are not
created, this energy flow does not change the internal energy of the channel
in steady state and one can safely ignore it. Also, it’s worth noting that
these thermoelectric effects are less significant at lower temperatures AM .
The gating property that is crucial to this analysis involves thermal
transport, though in a different way. This type of thermal transport is
indicated by the dashed arrows in Fig. 1. Heat flow perpendicular to the
electrical current is possible if electrically insulating (cladding) material
is present directly adjacent to the channel, as shown in Fig. 1(a). In Fig.
1(b) the direction of induced thermal transport is into or out of the anode
and cathode. Specifically, the variable $x_{GT}$ is the internal energy $U$ of
the conducting channel, and $a_{GT}$ is the change from the equilibrium value.
The conjugate force can be related to the temperature difference, $\Delta T$,
between the channel and the ambient temperature, $T$, of the electrical leads,
as Patitsas2015 ; mazur

$X_{GT}=-\frac{1}{T^{2}}\Delta T\,.$ (20)

The $c_{22}$ coefficient is directly related to the heat capacity $C_{GT}$ of
the channel by the relation:

$a_{GT}=-T^{2}C_{GT}X_{GT}~{}.$ (21)

When this thermal subsystem is out of equilibrium, the thermal flux is given
by $L_{22}X_{GT}=-h\Delta T$ where $L_{22}$ is the Onsager coefficient and $h$
is the thermal conductance of the channel Reif . The relaxation time scale for
this thermal transport is

$\tau_{GT}=\frac{C_{GT}T^{2}}{L_{22}}=\frac{C_{GT}}{h}~{}\,,$ (22)

and is not to be confused with the (shorter) fluctuation time scale
$\tau_{GT}^{*}$. This conventional heat flow, not illustrated in Fig. 1, tends
to oppose the induction effect and creates a balance after the stationary
state is achieved, as discussed further, below.

The gate and the DR become coupled when $M_{11}$ depends on the gate variable,
as described by Eq. (10). Using Eq. (19), the induction coefficient $\gamma$
defined by Eq. (10) can be expressed as

$\gamma=\frac{T}{e^{2}C_{GT}}\frac{\partial G}{\partial T}~{},$ (23)

where $c_{V}$ is the volume specific heat capacity of the channel material. It
is helpful to define a dimensionless version of $\gamma$ as:

$\kappa\equiv\frac{\partial\ln G}{\partial\ln T}=\frac{e^{2}C_{GT}}{G}\gamma.$
(24)

The dynamical equations for the approach to equilibrium are also affected by
simple Joule heating of the channel. For the DR the dynamics are still
determined by Eq. (9). For the gate, Eq. (12) is modified to become

${{\dot{a}}}_{GT}=P_{J}+L_{22}X_{GT}+\tau_{GT}^{*}\gamma L_{22}X_{DR}^{2}\,,$
(25)

where $P_{J}=GW^{2}$ is the Joule heating. The characteristic time scale
$\tau_{GT}^{*}$ for thermal fluctuations is generally very short Patitsas2014
. Here we consider only the case where the applied bias $W$ is slowly varying,
so the first and third terms in the right-hand side of Eq. (25) may be treated
as essentially constant. The induction term is the third term in the right-
hand side of Eq. (25), i.e., the existence of an induced heat flux given by

$J_{ind}=e^{2}h\tau_{GT}^{*}\gamma W^{2}=\frac{\tau_{GT}^{*}}{\tau_{GT}}\kappa
P_{J}\,.$ (26)

It is important to note that for negative $\gamma$ the channel is cooled with
a power level given by $P_{cool}=-J_{ind}$. For many systems the ratio, $r$,
of gate fluctuation time to relaxation time is very small. When looking for
systems that might exhibit measurable TI cooling one would therefore look at
systems with very fast relaxation times. These systems might typically be
small in physical size; In fact for very small systems with a gating
characteristic relaxation rates approaching $1/\tau_{GT}^{*}$ are certainly
possible. In these systems, net cooling, where $P_{cool}$ exceeds $P_{J}$, is
a possibility if $\kappa<-1$. This important point will be re-visited below in
the context of both conventional and ballistic transport.

##  4 Stationary State

In the stationary state, ${{\dot{a}}}_{GT}=0$, and a temperature difference
$\Delta T$ is achieved for the channel relative to the anode and cathode which
are held at ambient temperature $T$. The net change $\Delta T$ is the sum of
two contributions $\Delta T_{ind}$ from TI and $\Delta T_{J}$ from Joule
heating. From here on I analyze specifically the bare channel system
illustrated in Fig. 1(b) where heat flows between the channel and the leads
only. Making use of Eqs. (23, 24, 25, 26) one obtains

$\Delta T_{ind}=\tau_{GT}^{*}\gamma
e^{2}W^{2}=\frac{\kappa\tau_{GT}^{*}GW^{2}}{C_{GT}}\,,$ (27)

which describes the effect of TI cooling. Accounting for Joule heating leads
to a fully-balanced stationary state as
$P_{cool}-P_{J}=-h\left(T_{GT}-T\right)$, or:

$T_{GT}-T=\left(1+r\kappa\right)\frac{GW^{2}}{h}\,.$ (28)

One surprising ramification of Eq. (27) is the possibility of $\Delta T$ being
negative under some conditions. This may at first seem counterintuitive, i.e.,
for one to run current through a conductor and have it get cooler. Equation
(17) accounts for this possibility and it is clear that only part of the total
system has reduced entropy. Without the DR, having $\Delta S_{GT}<0$ would
violate the 2nd law of thermodynamics. The reservoir concept becomes evident;
The DR produces entropy at such a great rate that subsystems are allowed to
lend, or dump, positive amounts of entropy production to the DR. There are,
however, limits to how far this lending can go. The gate cannot continue to
get further and further from equilibrium as the $L_{22}X_{GT}^{2}$ entropy
production term builds up as the magnitude of $X_{GT}$ gets larger.
Eventually, balance is achieved at the stationary state. Never, during the
formation of the stationary state, is the total entropy production rate
negative. In fact, induction cooling actually enhances the total rate of
entropy production to a more positive value. This is the physical meaning of
Eq. (15) which itself constitutes an example of Theorem 2 from Ref.
Patitsas2014  and can be thought of as a dynamical, irreversible, version of
Le Chatelier’s principle. In Le Chatelier’s principle, a variable is chosen to
be displaced away from equilibrium and other variables relax in such a way as
to reduce the magnitude of change in the force conjugate to the chosen
variable LandauL . The change in the system entropy is also reduced in
magnitude by the relaxation. Internal degrees of freedom respond in such a way
as to push the system closer towards full equilibrium. In the irreversible
case, the DR plays the role of the chosen variable. Other variables, here the
gate, respond in such a way as to increase the rate of entropy production by
the gate, and to get the entire system to equilibrium faster Patitsas2014 . In
fact, under the stationary restriction, ${{\dot{a}}}_{GT}=0$, the total
entropy production of the system is maximized. Clearly there is more to these
stationary states than merely being stationary. There is also the physical
interpretation of the state as one that facilitates, or induces, the approach
to equilibrium for the entire system. In this case, internal degrees of
freedom respond in such a way as to push the system faster towards full
equilibrium.

To further understand these stationary states, or inducer states, it’s helpful
to look at the probability distribution for the gate variable. One takes the
equilibrium Gaussian function $P_{0}$ from Eq. (3) and multiplies by
$\exp{\left(\dot{S}_{DR}\tau_{GT}^{*}/k_{B}\right)}$. Using Eq. (16), this
gives

$P(a_{GT})=c^{\prime}~{}\exp{\left(\frac{-a_{GT}^{2}}{2k_{B}c_{22}}+\frac{\tau_{GT}^{*}\gamma
X_{DR}^{2}a_{GT}}{k_{B}}\right)}~{}\,,$ (29)

where $c^{\prime}$ is a normalization constant. By inspection, this
distribution is skewed and the mean value is readily calculated to give the
same result as in Eq. (14). The ubiquitous $\tau_{GT}^{*}$ factor suggests a
physical interpretation as a fundamental time step. After each such time step
the DR entropy increases by an amount $\Delta
S_{DR}=\dot{S}_{DR}\tau_{GT}^{*}$ and the number of accessible microstates
increases by a factor of $\exp{\left(\Delta S_{DR}/k_{B}\right)}$. Sampling
all microstates with equal weight is the essential idea behind Eq. (6), and it
is not surprising that the probability distributions for some variables become
skewed. In the short time step $\tau_{GT}^{*}$ some values of $a_{GT}$ will
facilitate/induce higher levels of entropy production by the DR, and this will
give more statistical weight to those $a_{GT}$ values. It’s worth pointing out
that this simple time step interpretation seems to hold even though
$\tau_{GT}^{*}$ is below the expected threshold for coarse graining. It’s also
worth mentioning that even though $\tau_{GT}^{*}$ is very small, the factor
$\exp{\left(\dot{S}_{DR}\tau_{GT}^{*}/k_{B}\right)}$ can be much larger than
unity under the right conditions. It’s also helpful to add in some reasoning
commonly applied to chemical systems, since it’s well known that for a
chemical reaction, the $\Delta S$ value for the reaction can play an important
role in guiding the outcome. For this reason these values are measured and
tabulated for many reactions. Here, we apply that idea for each time step, and
the $\Delta S$ repeatedly guides and skews the outcome Patitsas2014  via Eq.
(6). Thermodynamic induction has recently been used as an attempt to explain
certain STM-based manipulation techniques, for example why Xe atoms can be
dragged so effectively across cold metal surfaces Patitsas2015 . The chemical
thinking just described seems to apply well to the adsorbate-dragging case. I
believe the same type of thinking is applicable to TI cooling as well. Despite
the entropy penalty given by Eq. (17), the DR conductance is greater and the
system overall is allowed to approach equilibrium faster by having the gate
induced to lower temperature. The time step interpretation also helps to
understand Eq. (27), since during each time step the DR dissipates an amount
of energy equal to $\tau_{GT}^{*}GW^{2}$. A fraction $\kappa$ of that energy
results in the induced temperature change. Finally, since the factor
$\exp{\left(\dot{S}_{DR}\tau_{GT}^{*}/k_{B}\right)}$ applies to a
thermodynamic variable it might also apply to individual microstates for the
gate. If these microstates are labelled $i$ then I postulate the following
probability distribution to be applied as a weighting factor for each such
microstate:

$P_{i}=\frac{1}{Z}e^{\tau_{GT}^{*}\dot{S}_{i}/k_{B}}\,,$ (30)

where $\dot{S}_{i}$ is the rate of change of the DR entropy when the gate is
in state $i$, and $Z$ is a normalization factor. I propose that this
probability distribution could have wide application in analyzing
nonequilibrium systems beyond the type of cooling considered here. This
weighting factor would be factored into general thermodynamic calculations
much like the Boltzmann factor is in equilibrium studies.

At this point, analysis of specific systems is in order. As it turns out,
analysis of conventional transport at low temperatures leads naturally to
analysis of ballistic transport so I will treat the ballistic case first.

##  5 Ballistic Transport

At very low temperatures, carrier scattering becomes less important in
determining the electrical conductance, and the boundaries of the conductive
channel assume a prominent role Datta2005 . In the absence of impurity
scattering, the channel conductance $G=I/W$ is determined by an approach taken
by Landauer which gives the electrical current as a summation over all
electronic states, with weighting in the summation given by the transmission
$\bar{T}$ for each such state, as well as the occupancy for both anode and
cathode given by the Fermi-Dirac distribution, $f_{0}(x)=(1+e^{x})^{-1}$
Datta2005 ; Landauer1970 ; Anderson1981 ; Azbel1984 ; Landauer1985 . The
Landauer formula specifies the current $I$ in terms of the applied bias $W$ as

$I=\frac{2e}{h}\int_{-\infty}^{\infty}dE\,\bar{T}(E)\left[f_{0}(\beta E-\beta
eW)-f_{0}(\beta E)\right]\,,$ (31)

where $\beta=1/k_{B}T$. Note that $\bar{T}$ includes summation over subbands
arising from the three dimensional nature of the channel. This function also
includes information on band edges. Even in the case where there is no
tunneling barrier to inhibit carrier transmission, $\bar{T}$ will be zero for
energies in band gaps. When the temperature is low enough, the step-like
nature of $f_{0}$ can be exploited to extract the transmission function as

$\bar{T}\approx\frac{h}{2e^{2}}\frac{\partial I}{\partial W}\,.$ (32)

The temperature dependence of $I$ and hence $G$ being contained in the two
instances of the $f_{0}$ function allows for straightforward calculation of
$\gamma$ using Eq. (23). If $\bar{T}(E)$ is constant over all energies then by
inspection of Eq. (31), the current is independent of temperature. This
implies a complete absence of thermodynamic induction. This observation has
significant bearing on the following analysis. For systems to exhibit
induction effects, the transmission must have structure. The existence of band
edges, resonant tunneling, and subbands provide some ways to produce such
important structure.

The dimensionless parameter $\kappa$ is a good measure of the extent of TI.
Using Eq. (24), then for fixed applied bias

$\kappa=\frac{T}{I}\frac{dI}{dT}=-\frac{\beta}{I}\frac{dI}{d\beta}\,,$ (33)

$-\beta\frac{\partial
I}{\partial\beta}=\frac{2e}{h}\int_{-\infty}^{\infty}dE\,\bar{T}(E)\left[g(\beta
E-\beta eW)-g(\beta E)\right]\,,$ (34)

where $g(x)=-xf_{0}^{\prime}(x)=x[4\cosh^{2}(x/2)]^{-1}$.

###  5.1 Slowly Varying Transmission Function

Integration by parts in Eq. (34) gives:

$-\beta\frac{\partial
I}{\partial\beta}=\frac{2e}{h\beta}\int_{-\infty}^{\infty}dE\,\frac{d\bar{T}}{dE}\left[+h(\beta(E-eW))-h(\beta(E))\right]\,,$
(35)

where

$h(x)=-\frac{xe^{x}}{1+e^{x}}+\ln(1+e^{x})=\ln(\cosh\frac{x}{2})-\frac{x}{2}\tanh\frac{x}{2}+\ln
2\,.$ (36)

The function $h(x)$ is even and peaked at the origin with inflection points
near $x=\pm 1.534$ at which the function falls to 67% of its peak value. For
large $x$, $h(x)$ falls off rapidly as $xe^{-x}$. The area under the curve is
$\pi^{2}/3$.

If $\bar{T}$ varies slowly with energy (compared to $k_{B}T$) then the $h$
functions in Eq. (35) will act like filter functions with the the result:

$-\beta\frac{\partial
I}{\partial\beta}=\frac{2\pi^{2}e}{3h\beta^{2}}\left[\frac{d\bar{T}}{dE}(eW)-\frac{d\bar{T}}{dE}(0)\right]\,,$
(37)

This confirms that $\gamma$ will be small when $\bar{T}$ varies slowly. A
linear term correcting for slowly-changing variations in an otherwise constant
transmission will still not result in any TI. If $W$ is small enough such that
the transmission does not vary appreciably over the range 0 to $eW$, then the
channel conductance is given by $G=e^{2}\bar{T}(0)/h$ and the dimensionless TI
parameter $\kappa_{ball}$ is given by

$\kappa_{ball}=\frac{\pi^{2}k_{B}^{2}T^{2}}{3\bar{T}(0)}\frac{d^{2}\bar{T}(0)}{dE^{2}}\,.$
(38)

####  5.1.1 Low Temperature Limit

When $k_{B}T$ is smaller than the energy scale at which the finest structure
is found in the transmission, then the transmission will appear to be slowly
varying and Eq. (37) will apply. Thus, in all systems at very low
temperatures, $\kappa_{ball}$ goes as $T^{2}$ and TI becomes progressively
weaker as the temperature is lowered.

###  5.2 Rapidly Varying Transmission Function

To find cases where the effects of TI will be strong in ballistic systems, it
makes good sense to look for systems where $\bar{T}$ varies rapidly, for
example, as a sharp peak or a sharp step.

####  5.2.1 Sharp Peaks

Here I first consider the transmission to take the form
$\bar{T}(E)=c\delta(E-E_{p})$, using the Dirac delta function. By Eq. (31) the
current is given by

$I=\frac{2ec}{h}\left[f_{0}(\beta E_{p}-\beta eW)-f_{0}(\beta
E_{p})\right]\,,$ (39)

and by Eq. (34)

$-\beta\frac{\partial I}{\partial\beta}=\frac{2ec}{h}\left[g(\beta E_{p}-\beta
eW)-g(\beta E_{p})\right]\,.$ (40)

For the following analysis, the applied bias $W$ is taken as a positive
quantity, with no loss in generality.

In the limiting case where $\beta eW<<1$, Eq. (39) gives

$I=-\frac{2ec}{h}\beta eWf_{0}^{\prime}(\beta E_{p})\,,$ (41)

and Eq. (40) gives

$-\beta\frac{\partial I}{\partial\beta}=-\frac{2ec}{h}\beta eWg^{\prime}(\beta
E_{p})\,.$ (42)

The ratio gives

$\kappa_{ball}=\frac{g^{\prime}}{f_{0}^{\prime}}=-1-\frac{xf_{0}^{\prime\prime}}{f_{0}^{\prime}}=-1+\beta
E_{p}\tanh(\beta E_{p}/2)\,.$ (43)

Thus $\kappa_{ball}=-1$ is achieved if $E_{p}=0$, and -1 is the global
minimum.

When the sharp peak at $E=0$ is accompanied by another distinct and equally
weighted peak at higher energy then one can show that at $eW$ values near this
2nd peak $\kappa_{ball}=g^{\prime}/(0.5+f_{0}^{\prime})$. The lowest
$\kappa_{ball}$ value of $\approx-0.1706$ is achieved at an applied bias
$\approx 1.384~{}k_{B}T/e$ above the peak position. Finally, for the non-
metallic case of just one peak well-removed from $E=0$,
$\kappa_{ball}=g/f_{0}$ and the lowest $\kappa$ value of $\approx-0.2785$ is
achieved at an applied bias $\approx 1.2785~{}k_{B}T/e$ above the peak
position.

To summarize, I have arrived at three general-purpose rules (the -1 rule, the
-0.17 rule and the -0.28 rule) for predicting $\kappa_{ball}$ values
favourable for TI cooling when the transmission has a sharp peaked structure.

####  5.2.2 Sharp Steps

Here I consider specifically the case where $\bar{T}=1$ at energies below
$E_{S}>0$ above which it equals 2. From Eq. (31), $I=e^{2}W/h$ right up to
$W=E_{s}/e$, while Eq. (35) is used to show that $\partial I/\partial\beta$ is
very small along the flat section of the step, and that near the step edge

$-\beta\frac{\partial I}{\partial\beta}=\frac{2e}{h\beta}h(\beta E_{s}-\beta
eW)\,.$ (44)

This gives a strictly positive result $\kappa=(k_{B}T/E_{s})h(\beta
E_{s}-\beta eW)$ that peaks right at $W=E_{s}/e$ with value $+\ln 2$. Only
downward steps in $\bar{T}(E)$ give negative $\kappa$. Also, the magnitude of
$\kappa_{ball}$ will be small for steps located at energies much larger that
$k_{B}T$, because of the normalization by $I$.

###  5.3 General Result

Summarizing so far, negative and significant $\kappa$ values as low as -1 can
be produced when the transmission as a function of energy is highly peaked,
whereas, in contrast, a slowly varying transmission will not give significant
$\kappa_{ball}$ values. Here, a variational approach is taken to show that -1
is the lower bound for all possible transmission functions. Equation (31) can
be written concisely in the form
$I=\frac{2e}{h}\int_{-\infty}^{\infty}dE\phi(E)\bar{T}$ and
$-\beta\frac{\partial
I}{\partial\beta}=\frac{2e}{h}\int_{-\infty}^{\infty}dE\gamma(E)\bar{T}$. The
ratio $\kappa$ is a functional of $\bar{T}(E)$ and the variation
$\delta\kappa$ is given by

$\delta\kappa_{ball}=\frac{\int_{-\infty}^{\infty}dE\gamma\delta\bar{T}}{\int_{-\infty}^{\infty}dE\phi\bar{T}}-\frac{\int_{-\infty}^{\infty}dE\gamma\bar{T}}{\left[\int_{-\infty}^{\infty}dE\phi\bar{T}\right]^{2}}\int_{-\infty}^{\infty}dE\phi\delta\bar{T}\,.$
(45)

Setting $\delta\kappa_{ball}=0$ gives

$\frac{\int_{-\infty}^{\infty}dE\phi\delta\bar{T}}{\int_{-\infty}^{\infty}dE\phi\bar{T}}=\frac{\int_{-\infty}^{\infty}dE\gamma\delta\bar{T}}{\int_{-\infty}^{\infty}dE\gamma\bar{T}}\,.$
(46)

Since $\delta\bar{T}$ is arbitrary

$\phi(E)\int_{-\infty}^{\infty}dE^{\prime}\gamma\bar{T}=\gamma(E)\int_{-\infty}^{\infty}dE^{\prime}\phi\bar{T}\,,$
(47)

and since $\phi$ and $\gamma$ are different functions, this condition is
impossible to satisfy with any non-zero function except for the delta function
$\bar{T}=c\delta(E-E^{\prime})$.

This establishes a proof that I express as a theorem for TI under conditions
of ballistic transport:

Theorem:

For nonequilibrium electrical systems governed by ballistic transport,
$\kappa_{ball}\geq-1$.

The extremal value $\kappa_{ball}=-1$ is only achieved when the applied bias
$W$ is small, and the transmission $\bar{T}(E)$ has the structure of a sharp
peak located at $E=0$.

When this theorem is combined with Eq. (26) and the physical input that
$\tau_{GT}^{*}<\tau_{GT}$ the following important corollary is established:

$J_{ind}\geq-P_{J}\,$ (48)

i.e.,

Corollary (no net cooling):

Net cooling is not possible when electrical transport is ballistic.
Thermodynamic induction can provide substantial cooling but it will always be
matched or exceeded by Joule heating, at least in the realm exclusive to
ballistic transport.

###  5.4 Subbands

For a one-dimensional open channel, the transmission $\bar{T}$ in the Landauer
formula, Eq. (31), is set to unity, which would seem to give
$\kappa_{ball}=0$, by Eq. (37). Some discussion on this matter is warranted in
regards to the band edges.

####  5.4.1 One Dimensional Systems

If the lateral dimensions of the channel are very small then the channel
effectively becomes a chain and subband summation is not required. Here this
condition is referred to as subband inactive. While technically a high enough
bias can always be applied to access subbands no matter how thin the chain is,
practically speaking such conditions are difficult to access when the chain
thickness is right down to atomic dimensions. In this subband inactive case
the current can be given in terms of a state sum as

$I_{1d}=\frac{2e}{h}\frac{2\pi}{L_{x}}\sum_{k_{x}}\,\bar{T}_{1d}(E)\frac{\partial
E}{\partial k_{x}}\left[f_{0}(\beta(E-eW))-f_{0}(\beta E)\right]\,,$ (49)

where $L_{x}=l$ is the length of the channel. The transmission $\bar{T}_{1D}$
is actually the transmission coefficient and would equal unity for an open
channel. Also implied here is a dispersion relation $E(k_{x})$ which includes
lower and upper band edges. The physical significance of $\frac{\partial
E}{\partial k_{x}}$ in Eq. (49) is the group velocity along the channel, and
it must be stressed that proper use of this only occurs in the continuum
limit.

The summation $\sum_{k_{x}}$ can be replaced by $\frac{L_{x}}{2\pi}\int
dk_{x}$ if the channel length is much longer than the size of the primitive
unit cell $a$, a condition referred to here as an extended channel. When this
integral is converted to one over energy, the group velocity factor is exactly
canceled by a density-of-states factor. If $E_{1}$ ($E_{2}$) denotes the lower
(upper) energy band edge, then for an open channel

$I_{1d}=\frac{2e}{h}\int_{E_{1}}^{E_{2}}dE\,\left[f_{0}(\beta(E-eW))-f_{0}(\beta
E)\right]\,.$ (50)

where we note the key difference to Eq. (31) is the existence of the limits at
$E_{1}$ and $E_{2}$. Making use of Eq. (35) I obtain

$\frac{\partial
I_{1d}}{\partial\beta}=\frac{2e}{h\beta}\,\left[h(\beta(E_{1}-eW))-h(\beta(E_{2}-eW))-h(\beta
E_{1})+h(\beta E_{2})\right]\,.$ (51)

This exact result works well for wide bands, so with band edge values much
larger in magnitude than both $k_{B}T$ and $eW$, $\gamma$ will be very small,
and hence TI effects will be very small. This result holds even if there is
considerable structure in the $E(k_{x})$ function within the band edges, as
mentioned, a result of the group velocity canceling the density of states
function. This result holds as long as the density of states is a continuous
function, i.e. for the extended channel. If the channel becomes short enough
then the density of states is a series of discrete peaks, and the analysis
presented earlier takes over. The important parameter to decide if a channel
density of states is continuous or discrete is the so-called coupling energy.
This parameter $\gamma_{c}$ describes the amount of coupling between a given
discrete state of the channel and the leads (anode and cathode), and may be
evaluated by calculation of the self-energy component of the single-particle
Green’s function of the contact (leads) coupled to the channel. Calculation of
this coupling would follow the nonequilibrium Green’s function formalism
Schwinger1959 ; Baym1962 ; Keldysh1965 . If the coupling energy exceeds the
typical energy spacing between peaks then the channel can be considered as
extended, and the density of states function as continuous.

A similar discussion applies to an extended channel containing a resonant
tunneling barrier. If the width in energy of the resonance is less than
$k_{B}T$ then the transmission will be considered to have a single sharp peak
and values for $\kappa_{ball}$ near -0.28 or -1 are possible, depending on the
position of the peak. Figure 2 shows $\kappa$ plotted as a function of applied
bias for an extended one-dimensional system with with band edges at $\pm 1.0$
eV. The resonances is described by a transmission given by

$\bar{T}=\frac{\gamma_{1}\gamma_{2}}{(E-E_{r})^{2}+\bar{\gamma}^{2}}$ (52)

where $\bar{\gamma}=(\gamma_{1}+\gamma_{2})/2$ Datta2005 . In this calculation
the two coupling parameters to each lead are taken to have the same value of
3.0 meV. In Fig. 2 plots are made for two values of $E_{r}$, 0.5 eV and 0 eV.
At 300 K temperature, the thick (blue) solid curve is symmetric about $E=0$
and dips down to -0.94, which agrees well with the -1 rule. At the lower
temperature of 30 K, the transmission no longer appears as sharp compared to
$k_{B}T$ and the extent of the dip is reduced to -0.57. Also the width of the
dip is much smaller in the thick (blue) dashed curve. Three calculations were
made for a resonance at +0.5 eV at 300 K (thin solid), 100 K (thin dashed),
and 300 K (thin dash-dot), as shown in Fig. 2. The 300 K curve dips down to a
minimum value of -0.26, near the expectation of the -0.28 rule. Also, as
expected the minimum does not occur at $E_{r}/e$ but rather at higher bias.
The 100 K and 30 K plots show this shift is proportional to the temperature,
as expected.

Figure 2:  Plots of $\kappa_{ball}$ vs. applied bias for a resonant tunneling
channel. For a resonance at 0 eV plots are shown at 300 K (thick, blue, solid)
and 30 K (thick, blue, dashed), and for resonance at 0.5 eV plots are shown at
300 K (thin, black, solid), 100 K (thin, black, dashed), and 30 K (thin,
black, dot-dash).

In summary, significant TI is possible in 1-dimensional systems that are
either very short open channels, or have a resonant tunneling structure.

####  5.4.2 Two and Three Dimensional Systems

For an open 2-dimensional channel, Eq. (49) is generalized to

$I_{2d}=\frac{2e}{h}\frac{2\pi}{L_{x}}\sum_{k_{x}}\sum_{k_{y}}\,\frac{\partial
E}{\partial k_{x}}\left[f_{0}(\beta(E-eW))-f_{0}(\beta E)\right]\,.$ (53)

For the extended case

$I_{2d}=\frac{L_{y}}{2\pi}\frac{2e}{h}\int\int dk_{x}dk_{y}\,\frac{\partial
E}{\partial k_{x}}\left[f_{0}(\beta(E-eW))-f_{0}(\beta E)\right]\,.$ (54)

This double integral can be converted to an integral over energy alone by
considering the integral over $k_{x}$ and $k_{y}$ over a thin region of
k-space that includes only states with energies between $E$ and $E+dE$. The
thickness of this ribbon in the $x$ direction is $dE/|\frac{\partial
E}{\partial k_{x}}|$ so again there is exact cancellation of the x-component
of the group velocity. Integration over $k_{y}$ gives the parameter $\Xi(E)$
equal to the maximum extent of the shell in the $y$ direction, leaving

$I_{2d}=\frac{L_{y}}{2\pi}\frac{e}{h}\int
dE\,\Xi(E)\left[f_{0}(\beta(E-eW))-f_{0}(\beta E)\right]\,.$ (55)

For many of the allowed energy values in the band, one would expect $\Xi(E)$
to be on the order of $2\pi/a$ where $a$ is the typical unit cell dimension on
the nanometer scale. We see that $\frac{L_{y}}{2\pi}\Xi$ and $\bar{T}$ are the
same function for an open channel and on the order of $L_{y}/a$ which is
roughly the number of subbands expected for an open channel. The same
conclusion about subband number holds true for a 3-dimensional channel. Thus,
for an open channel, $\bar{T}$ is the integrated density of states function
$N_{d-1}(E)$ corresponding to the directions transverse to the current flow.
Since the transverse density of states is the derivative of this function,
i.e., $D_{d-1}=dN_{d-1}/dE$, making use of Eq. (35) gives

$-\beta\frac{\partial
I}{\partial\beta}=\frac{2e}{h\beta}\int_{-\infty}^{\infty}dE\,D_{d-1}(E)\left[+h(\beta(E-eW))-h(\beta(E))\right]\,.$
(56)

The subscript $d-1$ is a reminder that for a 3-dimensional channel, the
transverse density of states resembles that of a 2-dimensional system, etc.
Even though structure in the density of states in the flow direction is
canceled out, such structure in the transverse directions does come through to
affect TI through the $D_{d-1}$ function. I point out that since $N_{d-1}$
might sometimes decrease with energy, $D_{d-1}$ is not positive definite. For
temperatures low enough such that $D_{d-1}$ does not vary much over the energy
interval $k_{B}T$, the localized nature of $h(x)$ can be exploited to give:

$-\beta\frac{\partial
I}{\partial\beta}=\frac{2e}{h\beta^{2}}\frac{\pi^{2}}{3}\left[D_{d-1}(eW)-D_{d-1}(0)\right]\,.$
(57)

If the channel becomes subband active by having the lateral dimension
increased, one expects a simple staircase structure for $N_{d-1}$. From Eq.
(57) it’s apparent that an upwards step (while $W$ is positive) will always
give a positive spike in $\kappa_{ball}$ as $eW$ is scanned through the step
energy. Negative $\kappa_{ball}$ will only be found for downward steps in
$N_{d-1}$.

For the case where the channel is extended in all three directions, the
function $D_{2}$ plays the important role and it is known to be constant when
the dispersion relation is parabolic. In this case Eq. (57) shows that there
would be no TI heat transfer. In many systems the dispersion relation is not
parabolic over the entire band, so there may be some 3-dimensional fully
extended systems showing appreciable ballistic TI heating or cooling. For the
case where the channel is extended in the $x$ direction and one transverse
direction, but subband inactive in the other transverse direction, it is the
one-dimensional function $D_{1}$ which enters Eq. (57). In this case
$D_{1}(E)$ goes as $E^{-1/2}$, again for parabolic dispersion, with the result
that only TI heating is possible; $\kappa_{ball}$ will always be positive.
Thus in all channels with dimensions either extended or subband inactive it
seems the possibilities of observing TI cooling are small.

If however one, or both, of the transverse dimensions is subband active then
sharp peaks in $D_{d-1}$ are expected and near these peaks significant and
negative $\kappa_{ball}$ values are expected; These structures may be
described as ribbons and thick sheets.

Figure 3:  Plots of $\kappa$ vs. applied bias for ballistic conduction in
zigzag carbon nanotubes. Numerical computation was performed for the case
$m=6$ (0.93 nm diameter), with short (5.04 nm length, left axis, black curve)
and long (84 $\mu$m, right axis, blue plot) nanotubes. Both calculations were
done at 100 K temperature.

###  5.5 Numerical Calculations for Carbon Nanotubes

Many of these results presented in this section are well-illustrated by
specific numerical calculations performed on carbon nanotubes. The carbon
nanotube is formed by rolling up a sheet of graphene layer with carbon atoms
arranged in a hexagonal pattern sheet with bond length $a_{0}=0.28$ nm,
primitive unit cell vectors $\overrightarrow{a_{1}}=a\hat{x}+b\hat{y}$ and
$\overrightarrow{a_{2}}=a\hat{x}-b\hat{y}$, where $\hat{x}$ runs along the
tube axis, $a=0.42$ nm, and $b=0.24$ nm. The Brillouin zone is hexagonal-
shaped with vertices at $(\pm\pi/a,\pm\pi/3b)$ and $(0,\pm 2\pi/3b)$. A simple
tight-binding model, well-described in Ref. Datta2005 , is implemented with a
dispersion relation given by

$E(k_{x},k_{y})=\pm t\sqrt{1+4\cos k_{y}b\cos k_{x}a+4\cos^{2}k_{y}b}\,,$ (58)

where $t=1.43$ eV. States with $E=0$ are found at the vertices
$(\pm\pi/a,\pm\pi/3b)$. When the sheet is rolled up to form a zigzag nanotube,
periodic boundary conditions confine $k_{y}$ to discrete lines parallel to the
x-axis, i.e., $k_{y}=\pi\nu_{y}/mb$ where $\nu_{y}$ is a running integer index
and the integer $m$ determines the nanotube diameter as $d=2bm/\pi$. If $m$ is
an integer multiple of 3, then the $E=0$ states are accessed and the nanotube
channel will be metallic. Equation (53) is used to numerically sum up for the
current as a function of applied bias and temperature. The temperature
derivative is evaluated numerically to produce $\kappa_{ball}$. The
calculation leaves out electron-electron interactions: effects from possible
charge buildup in this inflow/outflow problem Datta2005 , and is simply
intended to emphasize the effects of band and subband structure on $\partial
I/\partial T$.

Figure 4:  Plots of temperature derivative of current density vs. applied bias
($\partial K/\partial T$ vs. W) for ballistic conduction calculations with
$m=24$ (3.7 nm diameter, left axis, thin black curve) and with $m=1601$ (247
nm diameter, right axis, thick blue curve). In both cases the zigzag carbon
nanotube length is 84 $\mu$m long and the temperature is 100 K.

Results for $\kappa_{ball}$ as a function of applied bias for a rather small
zigzag nanotube (0.93 nm diameter and 5.04 nm length) at 100 K temperature is
shown in Fig. 3. A prominent downwards peak (dip) at zero bias goes almost
down to $\kappa=-1$ just as the -1 rule predicts. The existence of such a peak
is consistent with the channel being metallic. Above 0.25 V, $\kappa_{ball}$
becomes very small until a peak-dip pair is encountered near 1.5 V and as the
-0.17 rule predicts, a positive peak is quickly followed by a second peak that
indeed dips down to $\approx-0.17$. Successive peaks are also found and these
fall off in magnitude as the electrical current increases. Overall this plot
is consistent with a molecular type system with discrete and well-spaced
energy levels. Though such short chains are difficult to produce and connect
into electrical circuits, STM-based techniques are known to produce very small
wires made of gold which span the space between two STM probes Lou2010 . The
STM probes play the role of the anode and cathode depicted in Fig. 1(b). The
length of these wires can be very short, from approximately 100 nm down to 10
nm or perhaps even shorter. These specialized gold wires may have electronic
structure resembling a series of discrete peaks and hence $\kappa$ vs. bias
curves that might resemble the 5.04 nm nanowire calculation.

As the channel length is increased in the computations, more peaks are
produced as expected until the length exceeds about 200 nm. For longer
nanotubes the peaks coalesce and dramatically weaken, including the $E=0$
peak. At the temperature of $100$ K, nanotubes longer than 200 nm may be
considered as extended in the flow direction. Also in Fig. 3, a
$\kappa_{ball}$ vs. $W$ plot is shown for a zigzag nanotube 84 $\mu$m in
length. The zero energy peak has completely disappeared. The only structure
left is 10 peaks all corresponding to subband structure, i.e, whenever the
subband peak number changes either upwards or downwards. Most striking is the
change in the scale in Fig. 3 with the strongest peak less than 0.003. Below
4.8 V applied bias only positive $\kappa_{ball}$ peaks are observed. Only
above this where the subband number decreases as the top of the band is
approached are negative value peaks observed where TI cooling may occur. The
lack of structure in the extended channel plot is noteworthy; All band
structure related to the $x$ direction is indeed filtered out by the
cancellation of the group velocity as discussed above.

Further calculations are presented in Fig. 4 in which the effect of increasing
the nanotube diameter is explored. In these plots the derivative with respect
to temperature is taken of the sheet current density $K$ calculated as the
current divided by the tube circumference. With an intermediate diameter
nanotube, (m=24, 3.7 nm diameter) and 34 $\mu$m long, the plot shows a series
of discrete, positive peaks up to a bias of 3 V, after which the peaks point
downwards. Each peak signifies the beginning or end of a subband. In light of
Eq. (57), given that there is one active dimension transverse to the direction
of current flow, it makes good sense to see the density of states function
$D_{1}$ as a series of equally-weighted spikes, consistent with Eq. (44). This
is indeed the case, barring degeneracies, for example the first four peaks
from 0.6 V to 1.7 V.

When $m$ is increased to a value of 2001, producing a tube with diameter 309
nm, the subband structure coalesces as demonstrated by the continuous nature
of the thick blue curve of Fig. 4. Given that this system is a large graphene
sheet rolled into a tube it should produce the same results as a graphene
sheet extended in both dimensions along the sheet. If electrical probes are
connected to large pieces of graphene, an applied bias of greater than 3 V
would be required to produce TI cooling; For smaller bias levels TI heating
would only add to Joule heating. The amount of cooling at $W=3.5$ V is very
small, where $P_{J}$ would only cancel off 0.0016% of the Joule heating. It
appears that it will not be easy to observe TI cooling in extended carbon
nanotubes and graphene. In practice it appears to be more likely to observe TI
cooling in very small carbon structures, in particular very short along the
flow direction.

###  5.6 Interpolation with Conventional Transport

For an open channel under low applied bias, the ballistic conductance is given
by

$G_{ball}=\frac{2e^{2}M}{h}\,,$ (59)

where $M=D_{d-1}$ is the subband number and is the same as $\bar{T}$. Since
longer channels will realistically exhibit carrier scattering some analysis
for the interpolation between purely ballistic and purely conventional
transport is warranted. The proper interpolation formula is

$\frac{1}{G_{net}}=\frac{1}{G_{ball}}+\frac{1}{G_{conv}}=\frac{h}{2e^{2}M}+\frac{h}{2e^{2}M}\frac{l}{\Lambda}\,,$
(60)

where $\Lambda$ is a length scale on the order of the mean free path Datta2005
. The second term in Eq. (60) must match the conventional conductance for a
channel $G_{conv}=\sigma A/l$, where $\sigma=ne^{2}\tau_{ep}/m$ is the
electrical conductivity and $\tau_{ep}^{-1}$ is the electron phonon scattering
rate. One would expect $M$ to scale with the channel cross-sectional area $A$,
and indeed the correct matching occurs if $M\approx k_{F}^{2}A$ where $k_{F}$
is the Fermi wavenumber. For conventional transport, TI will be connected to
how the electrical conductivity varies with temperature. Noting that $\sigma$
is proportional to $\Lambda$, and defining $\kappa_{0}=d\ln\sigma/d\ln T$, the
following interpolation formula is obtained for $\kappa$:

$\kappa_{net}=\kappa_{ball}+\kappa_{0}\left(1+\frac{l_{mfp}}{l}\right)^{-1}=\kappa_{ball}+\kappa_{conv}\,,$
(61)

where I have replaced $\Lambda$ with the mean free path defined from
$l_{mfp}=mv_{F}\sigma/ne^{2}$. Also, $\kappa_{conv}$ has been defined as the
product of $\kappa_{0}$ and $\left(1+\frac{l_{mfp}}{l}\right)^{-1}$. In the
limit $l<<l_{mfp}$, the reasonable result $\kappa_{net}=\kappa_{ball}$ is
obtained. More interestingly, in the limit $l>>l_{mfp}$ where conventional
conductance dominates, $\kappa_{net}=\kappa_{ball}+\kappa_{0}$, i.e., the
ballistic contribution persists in a surprising way. This result has nothing
to do with TI. If a bulk material with a small temperature dependence for the
conductivity is studied carefully, the ballistic $\kappa$ specified in Eq.
(38) may be significant and measurable, particularly at higher temperatures.
As far as the low temperature behaviour goes, if the Bloch-Grüneisen law holds
and the conventional resistivity goes as $T^{5}$, then $\kappa_{0}=-5$ and
$\kappa_{conv}$ goes as $T^{5}$, thus falling off much faster than
$\kappa_{ball}$.

The interpolation formula, Eq. (61), plays an important role in the following
discussion on conventional transport. In many of these systems, $r$ is small
and in the quest for systems with larger $r$ values, channels with mean free
paths near the channel length will be explored. In such systems the
$\left(1+\frac{l_{mfp}}{l}\right)^{-1}$ factor plays an important role.

##  6 Conventional Electrical Transport

In this study I focus on systems with negative values of $\gamma$ and
$\kappa$, since positive values will only result in induction heating adding
to the inherent Joule heating. For example, intrinsic semiconductors are a
poor choice for cooling even though they are expected to have a large value
for $\kappa$, ideally near $E_{g}/k_{B}T$ where $E_{g}$ is the band gap.

Conventional metals are good candidates for cooling the channel by induction
since their electrical conductivity decreases with temperature. In Fig. 5
calculations are presented that can be used to assess the level of inductive
cooling in a conductive channel made of highly purified silver. Silver is a
good choice of material as it is been extensively studied Ehrlich1974 ;
Caplin1975 ; Rumbo1976 ; Kos1990  and good low temperature results are
available after Fe impurities are removed by oxygenation Barnard1982 . In my
calculations, the electrical conductivity function was taken from Ref.
Matula1979  and the derivative with respect to temperature was evaluated
numerically. Matula’s $G_{1}(T)$ function fit for the Bloch-Grüneisen function
and supplied parameters were used to calculate $\sigma(T)$. From this function
both $\gamma$ and $\kappa$ were calculated using Eqs. (23, 24). For the
specific heat, which also enters into Eq. (26), the Debye model with Debye
temperature of 215 K was implemented to calculate the lattice contribution to
$c_{V}$ AM . The electronic contribution to the specific heat was also
included AM . In order to calculate the thermal conductivity a value of
$2.44\times 10^{-8}$ V2/K2 was used for the Lorenz number AM .

Figure 5:  For highly purified silver, plots of the logarithmic derivative
$\kappa_{0}=d\ln\sigma/d\ln T$ (thick, black, long-dash curve) and
$\kappa_{conv}$ for 1 mm (thick, blue, solid), 10 $\mu$m (thick, green, dash-
dot), and 100 nm (thick, red, short-dash) channel lengths, all on the left
axis. Also displayed as thin curves are $r\times\kappa_{conv}$ for the same
three channel lengths (1 mm: thin, blue, solid, 10 $\mu$m: thin, green, dash-
dot, 100 nm: thin, red, short-dash). Plotted on the right axis is the electron
mean free path (thin, black, dashed curve).

After producing an accurate calculation for the electrical conductivity vs.
temperature over the range from about 1 K up to temperatures well above room
temperature, the logarithmic derivative $\kappa_{0}$ is readily calculated.
This function, which is depicted as the thick (black) short-dashed curve in
Fig. 5, stays below -1 throughout the entire temperature range shown. Below
the Debye temperature of 215 K the phonon density as well the electron-phonon
scattering rate drop quickly. Indeed, below $\approx 100$ K, $\kappa_{0}$
drops down and eventually levels off at the value of -5 as expected from the
Bloch-Grüneisen law Matula1979 . It should be pointed out that for the case of
silver under 10 K, Matthiessen’s rule seems to break down and a $\kappa_{0}$
value near -4 may be more correct Barnard1982 . I will return to this low
temperature behaviour below.

Before obtaining $\kappa_{net}$, the interpolation formula, Eq. (61), and
hence $\kappa_{conv}$ is required which in turn depends on the channel length
$l$. To illustrate this sample geometry effect, a few representative plots are
displayed Fig. 5 as the thick curves. The thick (green) dot-dash curve was
calculated as $\kappa_{conv}$ for a sample length of 10 $\mu$m. At high
temperatures the mean free path is much less than $l$ and, as expected, there
is little change from the $\kappa_{0}$ curve. Below $\approx 35$ K the
$\kappa_{conv}$ curve bends upwards sharply and at 24 K the mean free path
matches the sample length. At temperatures well below this TI is effectively
quenched, at least for conventional transport, as scattering events (and
fluctuations) become rare. The minimum $\kappa_{conv}$ value of -2.85 suggests
that a highly purified silver channel of this length would be a good candidate
for exhibiting TI cooling near 34 K. For a longer channel (1 mm), the overall
story is similar, with the thick (blue) solid curve dipping down to even lower
values, as low as -4.7 at 18 K. For very small channel lengths TI is
manifested at more modest levels and is more significant at higher
temperatures, as exemplified by the thick (red) short-dash curve for a 100 nm
long channel.

Alone, the dimensionless quantity $\kappa_{conv}$ is a good indication of the
strength of TI but knowledge of the cooling power will better allow for
quantitative analysis. Equations (26, 28) can be used to quantify the strength
of TI cooling in terms of the Joule heating, if the ratio
$r=\tau^{*}_{GT}/\tau_{GT}$ is well-characterized. The ratio should always be
smaller than unity since the relaxation time is expected to exceed the
timescale for fluctuations that produce the relaxation. Also, for large
systems $r$ is expected to be small. In metals such as silver, electrons are
primarily responsible for thermal conduction so the fluctuation timescale must
be the electron-phonon scattering time, i.e.,
$\tau_{GT}^{*}=\tau_{ep}=l_{mfp}/v_{F}$. Using the Wiedemann-Franz law,
$\lambda=L_{0}T\sigma$, where here I take $L_{0}=\pi^{2}k_{B}^{2}/3e^{2}$,
then in the limit where the channel length greatly exceeds the mean free path:

$r=\left(\frac{L_{0}ne^{2}T}{2c_{V}E_{F}}\right)\frac{l_{mfp}^{2}}{L_{x}^{2}}\,,~{}~{}~{}~{}~{}~{}~{}l_{mfp}<<L_{x}\,.$
(62)

It is noted that the first factor in brackets in Eq. (62) is close to 1/3 at
very low temperatures where the specific heat is dominated by electrons, and
it never exceeds this value. When the channel length is less than the mean
free path, Eq. (62) must be modified; In order to keep $r<1$, the following
formula has the appropriate large and small sample limits:

$r=\left(1+\frac{2c_{V}E_{F}L_{x}^{2}}{L_{0}nTl_{mfp}^{2}}\right)^{-1}\,.$
(63)

With $P_{cool}=-r\kappa_{net}P_{J}$ and ignoring ballistic effects, the ratio
$P_{cool}/P_{J}$ is readily calculated, and is presented as the thin curves in
Fig. 5.

Figure 6:  Minimum values of the power ratio $r\kappa_{conv}=J_{ind}/P_{J}$
versus channel length for the case where the Bloch-Grüneisen law holds (thick,
black, solid curve on left axis) and the case where a constant resistivity
$\rho_{0}=8\times 10^{-12}~{}\Omega$m was added into the model (thin, red,
solid curve on left axis). On the right axis is plotted upper $T_{U}$, and
lower $T_{L}$, temperature limits for net cooling, again for the the case
where the Bloch-Grüneisen law holds (thin, black, solid and thin, blue, dashed
curves), and the case with the same offset resistivity (thick, black, solid
and thick, blue, dashed curves).

For the 10 $\mu$m channel, the thin (green) dash-dot curve represents
$r\kappa_{conv}$. This $r\kappa$ curve lies below -1 in the temperature range
18-73 K and net cooling is expected to be possible in this range (see Eq.
(28)). In general, the part of the $r\kappa$ curve below -1 represents the
window of opportunity for net cooling to occur. Observation of net cooling in
a rather ordinary electrical circuit would indeed be remarkable. A clear and
simple test for net cooling could involve a conductance experiment where the
applied bias is ramped up slowly. At high enough bias levels one would look
for decreases in the sample temperature, or for anomalous increases in sample
conductance. For the longer 1 mm channel, the thin (blue) solid curve shows
that net TI cooling will be possible in the somewhat narrower temperature
range 7 K to 20 K, though more will be said on this below. For channel lengths
below about 500 nm, the $r\kappa_{conv}$ curve does not dip below -1 and net
cooling by conventional transport will not be possible (see thin (red) dashed
curve, for example). For such short channels, net cooling might be marginally
possible if ballistic transport is favourable with $\kappa_{ball}\approx-1$
and then with $\kappa_{conv}$ added in, $\kappa_{net}$ might be less than -1.
Also, even if $\kappa_{net}$ is not less than -1, tests for the TI effect are
still possible. More modest levels of TI cooling (or even heating) might be
discerned after careful modeling of experiments made on such a system which
measure $\Delta T$ vs. applied bias $W$. If modeling for temperature rises due
to Joule heating alone are accurate enough then discrepancies with
experimental results might be explained by TI. It should be mentioned that the
temperatures reported here come from a discrete model presented here for TI
and must therefore be interpreted as an average over the entire channel.
Though the center of the channel could certainly be somewhat cooler than near
the leads, for example, working with the mean value suffices to illustrate the
important physics of the TI cooling that might be occurring in such channeled
conduction.

Continuing with the assumption of validity of the Bloch-Grüneisen law, in Fig.
6, I show a plot of the minimum value of the power ratio
$r\kappa_{conv}=-P_{cool}/P_{J}$ as a function of channel length, as the thin,
red, solid curve. Values below -1 are sustained over a wide range all the way
down to 390 nm. Net cooling for shorter samples fails essentially because the
mean free path is too long which makes $r$ small. This does not seem to be a
fundamental barrier and for materials other than silver this barrier may not
even exist. For channels longer than 390 nm, $P_{cool}$ will exceed $P_{J}$
over some range of temperature (limits $T_{U}$ and $T_{L}$ defined by
$r\kappa_{conv}=-1$ and displayed as thin curves on right axis), for example
from $T_{L}=3.3$ K to $T_{U}=7.3$ K for a 5 cm long channel. The lowest value
of -3.73 for $r\kappa_{conv}$ is found for a 300 $\mu$m channel.

Figure 7:  Power ratio $r\kappa_{conv}=J_{ind}/P_{J}$ versus temperature for a
10 $\mu$m long highly purified silver channel (solid, black curve), as well as
plots of the right-hand side of Eq. (64) for ambient temperature 65 K and
$W=4$ mV (blue, short-dash), 65 K and 20 mV (green, long-dash), and 295 K, 20
mV (red, dash-dot). Solutions for the stationary state are found at
intersection points with $r\kappa_{conv}$. These intersections are labelled a,
b, c, d, e.

The results discussed so far have not incorporated impurity scattering and
thus overestimate the relative strength of TI cooling at low temperatures. In
practice, the Bloch-Grüneisen $T^{5}$ law breaks down as impurity and surface
scattering become important when phonon scattering disappears. The
conductivity for highly purified silver samples has been found to flatten
below about 3 K to values around $10^{11}$ S Kos1990 ; Barnard1982 . With a
small constant added to the Bloch-Grüneisen resistivity, the logarithmic
derivative $\kappa_{conv}$ goes as $T^{5}$, causing induction cooling to fall
off at low temperatures. Incorporating this extra resistance into my numerical
model is straightforward and the results are also plotted in Fig. 6. For
channel lengths below about 10 $\mu$m there is no discernible change, but
above 100 $\mu$m there is a large detrimental effect, $r\kappa_{conv}$ is cut
down considerably, and net cooling is cut off completely for samples longer
than 0.9 mm. Being limited to purified samples no longer than about 1 mm is a
serious challenge for actual testing, as samples used in conductance work have
been over 10 cm in length Kos1990 . Nevertheless, producing highly purified
silver samples shorter than 1 mm is certainly possible and for such samples
the experiment appears to be straightforward. For example, if samples 100
$\mu$m long could be manufactured, the workable temperature range is
$T_{L}=12$ K to $T_{U}=35$ K. The widest temperature range of 120 K is
predicted to occur with 700 nm samples. This sample length also gives the
highest possible upper temperature limit for net cooling of $T_{U}=159$ K.
Similar numerical calculations have also been performed on channels composed
of highly purified copper, again using Matula’s fit parameters. For copper, I
find the greatest upper temperature limit to be $T_{U}=222$ K for a 360 nm
long channel, displayed as the cross in the upper left of Fig. 6. This change
in $T_{U}$ for copper is easily explained by noting that the electrical
conductivity in this temperature range is determined by the phonon number
density, and at temperatures below the Debye temperature the conductivity
changes rapidly. This is the reason for $\kappa_{0}$ values in the range -2 to
-3 at temperatures near 100 K to 200 K. The higher Debye temperature of 315 K
for copper as compared to 215 K for silver explains why $T_{U}$ is larger for
copper.

The extent of TI cooling can be better understood by re-expressing Eq. (28) in
terms of the electrical conductivity and the Lorenz constant, as:

$r\kappa_{conv}=-1+\frac{L_{0}T_{GT}}{W^{2}}\left(T_{GT}-T\right)\,.$ (64)

This produces a simple graphical method to understand how the gate temperature
behaves relative to the ambient temperature $T$ in terms of applied bias. As
long as $L_{0}$ remains constant, the right side of Eq. (64) is a parabola so
when plotted on the same graph as $r\kappa$ (solid curve in Fig. 7) one simply
searches for intersection points. For real metals the Lorenz constant does
vary slightly with temperature; Also, accounting for radiative transfer would
effectively increase both $h$ and $L_{0}$, especially at higher temperatures.
For the sake of simplicity, the parabolic right hand side was assumed for
three examples on a 10 $\mu$m long sample, plotted in Fig. 7. For an ambient
temperature of $T=T_{1}=65$ K, which lies 8 K below the upper limit $T_{U}$
for allowable net cooling, very small $W$ will produce a parabola which passes
through the point (65 K, -1) and from there heads downwards almost vertically
until intersecting the $r\kappa$ curve at a temperature slightly below
ambient.

For the first example, a bias of 4 mV is applied, still somewhat less than
$k_{B}/e=5.6$ mV. The situation is much more interesting now with a total of
three intersection points (labelled a, b, c), the nearest (a) to the starting
point occurring at 59 K, i.e. 6 K of net cooling. Physically, one would expect
this thermodynamic system to reside at this nearest intersection. As the bias
is increased further a tangent curve scenario would produce two intersections
and immediately after that just one, so the system would suddenly jump to that
remaining intersection point. The second example describes the situation at a
yet higher bias of 20 mV, with the single intersection labelled, d, at a gate
temperature of 18.8 K. Thermodynamic cooling is very strong at this bias level
with the system just 0.4 K above the theoretical limit of $T_{L}=18.4$ K. One
gets a clear sense of how yet larger $W$ would pin the gate temperature very
close to $T_{L}$. Equation (27) would seem to predict a temperature drop
greater than $T_{1}$ with large enough bias, but the more careful analysis
leading to Eq. (64) shows that the gate temperature gets clamped down to
$T_{L}$ at very large $W$, where $P_{cool}$ is only slightly larger than
$P_{J}$. The last example is the most striking; Imagine after taking the
system to d, the electrical current in the circuit is maintained with 20 mV
bias, while the ambient temperature is raised to $T_{2}$. Careful inspection
of the equations leading to Eq. (64) that if $T_{GT}$ stays unchanged, then
$P_{cool}$ and $P_{J}$ also stay the same, with only the thermal transfer to
the surroundings changing. This is true even if $T_{2}$ exceeds $T_{U}$. Of
course, if $T_{2}$ is too large with the curvature of the parabola staying
constant, eventually the intersection point will be lost and the entire system
will relax to ambient temperature. However, until this happens, TI cooling
continues to outpace Joule heating and a type of metastable (stationary) state
is maintained. In the third example $T_{2}$ is taken to room temperature with
$W=20$ mV, and the intersection point, e, occurs at a gate temperature of 20
K. In the laboratory, one could remove all sources of external cooling, let
all the visible components warm up to room temperature, and the silver channel
would stay at 20 K indefinitely!

This last example is strong testament to the potency of the type of cooling
being discussed here. The nonequilibrium nature of this thermoelectric system
opens up the applied bias as a variable capable of driving, and clamping, the
system with tremendous (thermodynamic) force under the right conditions.

##  7 Discussion

From my numerical calculations on highly purified silver and copper, the
prospects are very positive for observing TI cooling in the temperature range
15 K to $\approx$200 K, and for sample sizes in the range 400 nm to 1 mm. It
would be desirable to extend this temperature range. At the upper limit,
finding a system with $T_{U}$ above room temperature would create the
convenience of not requiring any other cooling such as liquid refrigerants, so
that one could thermoelectrically cool (and clamp down) the channel directly
from room temperature. I would expect similar results for other conventional
metals such as gold, chromium, etc. as well as highly doped semiconductors,
under degenerate conditions Sze , but the details remain to be worked out. It
may be that systems with very high Debye temperatures will have $T_{U}$ values
above room temperature. Materials with high Debye temperatures and high
mobility with long mean free paths should produce high $T_{U}$ values without
requiring very small sample sizes. One system that might meet these
requirements is the type of graphene sample that has been fabricated to
lengths as short as 5 $\mu$m and operated with very high observed mobilities
at 5 K Stormer2008 . Semiconductor systems at very low temperatures are also
worth exploring, in particular the two dimensional metals formed as
accumulated or inverted layers at interfaces. These semiconductor
heterostructures are known to produce conductive layers with very high
mobilities which increase with the lowering of temperature Stormer1999 . The
effective purity of these systems is enhanced tremendously by having the
dopants located at a distance from the conductive layer. For those materials
that do require very small channels, I note that constructing electronic
circuits with general purpose wires having features smaller than 1 mm and even
down to near 100 nm in width has become routine and under special
circumstances, electrical channels of length 15 nm have been demonstrated
Gervais2011 ; Kamp2015 .

Finding temperature and length ranges for purified silver, over which
$r\kappa_{net}$ is less than -1, and with values below -3, is significant (see
Fig. 6), but larger magnitudes of $\kappa_{net}$ are desirable. The same level
of $r\kappa_{net}$ values is expected for other conventional metals and for
degenerate semiconductors the magnitude of $\kappa_{conv}$ is not expected to
exceed a value of 3 Sze . Since larger negative values of $\kappa$ would
further enhance the extent of TI cooling, materials exhibiting metal-insulator
transitions (MIT) are worth investigation if a phase transition can occur with
changing temperature. Such sharp transitions should produce large values of
$\kappa_{conv}$ near the transition temperature. For example PrNiO3 and
NiS1-ySey both have $\kappa_{conv}$ values near +50 at 95 K and 225 K
respectively Obradors1993 ; Tokura1998 . However these systems have a metallic
state at high temperatures which changes to an insulating state at lower
temperature where electron-electron interactions lead to blockage of charge
flow. This is the case for most MIT systems but the double-exchange system
La1-xSrxMnO3 is an interesting exception Tokura1998 ; Urushibara1995 ;
Kawano1996 . For x=0.15 the resistivity drops sharply below the transition
temperature of 230 K. Over a modest range in temperature $\kappa_{conv}$ is
negative with values near -20. Materials with large $\kappa_{conv}$ values
could increase $T_{U}$, decrease $T_{L}$, or make TI cooling more feasible on
longer channels that would likely be easier to fabricate. The caveat with MIT
systems is the narrow range of temperature for these transitions, i.e,
$|\kappa_{conv}|$ might be large only near the transition. Good candidates for
general purpose net TI cooling should have $\kappa_{net}$ values well below -1
over a wide temperature range. I note though that even with a narrow
$T_{U}-T_{L}$ range, the induction clamping effect discussed above should
still be exploited to hold such systems near the transition temperature even
after other forms of sample cooling have been removed. Superconductors have a
transition temperature where the conductivity changes a great deal over small
temperature changes, and $\kappa$ will take on extremely large negative
values. Its is difficult to say if the clamping procedure will work for
superconductors though since TI cooling requires current flow and the critical
current density is easily exceeded in practice.

On the issue of possible applications of TI cooling, the first conclusion is
that such applications are much more likely to be found in small systems.
Fortunately small electronic systems are used widely in modern technology,
most notably in microelectronic integrated circuits and optoelectronic
devices. In these systems circuit elements on the 100 nm scale and smaller are
routinely implemented. Moreover, cooling technology that matches these small
scales is highly sought after. Such on-chip Peltier cooling technology exists
Alley2009  and may be beneficial for devices operating near room temperature.
However, for lower temperature applications this type of cooling is known to
lose effectiveness rapidly as the temperature is lowered AM . It seems then
that TI cooling might create a breakthrough in attempts to achieve low
temperature thermoelectric cooling of small devices, to temperatures near 20 K
or even lower. High mobility materials are desired for producing fast
electronic devices Feng2005 ; Avouris2010  and since mobilities often increase
at lower temperatures, TI may hold the key to producing very fast integrated
circuitry with built-in cooling. Elements of any future quantum computers,
such as interacting quantum dots would likely require cooling to very low
temperatures to maintain coherence DiVincenzo1999 ; Affleck2011 ; TI may be
used to achieve cooling to temperatures well below 1 K without the need of
expensive cryostats. The thermodynamically induced temperature clamping effect
can also be exploited; Circuits might be cooled by other means initially but
after the temperature is clamped TI cooling alone will suffice and sub-
circuits running at low temperatures can be incorporated into larger room-
temperature circuitry.

A further word is in order on the temperature clamping by TI: in the example
discussed in Fig. 7 the purified silver channel is initially brought down to
65 K by some external and traditional method of cooling. Then the bias of 20
mV is applied and the temperature is clamped down to 18 K. Afterwards, the
external cooling is removed and ambient temperature is returned to room
temperature. The channel remains clamped at about 20 K. If the bias is then
turned to zero, the metastable stationary state collapses and one finds
oneself with a channel at room temperature, and re-applying will not re-
establish the clamp. One is forced to externally re-cool the channel. I point
out that this state of affairs has much in common with a scheme I have
specified for the potential construction of an atomic tether using a scanning
tunneling microscope, also exploiting the TI principle. In that construction,
electrical bias is applied to the tunnel junction and the tether is built up
one atom at a time. As the tether grows more applied bias may be required to
help stabilize the structure. If the bias is set to zero, the tether
collapses. Though neither scheme has been realized yet, it is interesting to
point out the common features. In both cases, a thermodynamic state is
bolstered by the creation of nonequilibrium conditions (applied bias), and
neither would ever arise simply by the application of bias; The correct
sequence of precise steps is required for formation. Though in both cases this
bolstering might at first seem counterintuitive, at a very basic level these
states are stabilized by my postulated statistical weighting factor in Eq.
(30). The function that describes a nonequilibrium system’s total rate of
entropy production would seem to play a vital role. The realization of one
scheme (tether or clamp) would compel one to attempt the other, and search for
more examples, as the TI principle seems to have great generality.

##  8 Conclusions

In conclusion, a method has been developed for achieving a new type of
thermoelectric cooling that makes use of the principle of thermodynamic
induction. The extent of this thermodynamically-induced cooling is determined
by how strongly the electrical conductance depends on temperature. Such
dependence has been investigated for both conventional and ballistic
electrical transport. For ballistic transport a no-net cooling theorem has
been established which accounts for induction cooling and Joule heating. Good
candidate nano-systems for observing TI cooling are very short chains,
resonant tunneling structures, ribbons, and thick nanosheets. For conventional
transport, investigation of bulk purified silver and copper channels shows
that net cooling is possible over wide temperature ranges. These tests would
be conducted for smaller samples, and under higher bias, than in previous
transport studies. Under the right conditions, induction cooling is very
potent and can be used to clamp the temperature of a metallic channel in a
metastable type of stationary state.

A successful search for this inductive cooling would demonstrate the general
validity of the induction principle. In a broader context, a simple
statistical weighting factor for nonequilibrium stationary-type inducer states
has been developed here, and would seem to have application to many
irreversible systems. Achieving TI cooling would go a long way towards
establishing TI as an essential and general scientific principle.

###### Acknowledgements.

I thank Cathy J. Meyer for her assistance in editing the manuscript.

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