# Quantum Maxwell’s Demon in Thermodynamic Cycles

H. Dong, D.Z. Xu and C.P. Sun<EMAIL_ADDRESS>http://power.itp.ac.cn/
suncp/index.html Institute of Theoretical Physics, Chinese Academy of
Sciences, Beijing, 100190, China

###### Abstract

We study the physical mechanism of Maxwell’s Demon (MD) helping to do extra
work in thermodynamic cycles, by describing measurement of position, insertion
of wall and information erasing of MD in a quantum mechanical fashion. The
heat engine is exemplified with one molecule confined in an infinitely deep
square potential inserted with a movable solid wall, while the MD is modeled
as a two-level system (TLS) for measuring and controlling the motion of the
molecule. It is discovered that the the MD with quantum coherence or on a
lower temperature than that of the heat bath of the particle would enhance the
ability of the whole work substance formed by the system plus the MD to do
work outside. This observation reveals that the role of the MD essentially is
to drive the whole work substance being off equilibrium, or equivalently
working with an effective temperature difference. The elaborate studies with
this model explicitly reveal the effect of finite size off the classical limit
or thermodynamic limit, which contradicts the common sense on Szilard heat
engine (SHE). The quantum SHE’s efficiency is evaluated in detail to prove the
validity of second law of thermodynamics.

###### pacs:

03.67.-a,05.70.Ln,05.30.-d,03.65.Ta

##  I Introduction

Maxwell’s demon (MD) has been a notorious being since its existence could
violate the second law of thermodynamics (SLoT) MD_book ; Nori2009 : the MD
distinguishes the velocities of the gas molecules, and then controls the
motions of molecules to create a difference of temperatures between the two
domains. In 1929, Leo Szilard proposed the “one molecular heat engine”(we call
Szilard heat engine(SHE)) Szilard1929  as an alternative version of heat
engine assisted by MD. The MD firstly measures which domain, the single
molecule stays in and then gives a command to the system for extracting work
according to the measurement results. In a thermodynamic cycle, the molecule
seems to extract heat from a single heat bath at temperature $T$, and thus do
work $k_{\mathrm{B}}T\ln 2$ without evoking other changes. This consequence
obviously violates the SLoT.

The first revival of the studies of MD is due to the recognition about the
trade-off between information and entropy in the MD-controlled thermodynamic
cycles. The milestone discovery is the “Landauer principle” Landauer1961 ,
which reveals that erasing one bit information from the memory in computing
process would inevitably accompany an increasing entropy of the environment.
In the SHE, the erasing needs work $k_{\mathrm{B}}T\ln 2$ done by the external
agent. It gives a conceptual solution for the MD paradox Bennett1982  by
considering the MD as a part of the whole work substance, thus the erasing
information stored in the demon’s memory is necessary to restart a new
thermodynamic cycle. This observation about erasing the information of the MD
finally saves the SLoT.

The recent revival of the studies of MD is due to the development of quantum
information science. The corresponding quantum thermodynamics concerns the
quantum heat engines (QHEs) Kieu2004 ; htquan  utilizing quantum coherent
system serving as the work substance. A quantum work substance is a quantum
system off the thermodynamic limit, which perseveres its quantum coherence
Scully2003 ; htquan2006  to some extent, and obviously has tremendous
influence on the properties of QHEs. Especially, when quantum MD is included
in the thermodynamical cycle Zurek1984 ; lloyd1997 ; Quan2006 , the whole work
substance formed by the work substance plus MD would be off the thermodynamic
limit and possesses some quantum coherence. There are many attempts to
generalize the SHE by quantum mechanically approaching the measurement process
Zurek1984 , the motion control Quan2006 , the insertion and the expansion
process Ueda2010 . However, to our best knowledge, a fully quantum approach
for all actions in the SHE integrated with a quantum MD intrinsically is still
lack. The quantum-classical hybrid description of the SHE may result in some
notorious observations about MD assisted thermodynamic process, which seems to
challenge the common senses in physics. Therefore, we need a fully quantum
theory for the MD-assisted thermodynamics.

In this paper, we propose a quantum SHE assisted by MD with a finite
temperature different from that of the system. In this model, we give a
consistent quantum approach to the measurement process without using the von
Neumann projection Pzhang . Then we calculate the works done by the insertion
of the movable wall in the framework of quantum mechanics. The controlled gas
expansion is treated with the quantum conditional dynamics. Furthermore, we
also consider the process of removing wall to complete a thermodynamic cycles.
With these necessary subtle considerations, the quantum approach for the MD-
assisted thermodynamic cycle will go much beyond the conventional theories
about the SHE. We show that the system off the thermodynamic limit exhibits
uncommon observable quantum effects due to the finite size of system , which
results in the discrete energy levels that could be washed out by the heat
fluctuation. Quantum coherence can assist the MD to extract more work by
reducing effective temperature, while thermal excitation of the MD at a finite
temperature would reduce its abilities for quantum measurement and conditional
control of the expansion. It means that, only cooled to the absolute zero
temperature, could the MD help the molecule to do maximum work outside.

Our paper is organized as follows: In Sec. II we firstly give a brief review
of classical version of SHE, and then present our model in quantum version
with MD included intrinsically. The role of quantum coherence of MD is
emphasized with the definition of the effective temperature for arbitrary two
level system. In the Sec. III of main part, we consider the quantum SHE with a
quantum MD at finite temperature, doing measurement for the position of the
particle confined in a one dimensional infinite deep well. The whole cycle
consists four steps: insertion, measurement, expansion and removing. Detailed
descriptions are performed subsequently in the whole cycle of SHE. We
calculate the work done and heat exchange in every sub-step. In Sec. IV, we
discuss quantum SHE’s operation efficiency in comparison with the Carnot heat
engine. We restore the well-known results in the classical case by tuning the
parameters in the quantum version, such as the width of the potential well.
Conclusions and remarks are given in Sec. VI.

##  II Quantum Maxwell’s Demon in Szilard Heat Engine

In this section, we firstly revisit Szilard’s single molecular heat engine
(SHE) in brief. As illustrated in Fig. 1(a), the whole thermodynamic cycle
consists three steps: insertion(i-ii), measurement(ii-iii) and controlled
expansion(iii-iv) by the MD. The demon inserts a piston isothermally in the
center of the chamber. Then, it finds which domain, the single molecule stays
in and changes its own state to register the information of the system.
Without losing generality, we assume the demon initially is in the state $0$.
Finding the molecule is on the right, namely $L/2<x<L$, the demon changes its
own memory to state $1$, while it does not change if the molecule is on the
left $\left(0<x<L/2\right)$. According to the information acquired in the
measurement process, the demon controls the expansion of the domain with the
single molecule: allowing the isothermal expansion with the piston moving from
$L/2$ to $L$ if its memory registers $0$, and moving from $L/2$ to $0$, if the
register is on the state $1$. In each thermodynamic cycle, the system does
work $W=k_{\mathrm{B}}T\ln 2$ to the outside agent in the isothermal
expansion. In an overall looking, the system extracts heat from a single heat
bath to do work, thus it would violate the SLoT if the MD were not treated as
a part of the work substance in the SHE. However, after the cycle, MD stores
one bit information as its final state and is in the mixture of $0$ and $1$
states with equal probability. Thus it does not return to its initial state.
Landauer’s principle states that to erase such a bit of information at
temperature $T$ requires the dissipation of energy at least
$k_{\mathrm{B}}T\ln 2$. The work extracted by the system just compensates the
energy for erasing the information. Therefore, the SLoT is saved. In this
sense, the classical version of MD paradox is only a misunderstanding, due to
the ignorance of some roles of the MD Bennett1982 .

Figure 1: Classical and quantum Szilard’s single molecular heat engine. (a)
Classical version: (i-ii) A piston is inserted in the center of a chamber.
(ii-iii) The demon finds which domain, the single molecule stays in. (iii-iv)
The demon controls the system to do work according to its memory; (b) Quantum
version: The demon is modeled as a two level system with two energy levels
$\left|g\right\rangle$ and $\left|e\right\rangle$ and energy spacing $\Delta$.
The chamber is quantum mechanically described as an infinite potential with
width $L$. (I-II) An impenetrable wall is inserted at arbitrary position in
the potential. (II-III) The demon measures the state of the system and then
record the results in its memory by flipping its own state or no action taken.
The measurement may result in the wrong results illustrated in the green
dashed rectangle. (III-IV) The demon controls the expansion for the single
molecule according to the measurement. (IV-V) The wall is removed from the
potential.

In the most of previous investigations about the MD paradox, it is usually
assumed the system and the MD possess the same heat bath. Thus the whole work
substance formed by the system plus the MD is in equilibrium, and no quantum
coherence exists. If the demon is in contact with a lower temperature heat
bath while the system’s environment possesses higher temperature $T$, the work
needed in the erasing process is smaller than $k_{B}T\ln 2$ Quan2006 . Under
this circumstance, we actually construct a quantum heat engine with non-
equilibrium or an equilibrium working substance work between two different
heat baths. Furthermore, when the MD is initially prepared with quantum
coherence, the quantum nature of the whole work substance results in many
exotic functions for QHE.

To tackle this problem, we study here a quantum version of Szilard’s model
with an MD accompanying it. In this model, the chamber is modeled as an
infinite square potential well with the width $L$, as illustrated in Fig.
1(b). And the demon is realized by a single two-level atom with energy levels
$\left|g\right\rangle$, $\left|e\right\rangle$ and level spacing $\Delta$.
Initially, the system is in thermal state with inverse temperature $\beta$.
And the demon has been in contact with the low temperature bath at the inverse
temperature $\beta_{D}$. Namely, the demon is initial prepared in the
equilibrium state

$\rho_{D}=p_{g}\left|g\right\rangle\left\langle
g\right|+p_{e}\left|e\right\rangle\left\langle e\right|,$ (1)

with the probability $p_{e}=1-p_{g}$ in the excited state and one in ground
state

$p_{g}=1/\left[1+\exp\left(-\beta_{D}\Delta\right)\right].$

Actually, the inverse temperature $\beta_{D}$ could represent an effective
inverse temperature of the MD with quantum coherence. For an environment being
a mesoscopic system, the number of its degrees of freedom is not so large.
Under this circumstance, the strong coupling to the MD leaves finite off-
diagonal elements in the reduced density matrixDong2007 . This remnant of
coherence can be utilized to improve the apparent efficiency of the heat
engine Scully2003 ; htquan2006 . For the demon with coherence, the density
matrix usually reads as

$\rho_{D}=\left[\begin{array}[]{cc}p_{g}&F\\\
F^{\ast}&p_{e}\end{array}\right],$ (2)

where the off-diagonal element $F$ measures the quantum coherence. The eigen-
values of the above reduced density matrix represent two effective population
probabilities as

$\displaystyle p_{+}\left(F\right)$ $\displaystyle\simeq$ $\displaystyle
p_{e}-\coth\left(\frac{\Delta}{2}\beta_{D}\right)\left|F\right|^{2},$
$\displaystyle p_{-}\left(F\right)$ $\displaystyle\simeq$ $\displaystyle
p_{g}+\coth\left(\frac{\Delta}{2}\beta_{D}\right)\left|F\right|^{2}.$ (3)

We can define an effective inverse temperature $\beta_{\mathrm{eff}}=\ln
p_{+}\left(F\right)/p_{-}\left(F\right)$ for the two-level MD, namely,

$\beta_{\mathrm{eff}}=\beta_{D}+\frac{4\left|F\right|^{2}}{\Delta}\cosh^{2}\left(\frac{\Delta}{2}\beta_{D}\right)\coth\left(\frac{\Delta}{2}\beta_{D}\right).$
(4)

The effective temperature $T_{\mathrm{eff}}=1/\beta_{\mathrm{eff}}$ here is
lower than the bath temperature $T_{D}$. As shown as follows, it is the lowing
of the effective temperature of the MD that results in an increasing of the
heat engine efficiency.

As for the modeling of the chamber as an infinite square potential well, the
eigenfunctions of the confined single molecule are

$\left\langle
x\right.\left|\psi_{n}\left(L\right)\right\rangle=\sqrt{\frac{2}{L}}\sin\left[n\pi
x/L\right],$ (5)

with the corresponding eigen-energies $E_{n}\left(L\right)=\left(\hbar
n\pi\right)^{2}/\left(2mL^{2}\right)$, where the quantum number $n$ ranges
from $1$ to $\infty$.

On this bases, the initial state of the total system is expressed as a product
state

$\rho_{0}=\frac{1}{Z\left(L\right)}\sum_{n}e^{-\beta
E_{n}\left(L\right)}\left|\psi_{n}\left(L\right)\right\rangle\left\langle\psi_{n}\left(L\right)\right|\otimes\rho_{0}^{D},$
(6)

where

$Z\left(L\right)=\sum_{n}\exp\left[-\beta E_{n}\left(L\right)\right]$ (7)

is the partition function of the system.

Here, we remark that the discrete spectrum of the system results from the
finite size of the width $L.$ As $L\rightarrow\infty$, the spectrum becomes
continuous as the energy level spacings is proportional to $1/L^{2}$. Then
heat excitation characterized by $k_{B}T$ can wash out the quantum effect so
that the system approaches a classical limit. Some of finite size effect based
quantum phenomenon could also disappear as $T\rightarrow\infty.$

With the above modelings, the MD-assisted thermodynamic cycle for the quantum
SHE is divided as four steps illustrated in Fig. 1(b): (I-II) the insertion of
a mobile solid wall into the potential well at a position $x=l$ (the origin is
$x=0$ ); (II-III) the measurement done by the MD to create the quantum
entanglement of its two internal states to the spatial wave functions of the
confined molecule; (III-IV) quantum control for the mobile wall to move the
according to the record in the demon’s memory; (IV-V) removing the wall so
that the next thermodynamic cycle can be restarted. Their descriptions will be
discussed subsequently in the next section and detailed calculations will be
found in the Appendix.

##  III Quantum Thermodynamic Cycles with Measurement

In this section we analyze in details the thermodynamic cycle of the molecule
confined in an infinite square potential well. The molecule’s position is
monitored and then controlled by the MD. The MD may have quantum coherence as
in Eq.2, or equivalently, possesses a lower temperature
$T_{\mathrm{D}}=1/\beta_{\mathrm{D}}$ than $T=1/\beta$ of the confined
molecule’s heat bath. In each step, we will evaluate the work done by outside
agent and heat exchange in detail. In order to concentrate on the physical
properties, we put the calculations in the Appendix.

Figure 2: (Color Online) Probability $P_{L}$ and the corresponding classical
one $P_{L}^{C}$ vs temperature $1/\beta$ for different piston position $l=1/3$
and $l=1/4$. Without losing generality, we set the parameters as $L=1$,
$m=\pi^{2}/2$ and $\hbar=1$.

### Step1: Quantum Insertion (I-II)

In the first process, the system is in contact with the heat bath $\beta$,
then a piston is inserted isothermally into the potential at position $l$. The
potential is then divided into two domains, denoted simply as $L$ and $R$,
with the length $l$ and $L-l$ respectively. The eigenstates change into the
following two sets as

$\displaystyle\left\langle x\right.\left|\psi_{n}^{R}(L-l)\right\rangle$
$\displaystyle=$
$\displaystyle\begin{cases}\sqrt{\frac{2}{L-l}}\sin\left[\frac{n\pi\left(x-l\right)}{L-l}\right]&l\leq
x\leq L\\\ 0&0\leq x\leq l\end{cases},$ (8)

and

$\left\langle x\right.\left|\psi_{n}^{L}(l)\right\rangle=\begin{cases}0&l\leq
x\leq L\\\ \sqrt{\frac{2}{l}}\sin(n\pi x/l)&0\leq x\leq l\end{cases},$

with the corresponding eigen-values $E_{n}\left(L-l\right)$ and
$E_{n}\left(l\right)$. In the following discussions we use the free
Hamiltonian $H_{T}=H+H_{D}$ for

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{n}[E_{n}\left(l\right)\left|\psi_{n}\left(l\right)\right\rangle\left\langle\psi_{n}\left(l\right)\right|$
$\displaystyle+E_{n}(L-l)\left|\psi_{n}\left(L-l\right)\right\rangle\left\langle\psi_{n}\left(L-l\right)\right|]$

for $0\leq l\leq L$ and $H_{D}=\Delta\left|e\right\rangle\left\langle
e\right|.$ Here, we take its ground state energy as the zero point of energy
of atom.

At the end of the insertion process, the system is still in the thermal state
with the temperature $\beta$ and the MD is on its own state without any
changes. With respect to the above splitted bases, the state of the whole
system is rewritten in terms of the new bases as

$\rho_{\mathrm{ins}}=[P_{L}\left(l\right)\rho^{L}\left(l\right)+P_{R}\left(l\right)\rho^{R}\left(L-l\right)]\otimes\rho_{0}^{D},$
(9)

where

$\rho^{L}\left(l\right)=\sum_{n}\frac{e^{-\beta
E_{n}\left(l\right)}}{Z\left(l\right)}\left|\psi_{n}^{L}(l)\right\rangle\left\langle\psi_{n}^{L}(l)\right|,$
(10)

and

$\rho^{R}\left(L-l\right)=\sum_{n}\frac{e^{-\beta
E_{n}\left(L-l\right)}}{Z\left(L-l\right)}\left|\psi_{n}^{R}(L-l)\right\rangle\left\langle\psi_{n}^{R}(L-l)\right|,$
(11)

refer to the system localized in the left and right domain respectively. With
respect to the their sum
$\mathcal{Z}\left(l\right)=Z\left(l\right)+Z\left(L-l\right),$ the temperature
dependent ratios

$P_{L}\left(l\right)=Z\left(l\right)/\mathcal{Z}\left(l\right)$

and

$P_{R}\left(l\right)=Z\left(L-l\right)/\mathcal{Z}\left(l\right).$

are the probabilities to find the single molecule on the left and the right
side respectively. For simplicity, we denote $P_{L}\left(l\right)$ and
$P_{R}\left(l\right)$ by $P_{L}$ and $P_{R}$ respectively in the following
discussions. We emphasize that the probabilities are different from the
classical probabilities, $P_{L}^{c}=l/L$ and $P_{L}^{c}=\left(L-l\right)/L$,
finding single molecule on the left and right side that is proportional to the
volume. We numerically illustrate this discrepancy between this classical
result and ours in Fig. 2 for different insertion position $l=1/3$ and
$l=1/4$. It is clearly in Fig. 2 that the probabilities $P_{L}$ approaches to
the corresponding classical ones $P_{L}^{c}$, as the temperature increases to
the high temperature limit. However, a large discrepancy is observed at low
temperature. This deviate from the classical one is due to the discreteness of
the energy levels of the potential well with finite width, which disappears as
level spacing becomes small with $L\rightarrow\infty$. In this case, the heat
excitation will erase all the quantum feature of the system and the classical
limit is approached.

In this step, work should be done to the system. In the isothermal process,
the work done by the outside agent can be expressed as
$W_{\mathrm{ins}}=\Delta U_{\mathrm{ins}}-T\Delta S_{\mathrm{ins}}$, with the
internal energy change

$\Delta
U_{\mathrm{ins}}=\mathrm{Tr}\left[\left(\rho_{\mathrm{ins}}-\rho_{0}\right)H_{T}\right]$

and the total entropy change

$\Delta
S_{\mathrm{ins}}=\mathrm{Tr}\left(-\rho_{\mathrm{ins}}\ln\rho_{\mathrm{ins}}+\rho_{0}\ln\rho_{0}\right).$

During this isothermal process, the work done by outside just compensates the
change of the free energy as

$W_{\mathrm{ins}}=T\left[\ln
Z\left(L\right)-\ln\mathcal{Z}\left(l\right)\right].$ (12)

The same result has been obtained in Ref. Ueda2010 . By taken inverse
temperature $\beta=1$ and $L=1$, we illustrate the work needed for the
insertion of the piston into the potential in Fig. 3. It is shown that to
insert the piston at the center of the potential needs the maximum work to be
done. Another reasonable fact is that no work is needed to insert the piston
at position $l=0$ and $l=L$. Classically, it is well known that no work should
be paid for inserting the piston at any position, while for a fixed $L$, we
notice that $W_{\mathrm{ins}}\rightarrow-\infty$ as $T\rightarrow\infty$. The
discrete property of the system due to the finite width of the potential well
results in the typical quantum effect, even at a high temperature, namely,
$\lim_{T\rightarrow\infty}W_{\mathrm{ins}}\neq 0$ and
$\lim_{T\rightarrow\infty}Q_{\mathrm{ins}}\neq 0$. This finite size induced
quantum effect is typical for mesoscopic system. To restore the classical
results, we simply take the limit $L\rightarrow\infty$ to make the spectrum
continuous, rather than $T\rightarrow\infty$. Under this limit
$L\rightarrow\infty$, we have
$\mathcal{Z}\left(l\right)/Z\left(L\right)\rightarrow 1$, which just recovers
the classical result that

$\lim_{L\rightarrow\infty}W_{\mathrm{ins}}=0,$ (13)

as illustrated in Fig. 3(b) for different insertion positions $l=0.1L$, $0.3L$
and $0.5L$.

Figure 3: (Color Online) Work done by the outside agent. (a)$W_{\mathrm{ins}}$
vs $l$ for different system inverse temperature $\beta=1$, $0.5$ and $0.1$.
Here, we choose the same parameter as that in Fig. 2. (b) $W_{\mathrm{ins}}$
vs $L$ for different insertion position $l=0.1L$, $0.3L$ and $0.5L$.

After the insertion of piston, the entropy of the system changes. The system
exchanges heat with the heat bath during this isothermal reversible process.
The heat is obtained by $Q_{\mathrm{ins}}=-T\Delta S_{\mathrm{ins}}$ as

$Q_{\mathrm{ins}}=\left(T-\frac{\partial}{\partial\beta}\right)\left[\ln
Z\left(L\right)-\ln\mathcal{Z}\left(l\right)\right].$ (14)

Similar to the asymptotic properties of the work in Eq. (13),
$Q_{\mathrm{ins}}$ approaches to zero when $L\rightarrow\infty$.

### Step2: Quantum Measurement (II-III)

In the second step, the system is isolated from the heat bath. The MD finds
which domain, the single molecule stays in and registers the result into its
own memory. In the classical way, the memory can also be imaged as a chamber
with single molecule. The classical state of single molecule on the right and
left side are denoted as the state $0$ and $1$. And the memory is architecture
always by two bistable states with no energy difference $\Delta=0$ and no
energy is needed in the measurement process. This setup based on “chamber ”
argument seems to exclude the possibility for quantum coherence in a
straightforward way. Therefore, we adopt the TLS as the memory to allow the
quantum coherence to take the role, as discussed in Sec. II. In the scheme
here, the demon performs the controlled-NOT operation Quan2006 . If the
molecule is on the left side, no operation is done. And the demon flips its
state, when finding the molecule on the right. This operation is realized by
the following unitary operator,

$\displaystyle U$ $\displaystyle=$
$\displaystyle\sum_{n}\left|\psi_{n}^{L}\left(l\right)\right\rangle\left\langle\psi_{n}^{L}\left(l\right)\right|\otimes\left(\left|g\right\rangle\left\langle
g\right|+\left|e\right\rangle\left\langle e\right|\right)$ (15)
$\displaystyle+\left|\psi_{n}^{R}\left(L-l\right)\right\rangle\left\langle\psi_{n}^{R}\left(L-l\right)\right|\otimes\left(\left|e\right\rangle\left\langle
g\right|+\mathrm{h.c}\right).$

After the measurement, the MD and the system are correlated. This correlation
enables the MD to control the system to perform work to the outside agent. The
density matrix of the whole system after measurement is

$\displaystyle\rho_{\mathrm{mea}}$ $\displaystyle=$
$\displaystyle\left[P_{L}p_{g}\rho^{L}\left(l\right)+P_{R}p_{e}\rho^{R}\left(L-l\right)\right]\otimes\left|g\right\rangle\left\langle
g\right|$ (16)
$\displaystyle+\left[P_{L}p_{e}\rho^{L}\left(l\right)+P_{R}p_{g}\rho^{R}\left(L-l\right)\right]\otimes\left|e\right\rangle\left\langle
e\right|.$

If the temperature of the demon is zero, namely $T_{D}=0$, the measurement
actually results in a perfect correlation between the system and the MD,

$\rho_{\mathrm{mea}}=P_{L}\rho^{L}\left(l\right)\otimes\left|g\right\rangle\left\langle
g\right|+P_{R}\rho^{R}\left(L-l\right)\otimes\left|e\right\rangle\left\langle
e\right|.$ (17)

Then the demon can distinguish exactly the domain where the single molecule
stays, e.g. state $\left|g\right\rangle$ representing the molecule on left
side and vice visa. At a finite temperature, this correlation gets ambiguous.
As illustrated in the dashed green box in Fig. 1(b), the demon actually gets
the wrong information about the domain, where the single molecule stays. For
example, the demon thinks the molecule is on the left with memory registering
$\left|g\right\rangle$, while the molecule is actually on the right. The MD
loses a certain amount of information about the system and lowers its ability
to extract work. For case $\Delta\neq 0$ at finite temperature, the above
imperfect correlation leads to a condition for the MD’s temperature, under
which the total system could extract positive work.

The worst case is that, when we first let the MD to become degenerate, i.e.,
$\Delta=0,$ then the temperature to approach zero. In this sense the demon is
prepared in s mixing state

$\rho_{0}^{D}\left(\Delta=0\right)=\frac{1}{2}\left(\left|g\right\rangle\left\langle
g\right|+\left|e\right\rangle\left\langle e\right|\right)$

and the state of the whole system after the measurement reads

$\rho_{\mathrm{mea}}=\left[\rho^{L}\left(l\right)+\rho^{R}\left(L-l\right)\right]\otimes\rho_{0}^{D}\left(\Delta=0\right).$
(18)

Thus, no information is obtained by MD. There exists another limit process
that the non-degenerate MD is firstly prepared in the zero-temperature
environment, and then let $\Delta$ approach zero. Thus, the state of the MD is
broken into $\left|g\right\rangle\left\langle g\right|$ of
$\rho_{0}^{D}\left(\Delta=0\right)$. In this case, we get a more cleaver MD as
mentioned above. The physical essence of the difference between the two limit
processes lies on the symmetry breakingjqliao2009  (we will discuss this again
later). With such symmetry breaking, the degenerate MD could also make an
ideal measurement. A intuitive understanding for the zero-temperature MD
helping to do work is that a more calm MD can see the states of the molecule
more clear, thus control it more effectively.

Next we calculate the work done in the measurement process by assuming the
total system being isolated from the heat bath of the molecule. The heat
exchange here is exactly zero, namely $Q_{\mathrm{mea}}=0$, since the
operation is unitary and the total entropy is not changed during this process.
However, the total internal energy changes, which merely results from the work
done by the outside agent

$W_{\mathrm{mea}}=P_{R}\left(p_{g}-p_{e}\right)\Delta.$

to register the MD’s memory. The work needed is actually a monotonous function
of the demon’s bath temperature $T_{D}$. If the temperature of the demon is
zero ( the MD is prepared in a pure state ), namely $T_{D}=0$, the work
reaches its maximum $W_{\mathrm{mea}}^{\mathrm{max}}=P_{R}\Delta$. The demon
can distinguish exactly the domain, where the single molecule stays, state
$\left|g\right\rangle$ representing molecule on the left, and vice visa. As
discussed as follows, the work done by the outside agent here is the main
factor to low down the efficiency of the heat engine. However, the low
temperature results in a more perfect quantum correlation between the MD and
the system, thus enables the MD to extract more work. Requirement of the work
done in the measurement and the ability of controlling free expansion are two
competing factors of the QHE. Finally, we prove that a low temperature of the
demon results in the high efficiency of quantum heat engine in the following
section. It is clear that less work is needed, if the insertion position is
closer to the right boundary of the potential. And the work needed in the
measurement process approaches to zero, namely $W_{mea}\rightarrow 0$, when
$l\rightarrow L$. Thus, the efficiency is promoted to reach the corresponding
Carnot efficiency when $l=L$ for this measurement.

### Step3: Controlled Expansion (III-IV)

In the third step, the system is brought into contact with the heat bath with
temperature $\beta$. Then the expansion is performed slowly enough to enable
the process to be reversible and isothermal. The expansion is controlled by
the demon according to its memory. Finding its state on
$\left|g\right\rangle$, the outside agent allows the piston to move right,
thus the single molecule performs work to the outside. However, the agent pays
some work to move piston to the right if the MD’s memory is inaccurate, e.g.
the situation in the green dashed box in Fig. 1(b). If in state
$\left|e\right\rangle$, the piston is allowed to move to the left side. Under
this description, we avoid the conventional heuristic discussion with adding
an object in the classical version of SHE. Here, we choose two arbitrary final
positions of the controlled expansion as $l_{g}$ and $l_{e}$ for the
corresponding MD’s state $\left|g\right\rangle$ and $\left|e\right\rangle$. We
will prove later that the total work extracted is independent on the expansion
position chosen here. After the expansion process, the density matrix of the
whole system is expressed as

$\displaystyle\rho_{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle\left[P_{L}p_{g}\rho^{L}\left(l_{g}\right)+P_{R}p_{e}\rho^{R}\left(L-l_{g}\right)\right]\otimes\left|g\right\rangle\left\langle
g\right|$ (19)
$\displaystyle+\left[P_{L}p_{e}\rho^{L}\left(l_{e}\right)+P_{R}p_{g}\rho^{R}\left(L-l_{e}\right)\right]\otimes\left|e\right\rangle\left\langle
e\right|.$

During the expansion, the system performs work $-W_{\mathrm{exp}}\geq 0$ to
the outside agent,

$\displaystyle W_{\mathrm{exp}}$ $\displaystyle=$ $\displaystyle
T\left[\ln\mathcal{Z}\left(l\right)+P_{L}\ln P_{L}+P_{R}\ln P_{R}\right.$ (20)
$\displaystyle-P_{L}p_{g}\ln Z\left(l_{g}\right)-P_{R}p_{e}\ln
Z\left(L-l_{g}\right)$ $\displaystyle\left.-P_{L}p_{e}\ln
Z\left(l_{e}\right)-P_{R}p_{g}\ln Z\left(L-l_{e}\right)\right].$

For a perfect correlation ($p_{g}=1$), the piston is moved to the side of the
potential, namely $l_{g}=L$ and $l_{e}=0$, and the work is simply

$W_{\mathrm{exp}}=T\left(P_{L}\ln P_{L}+P_{R}\ln
P_{R}\right)-W_{\mathrm{ins}},$

which is the maximum work one can be extracted in this process. In the
classical limit $L\rightarrow\infty$, and the work is

$W_{\mathrm{exp}}=T\left(P_{L}\ln P_{L}+P_{R}\ln P_{R}\right).$

We restore the well known result $W_{\mathrm{exp}}=-k_{\mathrm{B}}T\ln 2$,
when the piston is inserted in the center of the potential. If the demon is
not perfectly correlated to the position of the single molecule ($p_{g}<1$),
the work extracted $-W_{\mathrm{exp}}$ would be less. Therefore, it is clear
that the ability of MD to extract work closely depends on the accuracy of the
measurement.

In this step, the heat exchange is related to the change of entropy as

$\displaystyle\\!\\!\\!\\!Q_{\mathrm{exp}}=$ $\displaystyle
P_{L}\left(T-\frac{\partial}{\partial\beta}\right)\left[\ln
Z\left(l\right)-p_{g}\ln Z\left(l_{g}\right)-p_{e}\ln
Z\left(L-l_{e}\right)\right]$
$\displaystyle\\!\\!\\!\\!+P_{R}\left(T-\frac{\partial}{\partial\beta}\right)\left[\ln
Z\left(L-l\right)-p_{e}\ln Z\left(L-l_{g}\right)-p_{g}\ln
Z\left(l_{e}\right)\right].$ (21)

### Step4: Removing(IV-V)

To complete the thermodynamic cycle, the system and the MD should be reset to
their own initial states respectively. As for the system, the piston inserted
in the first step should be removed. In the previous studies, this process is
neglected, since the measurements are always ideal and the piston is moved to
an end boundary of the chamber. Thus no work is required to remove piston.
However, in an arbitrary process, we can show the importance of removing
piston in the whole cycle. During this process, the system keeps contact with
the heat bath with inverse temperature $\beta$ and the removing is performed
isothermally. The density matrix of the total system after removing the piston
reads

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\rho_{\mathrm{rev}}$ $\displaystyle=$
$\displaystyle\sum_{n}\frac{e^{-\beta
E_{n}\left(L\right)}}{Z(L)}\left|\psi_{n}\left(L\right)\right\rangle\left\langle\psi_{n}\left(L\right)\right|\otimes$
(22)
$\displaystyle\\!\\!\\!\\!\\!\left[\left(P_{L}p_{g}\\!+\\!P_{R}p_{e}\right)\left|g\right\rangle\\!\\!\left\langle
g\right|\\!+\\!\left(P_{L}p_{e}\\!+\\!P_{R}p_{g}\right)\left|e\right\rangle\\!\\!\left\langle
e\right|\right].$

In this process, the work done and the heat absorbed by the outside are

$\displaystyle\\!\\!\\!\\!\\!\\!\\!W_{\mathrm{rev}}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left[\left(\rho_{\mathrm{rev}}-\rho_{\mathrm{exp}}\right)\left(H+H_{D}\right)\right]$
(23)
$\displaystyle-T\mathrm{Tr}\left[-\rho_{\mathrm{rev}}\ln\rho_{\mathrm{rev}}\right]+T\mathrm{Tr}\left[-\rho_{\mathrm{exp}}\ln\rho_{\mathrm{exp}}\right],$

and

$Q_{\mathrm{rev}}=-T\mathrm{Tr}\left[-\rho_{\mathrm{rev}}\ln\rho_{\mathrm{rev}}\right]+T\mathrm{Tr}\left[-\rho_{\mathrm{exp}}\ln\rho_{\mathrm{exp}}\right],$
(24)

respectively. We refer the Appendix for the exact expression of those two
formula. The MD now is no longer entangled with the system. And the density
matrix of the demon is factorized out as

$\rho_{\mathrm{rev}}^{D}=\left(P_{L}p_{g}\\!+\\!P_{R}p_{e}\right)\left|g\right\rangle\\!\\!\left\langle
g\right|\\!+\\!\left(P_{L}p_{e}\\!+\\!P_{R}p_{g}\right)\left|e\right\rangle\\!\\!\left\langle
e\right|.$ (25)

In the ideal case $T_{D}=0$, the demon is on the state

$\rho_{\mathrm{rev}}^{D}=P_{L}\left|g\right\rangle\left\langle
g\right|+P_{e}\left|e\right\rangle\left\langle e\right|$

with entropy

$S_{\mathrm{rev}}^{D}=-P_{L}\ln P_{L}-P_{R}\ln P_{R}.$

According to Landauer’s Principal, erasing the memory of the MD dissipates at
least $T_{D}S_{\mathrm{rev}}^{D}=0$ work into the environment. In this sense,
we can extracted $k_{B}T\ln 2$ work with MD’s help. However, we does not
violate the SLoT, since the whole system functionalizes as a heat engine
working between high temperature bath and zero temperature bath. Actually, the
increase of entropy in the zero temperature bath is exactly
$S_{\mathrm{rev}}^{D}$. Therefore, the energy dissipated actually depends on
the temperature of environment, where the information is erased. In the
previous studies, people always set the same temperature for the system and
MD. Thus the exactly mechanism of MD was not clear to certain extent,
especially for SHE. Let’s consider another special case $\Delta=0$, which
directly results in $p_{e}=p_{g}=1/2$. MD is prepared on its maximum entropy
state $\rho_{0}^{D}\left(\Delta=0\right)$. At the end of the cycle, MD
actually is on the same state, namely
$\rho_{\mathrm{rev}}^{D}=\rho_{0}^{D}\left(\Delta=0\right)$. Thus, no work is
paid to erase the memory.

After this procedure, the MD is decoupled from the system and brought into
contact with its own thermal bath with inverse temperature $\beta_{D}$. Since

$P_{L}p_{e}+P_{R}p_{g}\geq p_{e},$ (26)

the MD releases energy into its heat bath. We will not discuss this
thermalization process here in details. The MD and the system are reset to
their own initial states $\rho_{0}$, which allows a new cycle to start.

Figure 4: (Color Online) Work vs insertion position $l$ and MD’s inverse
temperature $\beta_{D}$. (a) Total work as a function of $\beta_{D}$ for
different $l=0.2$, $0.5$ and $0.8$. (b)Total work as a function of insertion
position $l$ for different $\beta_{D}=2.0$, $3.0$ and $4.0$. (c) Contour plot
for total work as function of $l$ and $\beta_{D}$. The position for maximum
work extracted is denoted as white dashed line. Figure 5: (Color Online)
Efficiency vs insertion position $l$ and O inverse temperature $\beta_{D}$.
(a) Efficiency as a function of $\beta_{D}$ for different $l=0.4$, $0.5$ and
$0.6$. (b) Efficiency as a function $l$ for different $\beta_{D}=2$, $3$ and
$4$. (c) Contour plot of efficiency vs $l$ and $\beta_{D}$.

##  IV Efficiency of Szilard Heat Engine

For quantum version of the SHE, the quantum coherent based on the finite size
of the chamber results in various different properties from the classical one.
Work is required during the insertion and removing processes, while the same
process can be done freely in the classical version. The microscopic model
here relates the efficiency of the measurement by MD to the temperature of the
heat bath. In the whole thermodynamic cycle, the work done by the system to
outside is the sum of all the work done in each process,

$\displaystyle W_{\mathrm{tot}}$ $\displaystyle=$
$\displaystyle-\left(W_{\mathrm{ins}}+W_{\mathrm{mea}}+W_{\mathrm{exp}}+W_{\mathrm{rev}}\right)$
(27) $\displaystyle=$ $\displaystyle T\left[\left(p_{e}\ln p_{e}+p_{g}\ln
p_{g}\right)\right.$
$\displaystyle-\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)$
$\displaystyle\left.-\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)\right]$
$\displaystyle-P_{R}\left(p_{g}-p_{e}\right)\Delta.$

To enable the system to do work outside, the temperature of the MD should be
low enough to make sure $W_{\mathrm{tot}}\geq 0$, which is known as the
positive-work condition(PWC) htquan . To evaluate the efficiency of QHE, we
need to obtain the heat absorbed from the high temperature heat bath.
Different from the classical one, the exchange of heat with high temperature
source persists in each step. The total heat absorbed from the high
temperature source is the sum over that of all the four steps,

$\displaystyle Q_{\mathrm{tot}}$ $\displaystyle=$
$\displaystyle-\left(Q_{\mathrm{ins}}+Q_{\mathrm{mea}}+Q_{\mathrm{exp}}+Q_{\mathrm{rev}}\right)$
(28) $\displaystyle=$ $\displaystyle T\left[\left(p_{e}\ln p_{e}+p_{g}\ln
p_{g}\right)\right.$
$\displaystyle-\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)$
$\displaystyle\left.-\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)\right].$

Here, the absorbed energy is used to perform work to the outside, while only
the measurement process wastes $W_{\mathrm{mea}}$, which is released to the
low temperature heat bath. It is very interesting to notice that
$W_{\mathrm{mea}}\rightarrow 0$ as $\Delta\rightarrow 0$, while the total heat
$Q_{\mathrm{tot}}\rightarrow 0$ and $W_{\mathrm{tot}}\rightarrow 0$. To check
the validity of SLoT, one should concern the the efficiency of this heat
engine in a cycle,

$\displaystyle\eta$ $\displaystyle=$ $\displaystyle
1-\frac{P_{R}\left(p_{g}-p_{e}\right)\Delta}{Q_{\mathrm{tot}}}.$ (29)

As an example, we consider the special case $l=L/2$, which is similar to the
case of the ordinary SHE with the piston inserted in the center of the
chamber. In this special case, the probabilities for the single molecule
staying at the two sides are the same as that of the classical one, namely
$P_{L}=P_{R}=1/2$. The total work extracted here can be written in a simple
form

$W_{\mathrm{tot}}=T\left(\ln 2+p_{e}\ln p_{e}+p_{g}\ln
p_{g}\right)-\left(p_{g}-p_{e}\right)\Delta/2.$ (30)

In this special case, to make the system capable to do work on the outside,
there is a requirement to the temperature of the demon (low temperature bath).
For example, when we choose $\beta=1$ and $\Delta=0.5$, the PWC is
$\beta_{D}\geq 2.09$. This requirement is more strict than that of Carnot heat
engine, $\beta_{D}>1$. And the efficiency of this heat engine reads

$\eta=1-\frac{\left(p_{g}-p_{e}\right)\Delta}{2T\left(\ln 2+p_{e}\ln
p_{e}+p_{g}\ln p_{g}\right)},$ (31)

which is lower than the corresponding Carnot efficiency

$\eta_{\mathrm{Carnot}}=1-\frac{T_{D}}{T}.$

Here, the efficiency is a monotonic function of the energy spacing $\Delta$
and reaches its maximum

$\eta_{\mathrm{max}}=1-\frac{2T_{D}}{T}\leq\eta_{\mathrm{Carnot}}$

with $\Delta=0$.

In the general case, we show the work done by the system and efficiency of the
heat engine vs the position of the wall $l$ and the temperature of demon
$\beta_{D}$ in Fig. 4 and Fig. 5. As illustrated in Fig. 4(a), for small
insertion position, e.g. $l=0.16$ and $0.36$, the system can not extract
positive work. There exists a critical insertion position $l_{\mathrm{cri}}$
to extract positive work, namely

$T\left(P_{L}^{\mathrm{cri}}\ln P_{L}^{\mathrm{cri}}+P_{R}^{\mathrm{cri}}\ln
P_{R}^{\mathrm{cri}}\right)+P_{R}^{\mathrm{cri}}\Delta=0,$ (32)

where $P_{R}^{\mathrm{cri}}=P_{R}\left(l_{\mathrm{cri}}\right)$ and
$P_{L}^{\mathrm{cri}}=P_{L}\left(l_{\mathrm{cri}}\right)$. This critical value
of insertion position here is $l_{\mathrm{cri}}=0.447$ for the typical
parameter chosen here. Due to the requirement of work in the measurement
process, the work extracted is not a symmetric function of the insertion
piston $l$, namely $W_{\mathrm{tot}}\left(0.5-l\right)\neq
W_{\mathrm{tot}}\left(0.5+l\right)$, as illustrated in Fig. 4(b,c). Since the
high energy state $\left|e\right\rangle$ of the demon is utilized to register
the right side for single molecule, more work is need when $l<L/2$. Due to the
requirement of work done by outside agent in the measurement process, the
optimal position to extract maximum work is not at the center of the
potential. The maximum work can be extracted for a given MD’s inverse
temperature is reached, when

$\frac{P_{L}^{\mathrm{wmax}}p_{e}+P_{R}^{\mathrm{wmax}}p_{g}}{P_{L}^{\mathrm{wmax}}p_{g}+P_{R}^{\mathrm{wmax}}p_{e}}=e^{-\beta\Delta},$
(33)

where $P_{L}^{\mathrm{wmax}}=P_{L}\left(l_{\mathrm{wmax}}\right)$ and
$P_{R}^{\mathrm{wmax}}=P_{R}\left(l_{\mathrm{wmax}}\right)$. It is clear that
the position for the maximum work depends on the temperature of the demon
$\beta_{D}$.

In Fig. 5, we show the efficiency of this single molecular heat engine. We
consider only the positive work situation, and set efficiency as $0$ for all
the negative work area. Fig. 5(a) shows the monotonous behavior of efficiency
as the MD’s inverse temperature. Efficiency is also a monotonous function of
the insertion position $l$, illustrated in Fig. 5(b,c), which is not similar
to the total work extracted. It worth noticing that the efficiency reaches its
maximum at $l=1$, while no work can be extracted. Since the measurement is the
only way of wasting energy, it is the only way to improve the efficiency by
reducing $W_{\mathrm{mea}}$ with decreasing $P_{R}$. The efficiency of QHE
reaches the well-known Carnot efficiency $\eta_{\mathrm{Carnot}}$, when
$P_{R}=0$. At the same time, the total work extracted approaches to zero,
namely $W_{\mathrm{tot}}=0$. We meet this dilemma, since the measurement
results in an imperfect correlation between MD and the system.

Before concluding this paper, we draw our attention to two limit
processesjqliao2009  again

$\displaystyle\lim_{\beta_{D}\rightarrow+\infty}\lim_{\Delta\rightarrow
0}\rho_{D}$ $\displaystyle=\left(\left|g\right\rangle\left\langle
g\right|+\left|e\right\rangle\left\langle e\right|\right)/2,$ (34)
$\displaystyle\lim_{\Delta\rightarrow
0}\lim_{\beta_{D}\rightarrow+\infty}\rho_{D}$
$\displaystyle=\left|g\right\rangle\left\langle g\right|.$ (35)

Note that taking the two limits in different orders leads to completely
different results, the latter being a reflection of the spontaneous symmetry
breaking phenomenon. This difference for the MD’s initial state results in the
different work extracted, namely,

$\displaystyle\lim_{\beta_{D}\rightarrow+\infty}\lim_{\Delta\rightarrow
0}W_{\mathrm{tot}}$ $\displaystyle=0,$ (36)
$\displaystyle\lim_{\Delta\rightarrow
0}\lim_{\beta_{D}\rightarrow+\infty}W_{\mathrm{tot}}$ $\displaystyle=k_{B}T\ln
2.$ (37)

The former one means that MD actually gets no information about the position
of molecule and extracts no work, while the latter one show that MD obtains
the exact information on the position of the molecule and enables the system
to perform maximum work to the outside agent. The same phenomenon has also
been revealed in the process of dynamic thermalizationjqliao2009 .

##  V Conclusions

In summary, we have studied a quantum version of SHE with a quantum MD with
lower finite temperature than that of the system. We overall simplified the MD
as a two-level system, which carries out measurement in quantum fashion and
controlling the system to do work to the out-side agent. In this sense, the MD
assisted thermodynamic cycle are clarified as the four steps, insertion,
measurement, expansion and removing, which are all described in the framework
of quantum mechanics. In each step, we also consider the special case to
restore the well-known results in classical version of SHE. We explicitly
analyzed the total work extracted and the corresponding efficiency. To resolve
the MD paradox, we compared the obtained efficiency of the heat engine with
that of Carnot heat engine. It is found the efficiency is always below that of
Carnot since the quantum MD is included as the a part of the the whole work
substance and its functions are also correctly “quantized”. Thus nothing
violates the SLoT.

In comparison with the classical version of SHE, the following quantum natures
were discovered in the quantum thermodynamic cycles: (1) The finite size
effect of the potential well was found as reason for the non-vanishing work
required in the insertion and removing of the middle walls, while the
corresponding manipulations could be achieved freely in the classical case;
(2) The quantum coherence is allowed to exist in the MD’s density matrix. It
is the decrease of effective temperature caused by this coherence that
actually improves the efficiency of SHE; (3) In the measurement process, the
finite temperature of MD actually results in the incorrect decision to control
the single molecule’s motion. This incorrectness decreased the MD’s ability to
extract work. To our best knowledge, even for in the classical case, the
similar investigation has never been carried out; (4) In the whole
thermodynamic cycle, the removing process is necessary in returning to the
initial state for the whole work substance. This fact is neglected in the
previous studies even for the classical SHE.

Finally, we should stress that the model studied here could help to resolve
many paradoxical observations due to heuristic arguments with hybridization of
classical-quantum points of views about thermodynamics. For instance, it could
be recognized that the conventional argument about the MD paradox only
concerns a classical version of MD at the same temperature as that of the
system. Our results can enlighten the comprehensive understandings about some
fundamental problems in thermodynamics, such as the relationship between
quantum unitarity and SLoTDong2010  .

###### Acknowledgements.

HD would like to thank J.N Zhang for helpful discussion. This work was
supported by NSFC through grants 10974209 and 10935010 and by the National 973
program (Grant No. 2006CB921205).

## Appendix

In this appendix, we present a detailed calculation for the work done and
efficiency of SHE. Following the calculations for the four steps listed in the
context step by step, the reader can deeply understand the physical essences
of the MD in some subtle fashion.

Step 1: Insertion. In this process, the changes of internal energy $\Delta
U_{\mathrm{int}}=\mathrm{Tr}\left[\left(H+H_{D}\right)\left(\rho_{\mathrm{ins}}-\rho_{0}\right)\right]$
and total entropy $\Delta
S_{\mathrm{ins}}=\mathrm{Tr}\left[-\rho_{\mathrm{ins}}\ln\rho_{\mathrm{ins}}\right]-\mathrm{Tr}\left[-\rho_{\mathrm{0}}\ln\left(\rho_{0}\right)\right]$
is explicitly given by

$\displaystyle\Delta U_{\mathrm{int}}$
$\displaystyle=\sum_{n}p_{n}\left(l\right)E_{n}\left(l\right)+$
$\displaystyle\sum_{n}p_{n}\left(L^{\prime}\right)E_{n}\left(L^{\prime}\right)-\sum_{n}p_{n}\left(L\right)E_{n}\left(L\right)$
(38) $\displaystyle=\frac{\partial}{\partial\beta}\left[\ln
Z\left(L\right)-\ln\mathcal{Z}\left(l\right)\right],$

where $L^{\prime}=L-l$ and

$\displaystyle\Delta S_{\mathrm{ins}}$
$\displaystyle=\left(\ln\mathcal{Z}\left(l\right)-\ln Z\left(L\right)\right)+$
$\displaystyle\beta\sum_{n}\left[\begin{array}[]{c}p_{n}\left(l\right)E_{n}\left(l\right)\\\
+p_{n}\left(L^{\prime}\right)E_{n}\left(L^{\prime}\right)-p_{n}\left(L\right)E_{n}\left(L\right)\end{array}\right]$
(41)
$\displaystyle=\left(1-\beta\frac{\partial}{\partial\beta}\right)\left(\ln\mathcal{Z}\left(l\right)-\ln
Z\left(L\right)\right),$ (42)

where

$p_{n}\left(y\right)=\frac{\exp\left(-\beta
E_{n}\left(y\right)\right)}{Z\left(y\right)}.$

For the isothermal process, the work done by outside agent and the heat
exchange are simply $W_{\mathrm{ins}}=\Delta U_{\mathrm{int}}-T\Delta
S_{\mathrm{ins}}$ and $Q_{\mathrm{ins}}=-T\Delta S_{\mathrm{ins}}$, namely,

$\displaystyle W_{\mathrm{ins}}$ $\displaystyle=T\left[\ln
Z\left(L\right)-\ln\mathcal{Z}\left(l\right)\right],$ (43) $\displaystyle
Q_{\mathrm{ins}}$
$\displaystyle=\left(T-\frac{\partial}{\partial\beta}\right)\left[\ln
Z\left(L\right)-\ln\mathcal{Z}\left(l\right)\right].$ (44)

Step2: Measurement. The measurement is realized by a controlled-NOT unitary
operation, which has been illustrated clearly in the Sec. II. After the
measurement process, the density matrix for the total system is

$\displaystyle\rho_{\mathrm{mea}}$ $\displaystyle=$
$\displaystyle\left[P_{L}p_{g}\rho^{L}\left(l\right)+P_{R}p_{e}\rho^{R}\left(L^{\prime}\right)\right]\otimes\left|g\right\rangle\left\langle
g\right|$
$\displaystyle+\left[P_{L}p_{e}\rho^{L}\left(l\right)+P_{R}p_{g}\rho^{R}\left(L^{\prime}\right)\right]\otimes\left|e\right\rangle\left\langle
e\right|.$

The entropy is not changed in this step. And the work done by outside is

$W_{\mathrm{mea}}=\Delta
U_{\mathrm{mea}}=P_{R}\left(p_{g}-p_{e}\right)\Delta.$ (45)

Step3: Controlled expansion. At the ending of expansion, the state for the
total system reads

$\displaystyle\rho_{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle\left[P_{L}p_{g}\rho^{L}\left(l_{g}\right)+P_{R}p_{e}\rho^{R}\left(L_{g}\right)\right]\otimes\left|g\right\rangle\left\langle
g\right|$
$\displaystyle+\left[P_{L}p_{e}\rho^{L}\left(L_{e}\right)+P_{R}p_{g}\rho^{R}\left(l_{e}\right)\right]\otimes\left|e\right\rangle\left\langle
e\right|.$

where $L_{g}=L-l_{g}$ and $L_{e}=L-l_{e}$

We move the wall isothermally. And the work done by out-side agent can be
obtain by the same methods used in insertion process as

$\displaystyle W_{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left[\rho_{\mathrm{exp}}\left(H+H_{D}\right)\right]-\mathrm{Tr}\left[\rho_{\mathrm{mea}}\left(H+H_{D}\right)\right]$
(46)
$\displaystyle-T\mathrm{Tr}\left[-\rho_{\mathrm{exp}}\ln\rho_{\mathrm{exp}}\right]+T\mathrm{Tr}\left[-\rho_{\mathrm{mea}}\ln\rho_{\mathrm{mea}}\right]$
$\displaystyle=$
$\displaystyle\sum_{n}[P_{L}p_{g}p_{n}\left(l_{g}\right)E_{n}\left(l_{g}\right)+P_{R}p_{e}p_{n}\left(L_{g}\right)E_{n}\left(L_{g}\right)+P_{L}p_{e}p_{n}\left(L_{e}\right)E_{n}\left(L_{e}\right)$
$\displaystyle+P_{R}p_{g}p_{n}\left(l_{e}\right)E_{n}\left(l_{e}\right)]+\left(P_{L}p_{e}+P_{R}p_{g}\right)\Delta$
$\displaystyle-\sum_{n}\left(P_{L}p_{n}\left(l\right)E_{n}\left(l\right)+P_{R}p_{n}\left(L^{\prime}\right)E_{n}\left(L^{\prime}\right)\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\Delta$
$\displaystyle-T\sum_{n}\left[P_{L}p_{g}p_{n}\left(l_{g}\right)\ln
P_{L}p_{g}p_{n}\left(l_{g}\right)+P_{R}p_{e}p_{n}\left(L^{\prime}\right)\ln
P_{R}p_{e}p_{n}\left(L_{g}\right)\right.$
$\displaystyle\left.\qquad\qquad+P_{L}p_{e}p_{n}\left(L_{e}\right)\ln
P_{L}p_{e}p_{n}\left(L_{e}\right)+P_{R}p_{g}p_{n}\left(l_{e}\right)\ln
P_{R}p_{g}p_{n}\left(l_{e}\right)\right]$
$\displaystyle-T\sum_{n}[p_{g}p_{n}\left(l\right)\ln
p_{g}p_{n}\left(l\right)+p_{e}p_{n}\left(L^{\prime}\right)\ln
p_{e}p_{n}\left(L^{\prime}\right)$
$\displaystyle\qquad\qquad+p_{e}p_{n}\left(l\right)\ln
p_{e}p_{n}\left(l\right)+p_{g}p_{n}\left(L^{\prime}\right)\ln
p_{g}p_{n}\left(L^{\prime}\right)]$ $\displaystyle=$ $\displaystyle
P_{L}T\left[\ln Z\left(l\right)-p_{g}\ln Z\left(l_{g}\right)-p_{e}\ln
Z\left(L_{e}\right)\right]+P_{R}T\left[\ln Z\left(L^{\prime}\right)-p_{e}\ln
Z\left(L_{g}\right)-p_{g}\ln Z\left(l_{e}\right)\right].$ (47)

The internal energy changes can be also evaluated as

$\displaystyle\Delta U_{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle\sum_{n}\left[P_{L}p_{g}p_{n}\left(l_{g}\right)E_{n}\left(l_{g}\right)+P_{R}p_{e}p_{n}\left(L_{g}\right)E_{n}\left(L_{g}\right)+P_{L}p_{e}p_{n}\left(L_{e}\right)E_{n}\left(L_{e}\right)+P_{R}p_{g}p_{n}\left(l_{e}\right)E_{n}\left(l_{e}\right)\right]$
(48) $\displaystyle+\left(P_{L}p_{e}+P_{R}p_{g}\right)\Delta$
$\displaystyle-\sum_{n}\left(P_{L}p_{n}\left(l\right)E_{n}\left(l\right)+P_{R}p_{n}\left(L^{\prime}\right)E_{n}\left(L^{\prime}\right)\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\Delta$
$\displaystyle=$
$\displaystyle\sum_{n}\left[P_{L}p_{g}p_{n}\left(l_{g}\right)E_{n}\left(l_{g}\right)+P_{R}p_{e}p_{n}\left(L_{g}\right)E_{n}\left(L_{g}\right)+P_{L}p_{e}p_{n}\left(L_{e}\right)E_{n}\left(L_{e}\right)+P_{R}p_{g}p_{n}\left(l_{e}\right)E_{n}\left(l_{e}\right)\right]$
$\displaystyle-\sum_{n}\left[P_{L}p_{n}\left(l\right)E_{n}\left(l\right)+P_{R}p_{n}\left(L^{\prime}\right)E_{n}\left(L^{\prime}\right)\right]$
$\displaystyle=$ $\displaystyle P_{L}\frac{\partial}{\partial\beta}\left[\ln
Z\left(l\right)-p_{g}\ln Z\left(l_{g}\right)-p_{e}\ln
Z\left(L_{e}\right)\right]+P_{R}\frac{\partial}{\partial\beta}\left[\ln
Z\left(L^{\prime}\right)-p_{e}\ln Z\left(L_{g}\right)-p_{g}\ln
Z\left(l_{e}\right)\right].$

Then, we obtain the heat exchanges in this process as
$Q_{\mathrm{exp}}=-T\Delta S_{\mathrm{exp}}=W_{\mathrm{exp}}-\Delta
U_{\mathrm{exp}}$ or

$\displaystyle Q_{\mathrm{exp}}$ $\displaystyle=$ $\displaystyle
P_{L}\left(T-\frac{\partial}{\partial\beta}\right)\left[\ln
Z\left(l\right)-p_{g}\ln Z\left(l_{g}\right)-p_{e}\ln
Z\left(L_{g}\right)\right]$
$\displaystyle+P_{R}\left(T-\frac{\partial}{\partial\beta}\right)\left[\ln
Z\left(L^{\prime}\right)-p_{e}\ln Z\left(L_{g}\right)-p_{g}\ln
Z\left(l_{e}\right)\right].$

Step4: Removing. The piston is removed in this process. After that, the system
returns to its initial state and is not entangled with MD as The last step
would be remove the wall in the trap. The system is on the state as

$\displaystyle\rho_{\mathrm{rev}}$ $\displaystyle=$
$\displaystyle\sum_{n}\frac{\exp\left[-\beta
E_{n}\left(L\right)\right]}{Z(L)}\left|\psi_{n}\left(L\right)\right\rangle\left\langle\psi_{n}\left(L\right)\right|\otimes$
$\displaystyle\left[\left(P_{L}p_{g}+P_{R}p_{e}\right)\left|g\right\rangle\left\langle
g\right|+\left(P_{L}p_{e}+P_{R}p_{g}\right)\left|e\right\rangle\left\langle
e\right|\right].$

Then, the work done and the heat absorbed is respectively

$\displaystyle W_{\mathrm{rev}}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left[\rho_{\mathrm{rev}}\left(H+H_{D}\right)\right]-\mathrm{Tr}\left[\rho_{\mathrm{exp}}\left(H+H_{D}\right)\right]$
$\displaystyle-T\mathrm{Tr}\left[-\rho_{\mathrm{rev}}\ln\rho_{\mathrm{rev}}\right]+T\mathrm{Tr}\left[-\rho_{\mathrm{exp}}\ln\rho_{\mathrm{exp}}\right]$

or

$\displaystyle W_{\mathrm{rev}}$ $\displaystyle=$
$\displaystyle\sum_{n}p_{n}\left(L\right)E_{n}\left(L\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\Delta$
(51)
$\displaystyle-\sum_{n}\left[P_{L}p_{g}p_{n}\left(l_{g}\right)E_{n}\left(l_{g}\right)+P_{R}p_{e}p_{n}\left(L_{g}\right)E_{n}\left(L_{g}\right)+P_{L}p_{e}p_{n}\left(L_{e}\right)E_{n}\left(L_{e}\right)+P_{R}p_{g}p_{n}\left(l_{e}\right)E_{n}\left(l_{e}\right)\right]$
$\displaystyle-\left(P_{L}p_{e}+P_{R}p_{g}\right)\Delta$
$\displaystyle+T[\sum_{n}p_{n}\left(L\right)\ln
p_{n}\left(L\right)+\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)]$
$\displaystyle-T\sum_{n}\left\\{P_{L}p_{g}p_{n}\left(l_{g}\right)\ln\left[P_{L}p_{g}p_{n}\left(l_{g}\right)\right]+P_{R}p_{e}p_{n}\left(L_{g}\right)\ln\left[P_{R}p_{e}p_{n}\left(L_{g}\right)\right]\right.$
$\displaystyle\left.\qquad\qquad+P_{L}p_{e}p_{n}\left(L_{e}\right)\ln\left[P_{L}p_{e}p_{n}\left(L_{e}\right)\right]+P_{R}p_{g}p_{n}\left(l_{e}\right)\ln\left[P_{R}p_{g}p_{n}\left(l_{e}\right)\right]\right\\}$
$\displaystyle=$ $\displaystyle T\left[-\ln
Z\left(L\right)+\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)\right.$
$\displaystyle\qquad-P_{L}\ln P_{L}-P_{R}\ln P_{R}-p_{e}\ln p_{e}-p_{g}\ln
p_{g}$ $\displaystyle\left.\qquad+P_{L}p_{g}\ln
Z\left(l_{g}\right)+P_{R}p_{e}\ln Z\left(L_{g}\right)+P_{L}p_{e}\ln
Z\left(L_{e}\right)+P_{R}p_{g}\ln Z\left(l_{e}\right)\right].$

and

$\displaystyle Q_{\mathrm{rev}}$ $\displaystyle=$
$\displaystyle-T\mathrm{Tr}\left[-\rho_{\mathrm{rev}}\ln\rho_{\mathrm{rev}}\right]+T\mathrm{Tr}\left[-\rho_{\mathrm{exp}}\ln\rho_{\mathrm{exp}}\right]$
(52) $\displaystyle=$ $\displaystyle T\left[\sum_{n}p_{n}\left(L\right)\ln
p_{n}\left(L\right)+\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)\right]$
$\displaystyle-T\sum_{n}\left\\{P_{L}p_{g}p_{n}\left(l_{g}\right)\ln\left[P_{L}p_{g}p_{n}\left(l_{g}\right)\right]+P_{R}p_{e}p_{n}\left(L_{g}\right)\ln\left[P_{R}p_{e}p_{n}\left(L_{g}\right)\right]\right.$
$\displaystyle\left.\qquad\qquad+P_{L}p_{e}p_{n}\left(L_{e}\right)\ln\left[P_{L}p_{e}p_{n}\left(L_{e}\right)\right]+P_{R}p_{g}p_{n}\left(l_{e}\right)\ln\left[P_{R}p_{g}p_{n}\left(l_{e}\right)\right]\right\\}$
$\displaystyle=$ $\displaystyle T\left\\{-\ln
Z\left(L\right)+\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)\right.$
$\displaystyle\left.\qquad-P_{L}\ln P_{L}-P_{R}\ln P_{R}-p_{e}\ln
p_{e}-p_{g}\ln p_{g}+P_{L}p_{g}\ln Z\left(l_{g}\right)+P_{R}p_{e}\ln
Z\left(L_{g}\right)+P_{L}p_{e}\ln Z\left(L_{e}\right)+P_{R}p_{g}\ln
Z\left(l_{e}\right)\right\\}$
$\displaystyle-\sum_{n}\left[p_{n}\left(L\right)E_{n}\left(L\right)-P_{L}p_{g}p_{n}\left(l_{g}\right)E_{n}\left(l_{g}\right)-P_{R}p_{e}p_{n}\left(L_{g}\right)E_{n}\left(L_{g}\right)\right.$
$\displaystyle\left.\qquad\qquad-
P_{L}p_{e}p_{n}\left(L_{e}\right)E_{n}\left(L_{e}\right)-P_{R}p_{g}p_{n}\left(l_{e}\right)E_{n}\left(l_{e}\right)\right]$
$\displaystyle=$ $\displaystyle
T\left[\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)+\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)-P_{L}\ln
P_{L}-P_{R}\ln P_{R}-p_{e}\ln p_{e}-p_{g}\ln p_{g}\right]$
$\displaystyle-\left(T-\frac{\partial}{\partial\beta}\right)\ln
Z\left(L\right)+P_{L}p_{g}\left(T-\frac{\partial}{\partial\beta}\right)\ln
Z\left(l_{g}\right)+P_{R}p_{e}\left(T-\frac{\partial}{\partial\beta}\right)\ln
Z\left(L_{g}\right)$
$\displaystyle+P_{L}p_{e}\left(T-\frac{\partial}{\partial\beta}\right)\ln
Z\left(L_{e}\right)+P_{R}p_{g}\left(T-\frac{\partial}{\partial\beta}\right)\ln
Z\left(l_{e}\right).$

The total work extracted by outside agent is the sum of work extracted in each
step as

$\displaystyle W_{\mathrm{tot}}$
$\displaystyle=-\left(W_{\mathrm{ins}}+W_{\mathrm{mea}}+W_{\mathrm{exp}}+W_{\mathrm{rev}}\right)$
$\displaystyle=T[\left(p_{e}\ln p_{e}+p_{g}\ln
p_{g}\right)-\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)$
$\displaystyle\qquad\qquad-\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)]-P_{R}\left(p_{g}-p_{e}\right)\Delta.$
(53)

The total heat absorbed can also be obtained as

$\displaystyle Q_{\mathrm{tot}}$
$\displaystyle=-\left(Q_{\mathrm{ins}}+Q_{\mathrm{exp}}+Q_{\mathrm{rev}}\right)$
$\displaystyle=T\left[\begin{array}[]{c}\left(p_{e}\ln p_{e}+p_{g}\ln
p_{g}\right)\\\
-\left(P_{L}p_{g}+P_{R}p_{e}\right)\ln\left(P_{L}p_{g}+P_{R}p_{e}\right)\\\
-\left(P_{L}p_{e}+P_{R}p_{g}\right)\ln\left(P_{L}p_{e}+P_{R}p_{g}\right)\end{array}\right].$
(57)

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# Edge and Width Dependent Electronic Properties of Nanoribbons of Manganese
Oxide

Yigit Sozen  Department of Photonics, Izmir Institute of Technology, 35430,
Izmir, Turkey  Ugur C. Topkiran  Department of Physics, Izmir Institute of
Technology, 35430, Izmir, Turkey  Hasan Sahin  Department of Photonics, Izmir
Institute of Technology, 35430, Izmir, Turkey

###### Abstract

In the present work, the structural, magnetic, and electronic properties of
the two- and one-dimensional honeycomb structures of recently synthesized MnO
[Zhang et al. Nat. Commun., 20, 1073-1078 (2021)] are investigated by using
first principles calculations. Our calculations show that the single layer 2D
MnO crystal has a degenerate antiferromagnetic (AFM) ground state and a
relatively less favorable ferromagnetic (FM) state. In addition, magnetic
anisotropy calculations unveil that the easy-axis direction for magnetism
originating from unpaired electron states in manganese atoms is normal to the
crystal plane. Electronically, while the FM-MnO is a direct semiconductor with
a narrow bandgap, AFM phases display large indirect bandgap semiconducting
behavior. Moreover, calculations on nanoribbons of MnO reveal that zigzag
edged ribbons display metallic bahavior, whereas armchair edged nanoribbons
are semiconductors. Magnetically, for both zigzag- or armchair-edged
nanoribbons, AFM order perpendicular to the nanoribbon growth direction is
found to be favorable over the other AFM and FM orders. Moreover, depending on
the edge symmetry and ribbon width, forbidden band gap values of nanoribbons
display distinct family behaviors.

##  I Introduction

Over the past decade, it has been shown that the dimensional reduction not
only allows scaling of the device sizes but also brings outstanding physical
phenomena due to the quantized motion of electrons and holes, thus paving the
way for the development of compact devices with novel features Liu et al.
(2017). In 2004, nanotechnology has started to conduct beyond the nanoscale
with the realization of atomically thin derivative of two-dimensional (2D)
graphite sheet, graphene, which displays unconventional properties Novoselov
et al. (2004). The unique properties of graphene mainly stems from the
dominant quantum effects in atomic scale that brings enhanced electronic and
thermal transport Geim and Novoselov (2007); Neto et al. (2009). Graphene-
triggered research have also revealed the existence of brand new 2D materials
such as silicene Vogt et al. (2012); Cahangirov et al. (2009),
germaneneCahangirov et al. (2009), hexagonal BN Li and Chen (2016), transition
metal dichalcogenides (TMDs) Chhowalla et al. (2013); Mak et al. (2010); Wang
et al. (2012); Ross et al. (2014); Georgiou et al. (2013); Radisavljevic et
al. (2011); Tongay et al. (2014) and phosphorene Liu et al. (2014). Although
these materials are being studied extensively by researchers, the number of
ultrathin materials exhibiting magnetic properties is very few. Among the
ultra-thin 2D crystal materials that have been discovered so far, especially
CrI3 McGuire et al. (2015), Cr2Ge2Te6Xing et al. (2017), Fe2GeTe2 Deng et al.
(2018), FePS3 Wang et al. (2016), FeCl2 Zhou et al. (2020), VS2 Ma et al.
(2012), VSe2 Ma et al. (2012), VI3 He et al. (2016), VCl3 He et al. (2016),
MnSe2 O’Hara et al. (2018), and RuCl3 Banerjee et al. (2016) come to the fore.

Further in plane dimensional reduction of 2D crystal structures led to the
formation of nanoribbons (NRs) in which the propagation of charge carriers are
confined to one dimension (1D) along the ribbon growth direction
Yagmurcukardes et al. (2016); Son et al. (2006); Lee et al. (2005). The large-
scale fabrication of 1D derivatives of variety range of 2D materials, such as
graphene Barone et al. (2006); Cai et al. (2010); Vo et al. (2014);
Jacobberger et al. (2015), MoS2 Mak and Shan (2016); Chhowalla et al. (2013),
and phosphorene Masih Das et al. (2016), have been reported. These advances
promoted their integration into device applications of spintronics Ruffieux et
al. (2016); Kong et al. (2014), optoelectronics Blankenburg et al. (2012), and
quantum information technologies Guo et al. (2009); Chen and Chang (2015); Tan
et al. (2020).

Among the 1D family, graphene NRs have been most intensively studied
prospects, which are useful for future nanoelectronics Wang et al. (2021).
These materials are simply classified according to their edge geometries, and
each brings distinctive electronic and magnetic features. Resulting from the
hexagonal symmetry, zigzag and armchair edge formation is possible in graphene
NRs. While the edge states induces metallicity in zigzag type structure,
armchair edge termination opens a band gap, which is controllable changing the
ribbon width Lee et al. (2005); Son et al. (2006). Moreover, the passivation
edge states with H atoms in zigzag NR leads to direct band gap formation.
Unlike the nonmagnetic nature of armchair NR, intrinsic ferromagnetic order in
opposite edge states of zigzag NR were reported to couple
antiferromagnetically with each other Lee et al. (2005).

Although nanoribbons of various materials have been extensively studied, it
has not been investigated how the properties of a 2D material that exhibits
intrinsically magnetism changes with further confinement to 1D. As
demonstrated by a recent experimental study, single layer easy synthesis of
graphene-like 2D crystalline structures of TiO2, Fe2O3, CoO, Ni2O3, MnO and
Cu2O can be achieved by performing a strictly controlled oxidation at the
metal-gas interfaces Zhang et al. (2021). Among these materials, which have
been experimentally synthesized, manganese oxide (MnO), which offers a rich
playground in terms of its magnetic properties, was chosen as the target
material of our study. Here, we carry out first-principles calculations in
order to investigate the structural, electronic, and magnetic properties of
hexagonal MnO in one and two dimensions.

The paper is organized as follows: in Sec. II we provide the details of our
computational methodology. Magnetic and electronic properties of 2D MnO are
discussed in Sec. III(A). In addition, Sec. III(B) presents the edge- and
width-dependent properties of NRs.

##  II Computational Methodology

Structural relaxation, electronic dispersion and magnetic ground state
calculations were carried out on the basis of density functional theory (DFT)
implemented in the Vienna ab-initio Simulation Package (VASP)Kresse and Hafner
(1993); Kresse and Furthmüller (1996) based on plane wave projector augmented
wave (PAW) Kresse and Joubert (1999); Blöchl (1994) pseudopotential formalism
of spin-polarized density functional theory. Generalized gradient
approximation (GGA) in the form of Perdew-Burke-Ernzerhof (PBE) Perdew et al.
(1996) functional was used for exchange-correlation energy. The van der Waals
(vdW) forces were taken into account using the DFT-D2 method of Grimme Grimme
(2006). In addition, for determination of the eaxy axis of the magnetic
moments inside crystal structure non-collinear calculations were perfomed.
Bader charge technique was utilized to obtain the final charge density on
atoms constituting the crystal structure Henkelman et al. (2006).

To include the correlations between $d$-orbitals of Mn atoms, the difference
between the on-site Coulomb parameter (U) and the exchange parameter (J), Ueff
= U-J, was taken to be 3.9 eV on the basis of DFT+U method Dudarev et al.
(1998). The kinetic energy cutoff and the convergence criterion for the total
energy was set to 650 eV and 10-5 eV, respectively.

A suffieciently large vaccum space of 18 Å is inserted in order to avoid
adjacent layer-layer interactions along the z-axis. For the structural
optimization of the primitive unit cell of 2D and NR MnO, a
6$\times$5$\times$1 and 6$\times$1$\times$1 $\Gamma$-centered k-point meshes
were used respectively, and they were doubled for accurate density of states
calculations. Pressures on the unit cell were decreased to a value within
$\pm$ 1.0 kBar in all three directions to achieve a fully relaxed structure.

For various magnetic phases of 2D crystal, the cohesive energy per atom was
calculated using the equation
$E_{Coh}=\frac{1}{n_{tot}}[{n_{Mn}E_{Mn}+n_{O}E_{O}-E_{sys}}]$, where $n_{Mn}$
is the number of Mn atoms, $n_{O}$ is the number of O atoms, $n_{tot}$ is the
total number of atoms, $E_{Mn}$ is the energy of a single Mn atom, $E_{O}$ is
the energy of a single O atom and $E_{sys}$ is the total energy of the system.

##  III Results and discussion

###  III.1 2D Single Layers of Manganese Oxide

Figure 1: (a) Top and side views of the geometric structure of 2D MnO. Orange
and red atoms represent Mn and O, respectively. (b) Spin density plots of FM,
zz-AFM, and arm-AFM orders, at the isosurface value of $5\times 10^{-3}$ e/Å3.
$\Delta$E corresponds to energy difference per formula unit with respect to
the ground state structure. (c) Variation in the ground state energy of zz-AFM
and arm-AFM with respect to the applied strain along the armchair direction.

As shown in Fig. 1a, 2D MnO displays graphene-like planar hexagonal lattice
structure, which contains single Mn-O pair in its primitive unit cell.
Presence of unpaired electrons on each Mn requires investigation of possible
magnetic orders for determination of the ground state of the material.
Therefore, the repeating unit is considered as a rectangular unit cell, to
simulate ferromagnetic (FM), zigzag antiferromagnetic (zz-AFM) and armchair
antiferromagnetic (arm-AFM) orders in the crystal lattice (see Fig. 1b). Total
energy calculations reveal that, both AFM configurations are energetically
identical and more favorable than FM with the energy difference ($\Delta$E) of
0.09 eV per formula unit. 3D magnetic charge densities represented in Fig. 1b
indicate that AFM coupling inside the lattice is stabilized through the
superexchange interactions between magnetic ions mediated by O atoms.
Furthermore, it is found that each Mn atom induces a magnetic moment of 5
$\mu_{B}$ originating from the unpaired electrons of 3d orbitals.

Table 1: For the different magnetic phases of MnO, the optimized lattice parameters for rectangular unit cell, $a$ and $b$; atomic distances between Mn atom and neighboring O atoms, $d_{Mn-O_{1-3}}$; the internal angles within a single MnO hexagon, $\alpha$ and $\beta$; the charge transferred from a Mn atom to surrounding O atoms, ($\Delta\rho$); the magnetic moment per Mn atom, ($\mu$); the cohesive energy, $E_{Coh}$; the workfunction, $\Phi$; the band gap value $E_{gap}$, with the band gap type, D or I, corresponding to direct or indirect band gaps.  | $a$ | $b$ | $d_{Mn-O_{1}}$ | $d_{Mn-O_{2}}$ | $d_{Mn-O_{3}}$ | $\alpha$ | $\beta$ | $\Delta\rho$ | $\mu_{Mn}$ | $E_{Coh}$ | $\Phi$ | $E_{gap}$
---|---|---|---|---|---|---|---|---|---|---|---|---
| (Å) | (Å) | (Å) | (Å) | (Å) | (deg) | (deg) |  ($e^{-}$) |  ($\mu_{B}$) | (eV) | (eV) | (eV)
FM | 5.97 | 6.89 | 1.99 | 1.99 | 1.99 | 120 | 120 | 1.3 | 5 | 4.48 | 3.03 | 0.17 (D)
zz-AFM | 5.81 | 6.97 | 1.97 | 1.98 | 1.98 | 118.2 | 123.5 | 1.3 | 0 | 4.52 | 3.72 | 1.35 (I)
arm-AFM | 5.98 | 6.76 | 1.98 | 1.96 | 1.98 | 118.1 | 123.6 | 1.3 | 0 | 4.52 | 3.91 | 1.55 (I)
Figure 2: (a) Schematic illustration of the hexagonal lattice, and defined
spherical coordinate system, in which $\theta$ is the angle between spin
vector ($\vec{S}$) and x-axis and $\phi$ is the angle between $\vec{S}$ and
z-axis. x and y lattice vectors label armchair and zigzag directions. Magnetic
anisotropy energy (MAE) plots at spin vector directions for (b) AFM, (c) zz-
AFM and (d) arm-AFM ordered 2D MnO structures.

Optimized structural parameteres of 2D MnO are obtained by taking each
magnetic order into account. The calculated lattice parameters (a and b), and
distances between Mn-O atoms ($d_{Mn-O_{1-3}}$) are given in Table 1. It is
seen that while the FM phase has a hexagonal lattice symmetry, the lattice
favors a slightly distorted crystal structure when Mn atoms are
antiferromagnetically (AFM) coupled. In addition, the distortion within the
crystal lattice differs according to the prevailing AFM state. In zz-AFM, the
crystal lattice shows 2.7 % shrinkage and 1.2 % enlargening along the armchair
(a) and zigzag directions (b), respectively. On the other hand, in arm-AFM, a
and b enlargens and narrows of about 0.2 % and 1.9 %, respectively. The
variation in Mn-O-Mn angles with respect to the magnetic order are represented
as $\alpha$ and $\beta$ within the Table 1. The calculated bond length of 1.99
Å between Mn-O atoms for the FM state, is found to differ in between 1.97-1.98
and 1.96-1.98 within the lattice structures of zz-AFM and arm-AFM order,
respectively. Therefore, depending on the experimental procedure or synthesis
techniques one can expect presence of internal strain inside the crystal and
emergence of various magnetic phases in MnO single layers.

As seen in Fig. 1c, the magnetic phase transition from zz-AFM to arm-AFM, or
vice versa, is estimated by applying a tensile strain of above 1.37 % or a
compressive strain of above approximately 1.34 %, respectively. Therefore,
strain engineering is useful tool to acquire controllable properties in 2D
MnO. Furthermore, Bader charge analysis reveals that MnO lattice is formed by
the ionically bonded Mn-O atoms, resulting from a charge donation of 1.3
$e^{-}$ from Mn to a O atom.

In order to determine the stability of magnetization against the thermal
excitation, it is esential to investigate magnetocrystalline anisotropy in
magnetic materials. For this purpose, non-collinear calculations are carried
out for 2D MnO by rotating the spin vector through the angle $\phi$ and
$\theta$, corresponding to the angle that the spin vector makes with the z- or
x-axis (see Fig. 2a). Here, magnetic anisotropy energy (MAE) per conventional
cell is calculated by using the equation of $\Delta
E=E(\phi,\theta)-E(\phi=0^{\circ})$, where $E(\phi,\theta)$ is the total
energy of defined spin vector direction, and $E(\phi=0^{\circ})$ is the energy
of the easy axis. Fig. 2(b-d) shows that the MAE per cell, the easy axis is
perpendicular to the MnO basal plane ($\phi$ = 0∘) in all magnetic phases. The
hard axis parallel to the basal plane lies along the zigzag ($\theta$ = 90∘)
and armchair ($\theta$ = 0∘) directions for the structures consist of zz-AFM
and arm-AFM orders, respectively. In the case of FM order, the total energy is
independent from the varying $\theta$. 2D MnO displays small MAEs, which are
calculated to be 0.055 meV for FM, and 0.127 meV for zz-AFM and arm-AFM
ordered structures. The corresponding result reveals that single layer MnO is
a soft antiferromagnetic material, where the direction of magnetization can be
altered with small energies.

Figure 3: Electronic bandstructures calculated for FM, zz-AFM and arm-AFM
ordered MnO single layer structures.

Electronic bandstructure calculations show that depending on the magnetic
phase 2D MnO display quite different electronic characteristics. As shown in
Fig. 3, FM state causes a narrow direct band gap of 0.17 eV at the $\Gamma$
symmetry point. Conversely, AFM configurations causes an indirect band gap,
whereas depending on its type, valence and conduction band edges (VBM and CBM)
localize on distinctive zone points. In zz-AFM, the indirect band gap,
corresponding to the energy difference between X and $\Gamma$ point, is
estimated as 1.35 eV. The arm-AFM order, where VBM and CBM appears at Y and
$\Gamma$ point, respectively, yields comparably larger indirect band gap of
1.55 eV. Work functions ($\Phi$) are calculated to be 3.03, 3.72 and 3.91 for
FM, zz-AFM and arm-AFM orders.

###  III.2 Single Layer Nanoribbons of Manganese Oxide

####  III.2.1 Structural and magnetic properties

In this section, width dependent structural, magnetic and electronic
properties of MnO nanoribbons are analyzed considering zigzag and armchair
terminated edges. The width of the zigzag and armchair nanoribbons (ZNR and
ANR) are determined by the number of zigzag chains (Nz) and Mn-O dimer lines
(Na) within the confined lattice direction, respectively. Depending on the
width, ZNR and ANR structures are labelled as Nz-ZNR and Na-ANR, where Nz and
Na ranges from 10 to 5, and 17 to 6, respectively. The representative
illustration of each nanoribbon type is given in Fig. 4a.

Figure 4: (a) Structural illustrations of MnO nanoribbons having zigzag and
armchair edges. Nz and Na corresponds to the number of zigzag chains and Mn-O
dimer lines within the confined lattice, and indicate the ribbon width. The w
gives the distance between the opposite edge atoms. Plots in (b) and (c) show
the variation in total energy difference per formula unit between FM,
AFM-$\parallel$, and AFM-$\perp$ spin orders as a function of the ribbon
width.

To understand the effect of confinement of charge carriers into the 1D on the
magnetic properties, we investigate the energetics of FM and AFM
configurations with respect to the varying ribbon width. Depending on the edge
type, AFM orders with zigzag or armchair patterns can be perpendicular or
parallel to the direction of the NR structures. Here, AFM orders in ribbons
are labeled as AFM-$\parallel$ or AFM-$\perp$ with respect to the condition of
being parallel or perpendicular to the repeating direction, respectively.
Atomic structures of magnetic states and the variation in the total energy
difference per formula unit ($\Delta$E) between them as a function of the
ribbon width is represented in Fig. 4b and c, for ZNR and ANR structures,
respectively. As in the 2D structure of MnO, ribbon structures exhibit more
favorable energetics in AFM spin configurations compared to FM ones.

For ZNRs with larger surface area, the calculated total energies of AFM states
are nearly identical, however resulting from the edge effects that become
dominant in thinner structures, AFM-$\perp$ exhibits lower energetics compared
to AFM-$\parallel$. While the energy difference between them is $\sim$1.3 meV
in 10-ZNR, it increases to $\sim$9.3 meV in 5-ZNR. Even if ZNRs possess
greater energy values in FM state, the $\Delta$E with respect to the magnetic
ground state structure shows dramatic decrease with decreasing width.
Moreover, owing to the dangling bonds at edges, the AFM-$\perp$ ordered
ribbons display a small magnetic moment of 0.8 $\mu_{B}$ arising from the O
atoms. On the contrary, the total magnetization in the AFM-$\parallel$ order
oscillates from 1.2 to 8.8 $\mu_{B}$, depending on Nz being even or odd
number, respectively. In even case, the magnetism is purely from O atoms at
edges, however, in odd case, additional magnetism appears as the number of Mn
atoms that are aligned in the opposite direction is not equal.

In ANRs, depending on whether Na is even or odd, we obtain an unsual energy
trend resulting from the significant variation in the energy difference
between AFM states. The reason is attributed to the glide or reflection
symmetry between the opposite edges including even or odd number of dimer
lines, respectively. Variation in total energy difference between magnetic
states given in Fig 4c reveals that the total energy difference between AFM
orders becomes negligible if Na is even. However, when Na is odd, AFM-$\perp$
coupling displays lower energetics. The energy difference between
AFM-$\parallel$ and AFM-$\perp$ tends to show gradual increase with a
decreased width, such that, the $\Delta$E of $\sim$6.4 meV in 17-ANR is
estimated to be $\sim$14.7 meV in 7-ANR. On the other hand, $\Delta$E between
FM and ground state AFM distribution remains almost steady, except the slight
drop in 6-ANR. Unlike the even-numbered ANR structures where the total
magnetization completely vanishes, in odd-numbered structures, an additional
Mn atom causes a magnetism of 5 $\mu_{B}$.

####  III.2.2 Electronic properties

Electronic properties of ZNR and ANR structures are investigated as a function
of the ribbon width. Calculated electronic band structures show that ZNR
structures exhibit metallic character due to the band crossing at the Fermi
level caused by edge states. (see Fig. S1 in Supplementary Material) Since the
magnetic ground state of ANRs depend on its width represented by Na, the
investigation of electronic properties is carried out considering each type of
AFM configuration. ANRs are predicted to be semiconductors whose band
dispersion characteristics show significant dependency to the symmetry type in
between the opposite edges, and the type of AFM state (see Fig. 5). While ANRs
with AFM-$\perp$ order are indirect band gap semiconductors, AFM-$\parallel$
order causes the formation of direct band gap at $\Gamma$ point. Apparently,
the presence of Mn related magnetism in ANRs with odd Na cancels the
degeneracy between minorty and majority states.

Figure 5: The calculated electronic band structures of 17-, 16-, 7-, and 6-ANR
structures with respect to AFM-$\parallel$ and AFM-$\perp$ states. (b) The
plot gives the variation in the band gap values as a function of the ribbon
width.

In the case of AFM-$\perp$, the band gap values are inversely proportional to
the width, and depending on whether the Na is even or odd, the change in the
energy gap presents two distinct family behaviors. According to the plot in
Fig 5, which gives the evolution of band gap as a function of width, band gap
in AFM-$\perp$ ordered structure widens with decreasing width by following a
decent upward trend. As material gets thinner, odd-even based fluctuation in
band gap value becomes more noticeable. The calculated band gap of 1.44 and
1.47 eV for 17-ANR and 16-ANR structures, enlarges to 1.60 and 1.80 eV for
7-ANR and 6-ANR structures, respectively. Furthermore, the charge density of
lowermost conduction and topmost valence bands, labelled as CB and VB, are
calculated through the $\Gamma-X$ path and illustrated in Fig. 2S for the
specific ribbon widths. It is seen that CB is mostly dominated by the atomic
orbitals of Mn-d that localized at the edges. On the other hand, VB derived
from the mixed Mn-d and O-p orbitals, which distribute through the inner
lattice.

When AFM-$\parallel$ configuration is considered, the band gap values follow
more complex behavior with the decreased width. As seen in Fig. 4, even-
numbered ANRs induces significantly larger energy gap formation compared to
those in odd-numbered ones. On the other hand, the gap size follows two
distinct trends with respect to the condition of Na being equal to 4n+1 or
4n-1 (4n+2 or 4n-2) in odd (even) structures. In here, n is a positive
integer. The VB and CB in AFM-$\parallel$ order consist of atomic orbitals
mostly localized at edge atoms (see Fig. 3S). Due to the confinement of
electron and hole states at edges, the gap size follows an almost steady trend
down to the limit of the thinnest 6-ANR and 7-ANR structures, where it
dramatically increases to 1.78 and 1.41 eV, respectively, resulting from the
increased interaction between the opposite edge states. As it seen in Fig. 3S,
the CB dominantly has Mn-d atomic character, while the VB is Mn-d and O-p
like. Moreover,the multiple family like behavior in AFM-$\parallel$ ordered
ANR can be clarified with the spatial change in electron and hole states with
varying ribbon width.

##  IV Conclusions

In conclusion, this study is devoted to the first-principles investigation of
structural, magnetic, and electronic properties of single layers of infinite
2D and nanoribbon structures of hexagonal MnO. Our total energy calculations
revealed that while the ground state of MnO has a degenerate AFM character,
there is also an excited FM state with relatively larger energy. As a result
of their magnetic characteristics each phase also display distinct electronic
properties. While AFM phases have large bandgap indirect semiconducting
behavior, FM phase exhibits a direct narrow bandpag semiconducting behavior.
The effect of one-dimensional confinement in MnO was investigated by means of
zigzag and armchair edge terminated nanoribbon structures. Width-dependent
total energy calculations revealed that AFM ordering persists down to the
thinnest ribbon structures. In armchair nanoribbons having odd or even-
numbered Mn-O dimer lines, the opposite edges display glide or reflection
symmetry, which leads to a distinction in the comparable energetics of
considered AFM magnetic orders. Unlike the zigzag edge formation that induces
a metallicity, armchair-edged nanoribbons display semiconducting
characteristics. The band gap is inversely proportional to the ribbon width
and follows distinct trends originating from the variation in edge symmetry.
This variation brings two and four different family behaviors when AFM-$\perp$
and AFM-$\parallel$ orders are considered, respectively. Furthermore,
AFM-$\perp$ and AFM-$\parallel$ cause indirect and direct band gap formations
in armchair nanoribbon structures. Our findings show that the low-dimensional
structures of MnO with their edge and width dependent properties are promising
materials to be used in nanoscale device applications.

##  V Acknowledgments

Computational resources were provided by TUBITAK ULAKBIM, High Performance and
Grid Computing Center (TR-Grid e-Infrastructure).

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