# Superconductivity in dilute system of sites with strong electron-electron
attraction

A. Yu. Zyuzin  A. F. Ioffe Physical-Technical Institute of Russian Academy of
Sciences, 194021 St. Petersburg, Russia

###### Abstract

We consider the effect of strong electron-electron attraction on
superconductivity in the dilute system of the negative U-centers. They couple
to the conducting electronic states and mediate attraction between electrons.
We predict the formation of the local Cooper pairs provided attraction exceeds
a certain threshold value. The global coherent superconducting state appears
only due to the Andreev scattering between the centers at low temperatures.
The dependence of the superconducting transition temperature and the second
critical magnetic field on attraction strength is calculated.

###### pacs:

74.20.-z, 74.25.Op, 74.45.+c

##  I Introduction

Electron gas undergoes transition to the superconducting state even at weak
attraction between the particles. The formation of electron pairs and their
successive condensation takes place at the same temperature. As attraction
strength increases, the continuous transition to the state with strongly
coupled composite bosons arises bib:Leggett ; bib:Nozieres ; bib:Melo . Narrow
electron band structure favors the two-electron bound states bib:Micnas ,
which are characterized by the pair dissociation temperature. On the low-
temperature side, a coherent state similar to the Bose condensate emerges.
Here the coherent transition temperature is determined by the inverse time of
inter-particle exchange. For Bose particles, this time is of the order the
inter-particle distance flight time (here and below $\hbar=k_{B}=1$).

Semiconductors at a low carrier concentration have relatively large values of
the superconducting transition temperature. An original feature of such
systems is the negative U-centers at resonance levels that arise in the main
electronic band upon doping with certain impurities. The
$A^{\mathrm{IV}}B^{\mathrm{VI}}$ compounds bib:Chernik ; bib:Kaidanov ;
bib:Nemov  doped with the so-called valence skippers bib:Varma  are particular
examples of such semiconductors.

Superconducting models with local on-impurity attraction were considered in
several articles bib:Mal'shukov . The recent theoretical and experimental
studies of these compounds can be found in Refs. bib:Koba ; bib:Haldol ;
bib:Girado-Gallo ; bib:kriener . It was noted that the highest superconducting
transition temperature can be reached in the regime when the Fermi level is
pinned at the resonances provided the degeneracy of energy levels with the
number of electrons differs by two $E(N)\simeq E(N+2)$. Here $E(N)$ is the
energy of the center with $N$ electrons. The presence of such degeneracy
raises the question of the influence of the charge Kondo effect on
superconductivity bib:Dzero ; bib:Matsushita .

Using the Hubbard-Stratonovich transformation we study negative U-centers
mediated superconductivity at the vicinity as well as far beyond the level
degeneracy limit. A weak coupling regime exists up to the emergence of
degeneracy and can be described by the BCS approach. As attraction becomes
stronger, one notices the resemblance with the problem of composite bosons
formation, where Bose pairs are formed locally. Although, the inter-particle
exchange of bosons, which leads to the emergence of a coherent state, occurs
in our case due to Andreev scattering. Coherence is established at
temperatures much lower than the pair breakdown temperature. We also calculate
the upper critical magnetic field. It is shown that in the strong coupling
regime magnetic field does not lead to dissociation of bosons, but rather
destroys their coherence.

##  II Main Definitions

We begin with the system of negative U-centers. They are formed on the dopants
whose electronic resonance states are weakly overlaped with the conducting
states. The Hamiltonian of the system in terms of electron creation and
annihilation operators has a well-known form

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{\textbf{p},\sigma}[\epsilon(\textbf{p})-\mu]a^{+}(\textbf{p},\sigma)a(\textbf{p},\sigma)+U_{imp}$
(1) $\displaystyle+$
$\displaystyle\sum_{i,\sigma,\textbf{p}}V\big{[}a^{+}(\textbf{p},\sigma)a_{i}(\sigma)+a_{i}^{+}(\sigma)a(\textbf{p},\sigma)\big{]}$
$\displaystyle+$
$\displaystyle\sum_{i}\big{[}(\epsilon_{R}-\mu)(n_{i,\downarrow}+n_{i,\uparrow})-Wn_{i,\downarrow}n_{i,\uparrow}\big{]}.$

Here $a^{+}(\textbf{p},\sigma),a(\textbf{p},\sigma)$ are the creation and
annihilation operators of the conducting electrons with spectrum
$\epsilon(\textbf{p})$, $\mu$ is the chemical potential, and $U_{imp}$
describes random scattering of the conducting electrons. Third term in
Hamiltonian (1) describes hybridization between the conducting and the
resonance states. Here $a^{+}_{i}(\sigma),a_{i}(\sigma)$ are creation and
annihilation operators of electrons on the resonance site denoted by index
$i$. The last term in (1) contains a sum of Hamiltonians, which describe the
resonance sites, where $E_{R}\equiv\epsilon_{R}-\mu$ is the resonance energy
measured with respect to the chemical potential, and $W>0$ is the energy of
the electron-electron attraction on site $i$.

We consider a situation where the resonance site contains one resonance state
in the relevant energy region near the Fermi level. Hence the Green function
of the resonance site, which is coupled to the band states, might be
approximated as

$g_{i}(\textbf{r},\textbf{r}^{\prime},\omega_{n})=\psi(\textbf{r}-\textbf{r}_{i})\psi(\textbf{r}^{\prime}-\textbf{r}_{i})g(\omega_{n}),$
(2)

where

$g(\omega_{n})=\frac{1}{i\omega_{n}-E_{R}+i\gamma sign(\omega_{n})},$ (3)

with the Matsubara frequency $\omega_{n}=(2n+1)\pi T$, the wavefunction of
localized state $\psi(\textbf{r})$. We assume that the wave-function is
constant $\sim a^{-3/2}$ in a cube of size $a$ of the order wave-function
localization length. Smearing of the resonance due to hybridization with the
band states is described by

$\gamma=\pi a^{3}\nu_{0}V^{2},$ (4)

where $\nu_{0}$ is the one-spin density of the conducting states at the Fermi
level.

We consider the disordered conducting electrons, where mean free time $\tau$
is determined mostly by scattering on non-resonance sites. The Green function
of conducting electrons averaged over the impurities scattering bib:abricos
is given by

$G(\textbf{r},\textbf{r}^{\prime},\omega_{n})=\int\frac{d^{3}\textbf{p}}{(2\pi)^{3}}\frac{\exp[i\textbf{p}(\textbf{r}-\textbf{r}^{\prime})]}{i\omega_{n}-\epsilon(\textbf{p})+\mu+i\mathrm{sign}(\omega_{n})/2\tau}.$
(5)

##  III Superconducting instability at weak attraction

The instability of the system with respect to the transition to the
superconducting state is described by the ladder $L(i,k)$, which is shown in
Fig.1, bib:abricos . Note that we consider electron-electron interaction on
the resonance sites only. In the dilute system of resonance sites upon the
propagation between them, conducting electrons experience multiple scattering
events by impurities. These scattering processes can be described by the
cooperon-diffusion ladder, which is schematically shown in Fig.2 and is given
by

$C(\textbf{q},\omega_{1},\omega_{2})=\frac{\Theta(-\omega_{1}\omega_{2})}{Dq^{2}+|\omega_{12}|},$
(6)

where $D$ is the diffusion constant of conducting electrons due to impurity
scattering.

The ladder equation for $L(i,k)$ has discrete structure, where summation has
to be performed over the resonance cites

$[1-W\Pi(0)]L(i,k)=W\delta_{i,k}+W\sum_{i\neq m}\Pi(i-m)L(m,k).$ (7)

$W\Pi(0)$ is onsite contribution to ladder, determined by equation (10).

Figure 1: Graphic equation for superconducting ladder $L(i,k)$, shown as
rectangle with letter L. Single lines correspond to Green functions of
resonance sites (3). Black square corresponds to hybridization to conducting
band. Double line corresponds to Green function (5) of conducting electrons.
Zigzag vertical lines correspond to on resonance site electron-electron
interaction $W$. Propagation between sites $i$ and $m$ is described by the
cooperon diffusion pole, shown in Fig.2.

Noting the long range nature of the cooperon-diffusion pole (6) in the
definition of $\Pi(i-m)$, we replace the summation in (7) over sites $k\neq i$
by the integration over position coordinate $\textbf{r}_{k}$ weighted with the
density of resonance sites $n_{R}$. As a result we obtain

$\sum_{i\neq m}\Pi(i-m)L(m,k)=n_{R}\int
d\textbf{r}_{m}\Pi(\textbf{r}_{i}-\textbf{r}_{m})L(\textbf{r}_{m},\textbf{r}_{k}).$
(8)

In this approximation, the superconducting transition temperature is
determined by equation

$1-W\Pi(0)=Wn_{R}\int d\textbf{r}\Pi(\textbf{r}),$ (9)

where the right hand side is taken at zero momentum.

Dispersion equation (9) can be written through the Green functions (3) in the
form

$\displaystyle 1-WT\sum_{\omega_{n}}g(\omega_{n})g(-\omega_{n})=$ (10)
$\displaystyle=\frac{2Wn_{R}\gamma^{2}}{\pi\nu_{0}}T\sum_{\omega_{n}}[g(\omega_{n})g(-\omega_{n})]^{2}C(\textbf{q}=0,\omega_{n},-\omega_{n}).$

The right hand side in Eq. (10) can be calculated as

$Wn_{R}\frac{\gamma^{2}\Lambda}{\pi^{2}\nu_{0}}\ln(E_{0}/T)$

with logarithmic cutoff given by $E_{0}\sim\mathrm{min}(\gamma,|E_{R}|)$. Here
we also introduce

$\Lambda\equiv\frac{1}{(E_{R}^{2}+\gamma^{2})^{2}},$ (11)

which appeared due to the term $[g(\omega_{n})g(-\omega_{n})]^{2}$ taken in
the limit $\omega_{n}\rightarrow 0$ and will arise in all definitions of the
transition temperature and critical magnetic fields.

Figure 2: Graphic equation for the cooperon-diffusion pole. Electron-impurity
scattering is described by the vertical dotted line.

The superconducting transition temperature can be found from Eq. (10) in the
weak coupling regime in the form

$T_{c}=E_{0}\exp\bigg{\\{}-\frac{\pi^{2}\nu_{0}(1-W/W_{c})}{\gamma^{2}\Lambda
Wn_{R}}\bigg{\\}},$ (12)

where we introduce

$W^{-1}_{c}=T\sum_{\omega_{n}}g(\omega_{n})g(-\omega_{n}).$ (13)

At $T\ll\gamma,|E_{R}|$ we find

$W^{-1}_{c}=\frac{1}{2|E_{R}|}\left(1-\frac{2}{\pi}\arctan\frac{\gamma}{|E_{R}|}\right).$
(14)

Evidently, the weak coupling regime corresponds to $W/W_{c}<1$. In this case
equation (9) is satisfied due to the long-range nature of
$\Pi(\textbf{r}_{i}-\textbf{r}_{m})$, which leads to the logarithmic
contribution in (10). At $W\ll W_{c}$ expression (12) resembles the one
obtained in bib:Shelankov .

Note that for positive $E_{R}>\gamma$, the strong coupling regime $W\sim
W_{c}=2E_{R}$ corresponds the valence skipping condition bib:Varma .

##  IV Strong interaction $W/W_{c}\geq 1$

It is convenient to consider the intermediate as well as strong coupling
regimes within the framework of the Ginzburg-Landau (G-L) functional. We
separate the functional into the sum of local contributions of individual
resonance sites and contributions, which are associated with the interaction
between different sites.

###  IV.1 Local part of G-L functional

The Green function, which is related to the Hubbard-Stratonovich
transformation near the resonance site $i$, in saddle point approximation is
given by

$\displaystyle
g_{S,i}=-\frac{1}{\bar{\omega}_{n}^{2}+E^{2}_{R}+|\Delta_{i}|^{2}}\begin{pmatrix}i\bar{\omega}_{n}+E_{R}&-\Delta_{i}\\\
-\Delta^{*}_{i}&i\bar{\omega}_{n}-E_{R}\end{pmatrix},~{}~{}~{}$ (15)

where

$\bar{\omega}_{n}=\omega_{n}\left(1+\frac{\gamma}{|\omega_{n}|}\right).$ (16)

The self-consistent equation for the local
$\Delta_{i}(\textbf{r})=\Delta_{i}|\psi(\textbf{r}-\textbf{r}_{i})|^{2}$ is
given by

$\Delta_{i}=WT\sum_{\omega_{n}}\frac{\Delta_{i}}{\bar{\omega}_{n}^{2}+E^{2}_{R}+|\Delta_{i}|^{2}}.$
(17)

This equation determines the dependence of $\Delta_{i}(T,W)$ on the
temperature and attraction strength $W$. There is a critical value of
attraction $W_{c}$, which is given by (13) so that $\Delta_{i}(T,W)=0$ at
$W<W_{c}$.

The local part of G-L functional is given by a sum over the resonance sites

$F_{loc}=\sum_{i}(W^{-1}-W_{c}^{-1})\Delta^{2}_{i}+b\Delta^{4}_{i},$ (18)

where $W_{c}$ is determined by expression (13), the coefficient $b$ is given
by

$b=T\sum_{\omega_{n}>0}\left[(\omega_{n}+\gamma)^{2}+E^{2}_{R}\right]^{-2},$
(19)

Note that both $W_{c}$ and $b$ are functions of $T$, $E_{R}$, and level
broadening $\gamma$. To estimate, one can show that $b^{-1}\sim W^{3}_{c}\sim
max(|E_{R}|^{3},\gamma^{3})$.

We would like to emphasize that in the strong coupling regime at $W>W_{c}$ the
presence of nonzero value of $\Delta$ does not mean the existence of
superconductivity. To have superconducting coherence we need the nonzero
quantity

$\langle\Delta_{i}\Delta_{k}\exp{i(\phi_{i}-\phi_{k})}\rangle\neq 0$ (20)

at large $|\textbf{r}_{i}-\textbf{r}_{k}|$ compared to the average distance
between the centers. This condition is determined by the nonlocal contribution
to the functional G-L functional.

###  IV.2 Nonlocal part of G-L functional

It is instructive to consider the nonlocal part of G-L functional as the
result of Andreev reflections in a system with a given distribution of
$\Delta_{i}$.

Using Nambu operators

$\Psi(\textbf{r})=\begin{pmatrix}\varphi_{\uparrow}(\textbf{r})\\\
\varphi^{+}_{\downarrow}(\textbf{r})\end{pmatrix},\Psi^{+}(\textbf{r})=\begin{pmatrix}\varphi^{+}_{\uparrow}(\textbf{r})&\varphi_{\downarrow}(\textbf{r})\end{pmatrix}$
(21)

the Hamiltonian of the system can be written as

$\displaystyle H_{\Delta}$ $\displaystyle=$ $\displaystyle
H_{\mathrm{cond}}+\int
d\textbf{r}\sum_{i}\bigg{\\{}E_{R}\Psi^{+}_{i}(\textbf{r})\tau_{z}\Psi_{i}(\textbf{r})-$
$\displaystyle-$
$\displaystyle\frac{1}{2}\Psi^{+}_{i}(\textbf{r})[\Delta_{i}(\textbf{r})\tau_{x}+\Delta^{*}_{i}(\textbf{r})\tau_{y}]\Psi_{i}(\textbf{r})$
$\displaystyle+$
$\displaystyle{V(\textbf{r})[\Psi^{+}_{i}(\textbf{r})\tau_{z}\Psi(\textbf{r})+\Psi^{+}(\textbf{r})\tau_{z}\Psi_{i}(\textbf{r})}]\bigg{\\}},$

where $H_{\mathrm{cond}}$ is the Hamiltonian of conducting electrons written
in Nambu representation $\Psi(\textbf{r})$ and $\Psi^{+}(\textbf{r})$.

Let us consider a situation in which the order parameters is finite on sites
$\textbf{r}_{i}$ and $\textbf{r}_{k}$ only. In this case correction to the
Hamiltonian of free electron gas is given by

$\delta H=\delta H_{i}+\delta H_{k},$ (23)

where

$\delta H_{i}=\int
d\textbf{r}\Psi^{+}(\textbf{r}-\textbf{r}_{i})\begin{pmatrix}0&\Delta_{i}\\\
\Delta^{*}_{i}&0\end{pmatrix}\Psi(\textbf{r}-\textbf{r}_{i}).$ (24)

The thermodynamic potential has a contribution, which is proportional to the
product $\Delta_{i}\Delta_{k}$. This term corresponds to the diagram shown in
Fig. 3 and is given by

$F(i,k)=-T\int^{1/T}_{0}d\tau_{1}d\tau_{2}\langle T_{\tau}\delta
H_{i}(\tau_{1})\delta H_{k}(\tau_{2})\rangle.$ (25)

Here the evolution of perturbation is determined as $\delta
H_{i}(\tau)=\exp(\tau H_{\Delta=0})\delta H_{i}\exp(-\tau H_{\Delta=0})$. Also
note that expression in (25) depends of the superconducting phase difference
$\phi_{ik}$. One obtains

Figure 3: Nonlocal correction to G-L functional. Multiple impurity scattering
is described by the Cooper diffusion pole, shown on fig.2

$F(i,k)=-B(i,k)|\Delta_{i}\Delta_{k}|\cos\phi_{ik},$ (26)

where

$\displaystyle B(i,k)$ $\displaystyle=$
$\displaystyle\frac{8\gamma^{2}}{\pi\nu_{0}}T\sum_{\omega_{n}>0}\int\frac{d\textbf{q}}{(2\pi)^{3}}\frac{\exp(i\textbf{qr}_{ik})}{2\omega_{n}+Dq^{2}}$
(27) $\displaystyle\times$
$\displaystyle\frac{1}{[(\omega_{n}+\gamma)^{2}+E^{2}_{R}]^{2}}.$

The first factor in the denominator in (27) is due to the cooperon-diffusion
ladder (6).

###  IV.3 Ginzburg-Landau functional at $W\sim W_{c}$

Expressions (18) and (26) determine Ginzburg-Landau functional. One has

$\displaystyle F(\Delta_{i}e^{i\phi_{i}})$ $\displaystyle=$
$\displaystyle\sum_{i}\bigg{[}\frac{1-W/W_{c}}{W}|\Delta_{i}|^{2}+b|\Delta_{i}|^{4}\bigg{]}$
(28) $\displaystyle-$ $\displaystyle\frac{1}{2}\sum_{i\neq
k}B(i,k)|\Delta_{i}\Delta_{k}|\cos(\phi_{ik}).$

##  V Temperature of transition to the coherent state

Without magnetic field, the average value of the order parameter
$\Theta\equiv\langle\Delta\rangle$ is a uniform real quantity. Mean field
equation for the complex order parameter $\Delta=\theta_{1}+i\theta_{2}$ in
this case is given by

$\Theta\equiv\langle\theta_{1}+i\theta_{2}\rangle=\frac{\int
d\theta_{1}d\theta_{2}\Delta\exp(-F_{MF}(\Theta)/T)}{\int
d\theta_{1}d\theta_{2}\exp(-F_{MF}(\Theta)/T)},$ (29)

where

$F_{MF}(\Theta)=\frac{1-W/W_{c}}{W}|\Delta|^{2}+b|\Delta|^{4}-\Theta\theta_{1}\sum_{k\neq
0}B(0,k).$ (30)

Expanding the right hand side of (29) over $\Theta$, we obtain equation for
the temperature of transition to the coherent state

$1=\frac{\langle|\Delta|^{2}\rangle}{2T}\sum_{k\neq 0}B(0,k),$ (31)

where

$\langle|\Delta|^{2}\rangle=\frac{\int^{\infty}_{0}d\rho\rho\exp\left[-\left(\frac{1-W/W_{c}}{W}\rho+b\rho^{2}\right)/T\right]}{\int^{\infty}_{0}d\rho\exp\left[-\left(\frac{1-W/W_{c}}{W}\rho+b\rho^{2}\right)/T\right]}.$
(32)

Performing a change $\sum_{k\neq 0}B(0,k)\rightarrow n_{R}\int
d\textbf{r}B(\textbf{r})$, from (27) we obtain

$\sum_{k\neq 0}B(0,k)=\frac{2\gamma^{2}\Lambda
n_{R}\ln(E_{0}/T)}{\pi^{2}\nu_{0}}.$ (33)

By analyzing equation (31), we obtain three regimes, which are schematically
shown in Fig. (4).

Firstly, calculating correlation function in the weak coupling regime
$W<W_{c}$ we can neglect the term $b\Delta^{4}$. From (31), (32), and (33) we
obtain the transition temperature in weak coupling limit (12).

Second, at $W\sim W_{c}$ and $T_{c}>(W-W_{c})^{2}/W_{c}$, we can neglect the
term $\sim\rho$ in the exponents in (32). In this case we obtain transition
temperature from (31)

$T_{c}=\bigg{[}\frac{n_{R}\gamma^{2}\Lambda}{\pi^{2}\nu_{0}\sqrt{\pi
b}}\ln|E_{0}/T_{c}|\bigg{]}^{2}.$ (34)

This expression is valid for both $W<W_{c}$ and $W>W_{c}$.

Third, at stronger interaction the thermodynamic average of $\Delta$ has
nonzero value. At $(W-W_{c})^{2}/W_{c}>T_{c}$, we can neglect fluctuation
contribution in (32) so that the transition temperature is given by expression

$T_{c}=\frac{(W/W_{c}-1)}{Wb}\frac{n_{R}\gamma^{2}\Lambda}{2\pi^{2}\nu_{0}}\ln|E_{0}/T_{c}|.$
(35)

We note that (35) coincides with expression obtained in Ref. bib:Mal'shukov .

At stronger interaction $W\gg W_{c}$, one gets
$\langle|\Delta|^{2}\rangle=W^{2}$ in equation (31). Therefore $T_{c}$ grows
with $W$.

Figure 4: Schematic dependence of the superconducting transition temperature
on the attraction strength

##  VI Upper critical magnetic field

Let us consider applied magnetic field with the vector potential
$\textbf{A}(\textbf{r})$. In this case the average order parameter is
nonuniform real quantity $\langle\Delta_{i}\rangle=\Theta_{i}$. Instead of
(31), the self-consistency equation is given by

$\Theta_{i}=\frac{\beta\langle|\Delta|^{2}\rangle}{2T}\sum_{k\neq
i}B(i,k)\Theta_{k}.$ (36)

Replacing summation in Eq. (36) with the integration and expanding the order
parameter as

$\displaystyle\Theta_{k}=\Theta(\textbf{r}_{i})+\frac{\textbf{r}^{2}_{ik}}{2}\left(\frac{d}{d\textbf{r}_{i}}-\frac{2e}{c}i\textbf{A}\right)^{2}\Theta(\textbf{r}_{i})$
(37)

near $\textbf{r}_{i}$, on the right hand side of (36) we obtain

$\displaystyle\sum_{k\neq i}B(i,k)\Theta_{k}$ $\displaystyle=$ $\displaystyle
n_{R}\int d\textbf{r}B(\textbf{r})\Theta(\textbf{r}_{i})+$ $\displaystyle+$
$\displaystyle\frac{n_{R}}{2}\int
d\textbf{r}\textbf{r}^{2}B(\textbf{r})\left(\frac{d}{d\textbf{r}_{i}}-\frac{2e}{c}i\textbf{A}\right)^{2}\Theta(\textbf{r}_{i}).$

It is well known that in order to determine the upper critical magnetic field
$H_{c2}$, it is necessary to choose an eigenstate $\Theta(\textbf{r})$ of (VI)
with the the largest eigenvalue. Finally, we find that the upper critical
field is determined by equation

$n_{R}\int
d\textbf{r}B(\textbf{r})-\frac{2T}{\langle|\Delta|^{2}\rangle}=\frac{n_{R}|e|H_{c2}}{c}\int
d\textbf{r}\textbf{r}^{2}B(\textbf{r}).$ (39)

According to (27) we have

$\int
d\textbf{r}\textbf{r}^{2}B(\textbf{r})=\frac{D\Lambda\gamma^{2}}{2\pi\nu_{0}T_{c}}.$
(40)

In the weak coupling case $W<W_{c}$ and at temperature near $T_{c}$, we obtain

$\frac{D|e|H_{c2}}{c}=\frac{4}{\pi}(T_{c}-T).$ (41)

At $W\sim W_{c}$, substituting $\langle|\Delta|^{2}\rangle=(T/\pi b)^{1/2}$
into (39), and for the system with the transition temperature (34) we obtain

$\frac{D|e|H_{c2}}{c}=\frac{2}{\pi}(T_{c}-T)\ln(E_{0}/T_{c}).$ (42)

At strong attraction, when $W>W_{c}$ and
$\langle|\Delta|^{2}\rangle=\langle|\Delta|\rangle^{2}=\left|\frac{1-W/W_{c}}{2bW}\right|$,
the critical field can be found from

$\frac{D|e|H_{c2}}{c}=\frac{4}{\pi}(T_{c}-T)\ln|E_{0}/T_{c}|.$ (43)

The critical field grows with the the attraction, approaching the value (43)
at $W>W_{c}$.

We emphasize that the magnetic field does not affect the local value of
$\Delta_{i}$. Rather, it reduces the effective values of the Josephson
coupling between different sites.

##  VII Conclusions

In this paper we have studied the superconducting transition in dilute system
of negative U-centers with strong electron-electron attraction.

We have shown that local densities of the Cooper pairs emerge at a temperature
exceeding the superconducting transition temperature. The state exists
provided attraction is stronger than some critical value $W_{c}$, which
depends on the position and width of the resonance levels. Global
superconductivity is established due to the Andreev reflection between the
resonance level at a lower temperature. This regime resembles the transition
from the BCS state to the liquid of composite Bose particles.

Results do not change under changing sign of $E_{R}$. Therefore they are valid
not only for valence skippers at small $|2E_{R}-W|<<E_{R}$ but also for system
of sites with negative $E_{R}$ at $|2E_{R}+W|<<|E_{R}$

##  VIII Acknowledgment

The author is thankful to A. L. Shelankov , V. Zyuzin and A. Zyuzin for
illuminating discussions and Pirinem School of Theoretical Physics for
hospitality.

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# Analysis of Multistage Feedforward Operational Transconductance Amplifiers
using Single-Pole Approximation

Taeju Lee  Department of Electrical Engineering, Columbia University, New
York, NY 10027, USA
E-mail<EMAIL_ADDRESS>
Abstract—A high-gain wideband operational transconductance amplifier (OTA) is
an essential block for applications requiring high data rates. As technology
nodes scale down, the minimum length of transistors shrinks and the intrinsic
gain is reduced. Therefore, design techniques that achieve a high-gain
wideband operation are essential in analog amplifier design. Also, appropriate
frequency compensation is required to guarantee the reliable operation of an
OTA. This paper presents analysis results of the OTAs that combine feedforward
paths and multistage amplifiers to achieve high-gain wideband operation as
well as frequency compensation. To analyze multistage feedforward OTAs and
provide an intuitive design method, the single-pole approximation model is
employed for each substage of the OTA. Using the single-pole approximation
model, the analysis is carried out from the two-stage OTA to the four-stage
OTA in this work. Index Terms—Analog front-end, amplifier, compensation
technique, feedforward, high gain, multistage, operational transconductance
amplifier (OTA), wideband.

##  I. INTRODUCTION

Semiconductor technology has continued to develop, which has significantly
advanced electronic devices in terms of power, speed, and size. As CMOS
technology has advanced rapidly, computing engines and communication devices
with high data rates become available. In these applications that require high
data rates, a high-gain wideband operational transconductance amplifier (OTA)
is essential for various analog blocks. As technology nodes shrink, digital
circuits can be developed to occupy less area, consume lower power, and enable
high-speed operation. However, when developing analog circuits such as OTAs
using advanced CMOS technologies, it must be carefully designed while
considering the performance such as stability and gain.

As CMOS technology advances, the minimum length of transistors decreases,
improving the speed and power efficiency of digital circuits such as flip-
flops and logic gates. However, an intrinsic gain of a single transistor is
degraded due to the minimized length, causing gain errors in negative feedback
systems. To overcome this gain degradation, the OTA can be developed by
cascading multiple gain stages [1]–[5]. Especially, a two-stage OTA has been
widely employed in bias circuits, analog front-ends, analog-to-digital
converters, and wireless receivers for applications such as communications
[6]–[9] and sensor interfaces [10]–[13]. As the number of cascaded gain stages
increases, additional poles are created, causing phase shift and degrading
stability in feedback systems. Therefore, in an OTA used in a feedback system,
the technique called Miller compensation is widely used to compensate for
stability by controlling phase shift [4] and [5].

This paper provides the analysis results of the multistage feedforward OTAs.
Section II reviews Miller compensation which is a widely used method to secure
stability for feedback systems. Section III describes the analysis results of
applying the single-pole approximation model from the two-stage design to the
four-stage design. Finally, the conclusion is drawn in Section IV. All
analysis results are verified using a 65-nm CMOS technology.

##  II. GENERAL CONSIDERATION

Figure 1: (a) Miller compensation using $C\textsubscript{{m}}$. (b) Addition
of $R\textsubscript{{z}}$ to remove the RHP zero.

Two-stage OTAs are widely used to provide a sufficient gain, but an additional
pole contributes to phase shift, ultimately degrading the phase margin (PM).
To mitigate the PM degradation, Miller compensation has been widely employed
in numerous OTA designs. As shown in Fig. 1(a), the Miller capacitor
$C\textsubscript{{m}}$ is placed between the input and output nodes of the
second stage $M\textsubscript{2}$, splitting poles and ultimately improving
the PM. At the interstage node $X$, the capacitance looking into the gate
terminal of $M\textsubscript{2}$ increases to
$(1+A\textsubscript{{v}2})C\textsubscript{{m}}$, shifting the dominant pole
toward the origin in low frequencies [4]. When looking into the output node of
$M\textsubscript{2}$ at very high frequencies, the resistance decreases from
$r\textsubscript{{o}2}||r\textsubscript{{o}3}$ to approximately
$R\textsubscript{{o}1}||(1/g\textsubscript{{m}2})||r\textsubscript{{o}2}||r\textsubscript{{o}3}\approx
1/g\textsubscript{{m}2}$ since $C\textsubscript{{m}}$ becomes a low-impedance
path at high frequencies [4]. Accordingly, when $C\textsubscript{{m}}$ is
employed, the pole at node $X$ moves toward the origin, the pole at the output
node moves away from the origin, and two poles are split [4].

However, employing $C\textsubscript{{m}}$ also provides the feedforward path
from node $X$ to the output node, contributing to a negative phase shift. In
other words, as $C\textsubscript{{m}}$ creates the feedforward path, the gain
becomes zero at $\omega\textsubscript{{z}}$ called a right-half-plane (RHP)
zero. In Fig. 1(a), the RHP zero is obtained by solving the condition of
$V\textsubscript{{X}}C\textsubscript{{m}}\omega\textsubscript{{z}}=V\textsubscript{{X}}g\textsubscript{{m}2}$
which means that the small-signal output becomes zero at
$\omega\textsubscript{{z}}$ [4]. Considering the gate-drain parasitic
capacitance of $M\textsubscript{2}$, $C\textsubscript{{gd}2}$,
$\omega\textsubscript{{z}}$ can be rewritten as
$g\textsubscript{{m}2}/(C\textsubscript{{m}}+C\textsubscript{{gd}2})$. The RHP
zero contributes to a negative phase shift as with a pole, resulting in the PM
degradation. As a solution to remove the RHP zero, the compensation resistor
$R\textsubscript{{z}}$ can be series-connected with $C\textsubscript{{m}}$ as
shown in Fig. 1(b). After adding $R\textsubscript{{z}}$,
$\omega\textsubscript{{z}}$ is obtained as
$1/[C\textsubscript{{m}}(g\textsubscript{{m}2}^{-1}-R\textsubscript{{z}})]$ by
solving
$V\textsubscript{{X}}g\textsubscript{{m}2}=V\textsubscript{{X}}[R\textsubscript{{z}}+(C\textsubscript{{m}}\omega\textsubscript{{z}})^{-1}]^{-1}$
[4]. Therefore, the RHP zero can move to the left-half plane by satisfying
$R\textsubscript{{z}}>g\textsubscript{{m}2}^{-1}$. The left-half-plane (LHP)
zero contributes to a positive phase shift, which can cancel out a pole
through appropriate design techniques and eventually improve the PM.

The following analysis covers the compensation technique with LHP zeros
generated by feedforward paths implemented using active devices. For
simplicity, the analysis is first conducted from the two-stage structure using
the single-pole approximation model and expanded to the four-stage structure
in the same manner.

##  III. MULTISTAGE FEEDFORWARD OTA

###  A. Two-Stage Design

For simplicity of multistage analysis, a two-stage feedforward OTA is first
analyzed using the diagram shown in Fig. 2(a). When neglecting the decoupling
capacitor $C\textit{\textsubscript{d}}$, the gains of the main path are
expressed as
$A\textsubscript{{v}1}=g\textsubscript{{m}1}R\textsubscript{{o}1}$ and
$A\textsubscript{{v}2}=g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})$.
The gain of the feedforward path is
$A\textsubscript{{vF}1}=g\textsubscript{{mF}1}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})$.
Then, the overall transfer function of the two-stage structure is given by

$\displaystyle\frac{V\textit{\textsubscript{out}}}{V\textit{\textsubscript{in}}}(s)\bigg{|}\textsubscript{2-{stage}}$
$\displaystyle=\frac{A\textsubscript{{v}1}A\textsubscript{{v}2}}{\Big{(}1+\frac{s}{\omega\textsubscript{{p}1}}\Big{)}\Big{(}1+\frac{s}{\omega\textsubscript{{p}2}}\Big{)}}+\frac{A\textsubscript{{vF}1}}{\Big{(}1+\frac{s}{\omega\textsubscript{{p}2}}\Big{)}}$
(1)
$\displaystyle=\frac{\big{(}A\textsubscript{{v}1}A\textsubscript{{v}2}+A\textsubscript{{vF}1}\big{)}\bigg{[}1+\frac{A\textsubscript{{vF}1}s}{\big{(}A\textsubscript{{v}1}A\textsubscript{{v}2}+A\textsubscript{{vF}1}\big{)}\omega\textsubscript{{p}1}}\bigg{]}}{\Big{(}1+\frac{s}{\omega\textsubscript{{p}1}}\Big{)}\Big{(}1+\frac{s}{\omega\textsubscript{{p}2}}\Big{)}}$
(2)

Figure 2: (a) Two-stage feedforward OTA and (b) its Bode plots.

The multistage feedforward OTAs used in this work are designed based on [1].
Eqs. (1) and (2) are obtained in the same way as [1]. From Eq. (2), the
voltage gain at low frequencies can be obtained as
$g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})+g\textsubscript{{mF}1}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})$
and approximated as
$g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})$.
In Fig. 2(a), the poles at the first and second output stages are obtained as
$\omega\textsubscript{{p}1}=1/(R\textsubscript{{o}1}C\textsubscript{{o}1})$
and
$\omega\textsubscript{{p}2}=1/(R\textsubscript{{o}2}||R\textsubscript{{oF}1})(C\textsubscript{{o}2}+C\textsubscript{{oF}1})$.
From Eq. (2), the LHP zero generated by the feedforward path of
$g\textsubscript{{mF}1}$ is expressed as
$\omega\textsubscript{{z}1}=\omega\textsubscript{{p}1}[(A\textsubscript{{v}1}A\textsubscript{{v}2}+A\textsubscript{{vF}1})/A\textsubscript{{vF}1}]=\omega\textsubscript{{p}1}[(g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}/g\textsubscript{{mF}1})+1]$.
If $g\textsubscript{{mF}1}\gg g\textsubscript{{m}1},g\textsubscript{{m}2}$,
$\omega\textsubscript{{z}1}$ can be approximated as
$\omega\textsubscript{{p}1}$. Therefore, the first pole
$\omega\textsubscript{{p}1}$ can be canceled out by choosing
$g\textsubscript{{mF}1}\gg g\textsubscript{{m}1},g\textsubscript{{m}2}$. This
ultimately improves the stability and PM.

Although the PM is improved by employing a feedforward path, the voltage gain
is degraded by $R\textsubscript{{oF}1}$. Therefore, to isolate the output
resistance of the feedforward path from the main path, the decoupling
capacitor $C\textit{\textsubscript{d}}$ can be placed between the feedforward
path output and the main path output [2]. Compared to the design that places
$C\textit{\textsubscript{d}}$ at all inputs and outputs of feedforward paths
[2], this work places $C\textit{\textsubscript{d}}$ only at the output of the
outer feedforward path to save the design area. By placing
$C\textit{\textsubscript{d}}$ as shown in Fig. 2(a), the output impedance of
the feedforward path becomes high at low frequencies and the effective output
resistance of the OTA is dominated by $R\textsubscript{{o}2}$. Therefore, the
voltage gain changes from
$g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})$
to
$g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}R\textsubscript{{o}2}$.
Also, the dominant pole $\omega\textsubscript{{p}2}$ approximately changes
from
$1/(R\textsubscript{{o}2}||R\textsubscript{{oF}1})(C\textsubscript{{o}2}+C\textsubscript{{oF}1})$
to $1/(R\textsubscript{{o}2}C\textsubscript{{o}2})$ as the output impedance of
the feedforward path is isolated from the main path. As the frequency
increases, $C\textit{\textsubscript{d}}$ operates almost as a short circuit.
Accordingly, $\omega\textsubscript{{p}1}$ and $\omega\textsubscript{{z}1}$
located at high frequencies are almost the same as in the case without
$C\textit{\textsubscript{d}}$.

Fig. 2(b) shows the simulation results of gain and PM based on the two-stage
OTA shown in Fig. 2(a): using only the two-stage OTA without the feedforward
path significantly degrades the PM while achieving a high gain (red line),
adding the feedforward path to the main path improves the PM but the gain is
reduced due to the output resistance of the feedforward path (blue line), and
combining the feedforward path with $C\textit{\textsubscript{d}}$ compensates
for the gain while ensuring sufficient PM (green line). Fig. 2(b) is obtained
using the schematic shown in Fig. 3 by setting $C\textsubscript{{d}}$ = 2 pF,
$V\textsubscript{{DD}}$ = 1.2 V, $V\textsubscript{{cm}}$ = 0.6 V, and
$I\textsubscript{{ref}}$ = 400 $\mu$A. The output load impedance is modeled as
10 M$\Omega$||2 pF on $V\textsubscript{{out}--}$ and
$V\textsubscript{{out}+}$.

Fig. 3 shows the overall schematics of the fully differential two-stage
feedforward OTA. The input stage $g\textsubscript{{m}1}$ is implemented by
employing the cross-coupled structure $M\textsubscript{2}$ and the diode-
connected load $M\textsubscript{3}$, which improves the voltage gain by
maximizing the output resistance of the first stage [3]. The output stages
$g\textsubscript{{m}2}$ and $g\textsubscript{{mF}1}$ are implemented as the
current reuse structure. The common-mode voltage of each output stage is set
using the common-mode feedbacks $CM\textsubscript{1}$ and
$CM\textsubscript{2}$. $g\textsubscript{{m}1}$ consumes 273 $\mu$A.
$g\textsubscript{{m}2}$ and $g\textsubscript{{mF}1}$ consume 1.48 mA and 3 mA,
respectively. Each $CM\textsubscript{{n}=1,2}$ consumes 104 $\mu$A. Throughout
this paper, the multistage feedforward OTAs are implemented by employing the
bias block and the common-mode feedback, as shown in Figs. 3(b) and (c).

Figure 3: (a) Fully differential two-stage feedforward OTA, (b) its bias
block, and (c) the common-mode feedback used in the output stage.

###  B. Three-Stage Design

Extending Eqs. (1) and (2) to a three-stage structure, the OTA produces three
poles and two zeros in Fig. 4(a). However, obtaining poles and zeros becomes
complex as the number of stages extends. Therefore, a three-stage feedforward
OTA is approximated by replacing the internal two-stage feedforward structure
with a single-pole system as shown in Fig. 4(b). As discussed in a two-stage
feedforward OTA shown in Fig. 2(a), $\omega\textsubscript{{p}1}$ can be
canceled out by $\omega\textsubscript{{z}1}$ through a feedforward path.
Accordingly, a two-stage feedforward OTA can be approximated as a single-pole
system with a pole of
$1/(R\textsubscript{{o}2}||R\textsubscript{{oF}1})(C\textsubscript{{o}2}+C\textsubscript{{oF}1})$
and a gain of
$g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})$
as shown in Fig. 4(b).

Figure 4: (a) Three-stage feedforward OTA and (b) its approximated schematic.

The Miller capacitors, $C\textsubscript{{m}1}$ and $C\textsubscript{{m}2}$,
are employed in Figs. 4(a) and (b). However, to simplify the circuit analysis,
$C\textsubscript{{m}1}$ and $C\textsubscript{{m}2}$ are first neglected.
Extending Eq. (1) to the three-stage structure while neglecting
$C\textsubscript{{d}}$, the overall transfer function of Fig. 4(b) is given by

$\displaystyle\frac{V\textit{\textsubscript{out}}}{V\textit{\textsubscript{in}}}(s)\bigg{|}\textsubscript{3-{stage}}$
$\displaystyle=\frac{A\textsubscript{2-{stage}}A\textsubscript{{v}3}}{\Big{(}1+\frac{s}{\omega^{\prime}\textsubscript{{p}1}}\Big{)}\Big{(}1+\frac{s}{\omega^{\prime}\textsubscript{{p}2}}\Big{)}}+\frac{A\textsubscript{{vF}2}}{\Big{(}1+\frac{s}{\omega^{\prime}\textsubscript{{p}2}}\Big{)}}$
(3)

where $A\textsubscript{2-{stage}}$ is an approximated voltage gain of the
internal two-stage structure, $A\textsubscript{{v}3}$ is the output stage gain
of the main path, $A\textsubscript{{vF}2}$ is the output stage gain of the
feedforward path, $\omega^{\prime}\textsubscript{{p}1}$ is a pole at the
output of the internal two-stage structure, and
$\omega^{\prime}\textsubscript{{p}2}$ is an output pole. Neglecting
$C\textsubscript{{d}}$ for simplicity, each stage has a gain as follows:
$A\textsubscript{2-{stage}}=g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})$,
$A\textsubscript{{v}3}=g\textsubscript{{m}3}(R\textsubscript{{o}3}||R\textsubscript{{oF}2})$,
and
$A\textsubscript{{vF}2}=g\textsubscript{{mF}2}(R\textsubscript{{o}3}||R\textsubscript{{oF}2})$.
To obtain poles and zero in Fig. 4(b), Eq. (3) can be rewritten as

$\displaystyle\frac{\big{(}A\textsubscript{2-{stage}}A\textsubscript{{v}3}+A\textsubscript{{vF}2}\big{)}\bigg{[}1+\frac{A\textsubscript{{vF}2}s}{\big{(}A\textsubscript{2-{stage}}A\textsubscript{{v}3}+A\textsubscript{{vF}2}\big{)}\omega^{\prime}\textsubscript{{p}1}}\bigg{]}}{\Big{(}1+\frac{s}{\omega^{\prime}\textsubscript{{p}1}}\Big{)}\Big{(}1+\frac{s}{\omega^{\prime}\textsubscript{{p}2}}\Big{)}}$
(4)

In Eq. (4), the voltage gain at low freqeuncies is defined as
$g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})g\textsubscript{{m}3}(R\textsubscript{{o}3}||R\textsubscript{{oF}2})+g\textsubscript{{mF}2}(R\textsubscript{{o}3}||R\textsubscript{{oF}2})$.
This gain can be approximated by
$g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})g\textsubscript{{m}3}(R\textsubscript{{o}3}||R\textsubscript{{oF}2})$.
Two poles are obtained as
$\omega^{\prime}\textsubscript{{p}1}=1/(R\textsubscript{{o}2}||R\textsubscript{{oF}1})(C\textsubscript{{o}2}+C\textsubscript{{oF}1})$
and
$\omega^{\prime}\textsubscript{{p}2}=1/(R\textsubscript{{o}3}||R\textsubscript{{oF}2})(C\textsubscript{{o}3}+C\textsubscript{{oF}2})$.
The zero by the feedforward path of $g\textsubscript{{mF}2}$ is expressed as

$\displaystyle\omega^{\prime}\textsubscript{{z}1}$
$\displaystyle=\omega^{\prime}\textsubscript{{p}1}\bigg{(}\frac{A\textsubscript{2-{stage}}A\textsubscript{{v}3}}{A\textsubscript{{vF}2}}+1\bigg{)}$
$\displaystyle=\omega^{\prime}\textsubscript{{p}1}\bigg{[}\frac{g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})g\textsubscript{{m}3}}{g\textsubscript{{mF}2}}+1\bigg{]}$
(5)

In Fig. 4(b), considering $C\textsubscript{{d}}$ at low frequencies, the
output impedance of the feedforward path becomes high. At low frequencies, the
output impedance is therefore dominated by $R\textsubscript{{o}3}$ and
$C\textsubscript{{o}3}$ while ignoring $R\textsubscript{{oF}2}$ and
$C\textsubscript{{oF}2}$. Accordingly, the voltage gain from Eq. (4) can be
expressed as
$g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})g\textsubscript{{m}3}R\textsubscript{{o}3}$.
Also, the dominant pole $\omega^{\prime}\textsubscript{{p}2}$, which is in a
relatively lower frequency range than $\omega^{\prime}\textsubscript{{p}1}$
and $\omega^{\prime}\textsubscript{{z}1}$, changes from
$1/(R\textsubscript{{o}3}||R\textsubscript{{oF}2})(C\textsubscript{{o}3}+C\textsubscript{{oF}2})$
to $1/(R\textsubscript{{o}3}C\textsubscript{{o}3})$. As the frequency
increases, $C\textsubscript{{d}}$ becomes almost a short circuit. Therefore,
$\omega^{\prime}\textsubscript{{p}1}$ and
$\omega^{\prime}\textsubscript{{z}1}$, which are in the relatively higher
frequency range, remain similar to the state in the absence of
$C\textsubscript{{d}}$.

Recall that
$\omega\textsubscript{{z}1}=\omega\textsubscript{{p}1}[(g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}/g\textsubscript{{mF}1})+1]$
from the two-stage feedforward OTA. If $g\textsubscript{{mF}1}$ is large
enough, $\omega\textsubscript{{z}1}$ can be designed as
$\omega\textsubscript{{p}1}$, and $\omega\textsubscript{{p}1}$ can be canceled
out by $\omega\textsubscript{{z}1}$. Similar to this operation, in Eq. (5),
$\omega^{\prime}\textsubscript{{z}1}$ can be designed as
$\omega^{\prime}\textsubscript{{p}1}$ if $g\textsubscript{{mF}2}$ is large
enough. However, in practical design, there is a limit to increasing
$g\textsubscript{{mF}2}$ and
$g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})$
is also large, so $\omega^{\prime}\textsubscript{{z}1}$ is generally designed
to be larger than $\omega^{\prime}\textsubscript{{p}1}$. Fig. 5(a) shows the
phase shift of the three-stage feedforward OTA and the phase peaking is
generated due to the difference between $\omega^{\prime}\textsubscript{{p}1}$
and $\omega^{\prime}\textsubscript{{z}1}$.

Figure 5: (a) Phase shift of the three-stage feedforward OTA, (b) Bode plots
considering feedforward path and $C\textsubscript{{d}}$, and (c) Bode plots
considering $C\textsubscript{{m}}$.

Fig. 5(b) presents the simulated Bode plots of the three-stage feedforward OTA
according to $g\textsubscript{{mF}2}$ and $C\textsubscript{{d}}$. By including
the feedforward amplifier $g\textsubscript{{mF}2}$ (blue line), the PM is
improved but the difference between $\omega^{\prime}\textsubscript{{p}1}$ and
$\omega^{\prime}\textsubscript{{z}1}$ accompanies the phase peaking. Also,
employing $C\textsubscript{{d}}$ improves the voltage gain at low frequencies
(green line). To further improve the PM, the Miller capacitors,
$C\textsubscript{{m}1}$ and $C\textsubscript{{m}2}$ shown in Fig. 4, are
employed. As shown in Fig. 5(c), the PM is improved through Miller
compensation [4], which exchanges the positions of
$\omega^{\prime}\textsubscript{{p}1}$ and
$\omega^{\prime}\textsubscript{{p}2}$, but the bandwidth is sacrificed.
$\omega^{\prime}\textsubscript{{z}1}$ also moves slightly toward the origin as
$\omega^{\prime}\textsubscript{{p}1}$ approaches the origin. In this three-
stage OTA design, $C\textsubscript{{d}}$ is set to 2 pF, and
$C\textsubscript{{m}1}$ and $C\textsubscript{{m}2}$ are set to 50 fF. This
means that $C\textsubscript{{d}}$ makes the low-impedance path than
$C\textsubscript{{m}1}$ and $C\textsubscript{{m}2}$. Therefore, the OTA is
still dominated by the feedforward path than the Miller compensation path at
high frequencies, which means that Eq. (5) is still valid to describe
$\omega^{\prime}\textsubscript{{z}1}$. Comparing the three-stage feedforward
OTA and the two-stage feedforward OTA, the voltage gain improves as the number
of stages increases. However, the three-stage structure shows a rapid phase
change because $\omega^{\prime}\textsubscript{{z}1}$ is not completely
designed as $\omega^{\prime}\textsubscript{{p}1}$ in an actual circuit design.

Figure 6: Fully differential three-stage feedforward OTA.

Figs. 5(b) and (c) are obtained using the fully differential three-stage
feedforward OTA shown in Fig. 6. The bias block and common-mode feedback are
the same as those shown in Figs. 3(b) and (c). The output load impedance is
modeled as 10 M$\Omega$||2 pF on $V\textsubscript{{out}--}$ and
$V\textsubscript{{out}+}$. The supply voltage $V\textsubscript{{DD}}$ is set
to 1.2 V. The common-mode voltages of $V\textsubscript{{out}+/--}$ and
$V\textsubscript{{in}+/--}$ are set to 0.6 V. The current consumption of each
stage is as follows: $g\textsubscript{{m}1}$ (274 $\mu$A),
$g\textsubscript{{m}2}$ (794 $\mu$A), $g\textsubscript{{mF}1}$ (821 $\mu$A),
$g\textsubscript{{m}3}$ (1.64 mA), $g\textsubscript{{mF}2}$ (3.28 mA), and
$CM\textsubscript{{n}=1,2,3}$ (104 $\mu$A). Note that each
$CM\textsubscript{{n}=1,2,3}$ sets the common-mode voltage using
$V\textsubscript{{cm}}$ (= 0.6 V) as shown in Fig. 3(c).

###  C. Four-Stage Design

Recall that
$\omega\textsubscript{{z}1}=\omega\textsubscript{{p}1}\big{[}(g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}/g\textsubscript{{mF}1})+1\big{]}$
in the two-stage OTA, then the two-stage feedforward OTA can be approximated
as a single-pole system if
$\omega\textsubscript{{z}1}=\omega\textsubscript{{p}1}$ by making
$g\textsubscript{{mF}1}$ large enough. By employing the approximated model of
the two-stage feedforward OTA, the three-stage design also can be approximated
as shown in Fig. 4(b). Then, the zero of $\omega^{\prime}\textsubscript{{z}1}$
is obtained as
$\omega^{\prime}\textsubscript{{p}1}\big{[}(g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})g\textsubscript{{m}3}/g\textsubscript{{mF}2})+1\big{]}$
as shown in Eq. (5). Similar to $\omega\textsubscript{{z}1}$ of the two-stage
design, if $g\textsubscript{{mF}2}$ is large enough,
$\omega^{\prime}\textsubscript{{z}1}$ becomes
$\omega^{\prime}\textsubscript{{p}1}$, and
$\omega^{\prime}\textsubscript{{p}1}$ can be canceled out by
$\omega^{\prime}\textsubscript{{z}1}$. Therefore, theoretically, the three-
stage OTA can be simplified into a two-pole system by making
$g\textsubscript{{mF}1}$ large and further simplified into a single-pole
system by making $g\textsubscript{{mF}2}$ sufficiently large.

Fig. 7 shows the four-stage feedforward OTA including the approximated circuit
model of the preceding stages. Note that the Miller capacitors
$C\textsubscript{{m}1,2,3}$ are employed in the four-stage design, but will be
first neglected in the analysis for simplicity and discussed in the Bode plot
results. In the red box of Fig. 7, if $g\textsubscript{{mF}1}$ is large
enough, the two-stage design can be simplified as a single-pole system
employing
$G^{\prime}\textsubscript{{m}}=(g\textsubscript{{m}1}R\textsubscript{{o}1})g\textsubscript{{m}2}$,
$R^{\prime}\textsubscript{{o}}=R\textsubscript{{o}2}||R\textsubscript{{oF}1}$,
and
$C^{\prime}\textsubscript{{o}}=C\textsubscript{{o}2}+C\textsubscript{{oF}1}$.
In the blue box of Fig. 7, if $g\textsubscript{{mF}1}$ and
$g\textsubscript{{mF}2}$ are large enough, the three-stage design can be
simplified into a two-pole system and further simplified into a single-pole
system by employing $G^{\prime\prime}\textsubscript{{m}}$,
$R^{\prime\prime}\textsubscript{{o}}$, and
$C^{\prime\prime}\textsubscript{{o}}$. In Eq. (4),
$G^{\prime\prime}\textsubscript{{m}}$ is approximately obtained as
$g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})g\textsubscript{{m}3}$.
$R^{\prime\prime}\textsubscript{{o}}$ and
$C^{\prime\prime}\textsubscript{{o}}$ are obtained as
$R\textsubscript{{o}3}||R\textsubscript{{oF}2}$ and
$C\textsubscript{{o}3}+C\textsubscript{{oF}2}$, respectively. Leveraging
approximated single-pole models of the two- and three-stage OTAs while
neglecting $C\textsubscript{{d}}$ in Fig. 7, the transfer function of the
four-stage OTA can be given by

Figure 7: Four-stage feedforward OTA with approximated models of two- and
three-stage OTAs.

$\displaystyle\frac{V\textit{\textsubscript{out}}}{V\textit{\textsubscript{in}}}(s)\bigg{|}\textsubscript{4-{stage}}$
$\displaystyle=\frac{A\textsubscript{3-{stage}}A\textsubscript{{v}4}}{\Big{(}1+\frac{s}{\omega^{\prime\prime}\textsubscript{{p}1}}\Big{)}\Big{(}1+\frac{s}{\omega^{\prime\prime}\textsubscript{{p}2}}\Big{)}}+\frac{A\textsubscript{{vF}3}}{\Big{(}1+\frac{s}{\omega^{\prime\prime}\textsubscript{{p}2}}\Big{)}}$
(6)
$\displaystyle=\frac{\big{(}A\textsubscript{3-{stage}}A\textsubscript{{v}4}+A\textsubscript{{vF}3}\big{)}\bigg{[}1+\frac{A\textsubscript{{vF}3}s}{\big{(}A\textsubscript{3-{stage}}A\textsubscript{{v}4}+A\textsubscript{{vF}3}\big{)}\omega^{\prime\prime}\textsubscript{{p}1}}\bigg{]}}{\Big{(}1+\frac{s}{\omega^{\prime\prime}\textsubscript{{p}1}}\Big{)}\Big{(}1+\frac{s}{\omega^{\prime\prime}\textsubscript{{p}2}}\Big{)}}$
(7)

where
$A\textsubscript{3-{stage}}=G^{\prime\prime}\textsubscript{{m}}R^{\prime\prime}\textsubscript{{o}}$,
$A\textsubscript{{v}4}=g\textsubscript{{m}4}(R\textsubscript{{o}4}||R\textsubscript{{oF}3})$,
and
$A\textsubscript{{vF}3}=g\textsubscript{{mF}3}(R\textsubscript{{o}4}||R\textsubscript{{oF}3})$.
The poles of the three- and four-stage outputs are expressed as
$\omega^{\prime\prime}\textsubscript{{p}1}=1/(R\textsubscript{{o}3}||R\textsubscript{{oF}2})(C\textsubscript{{o}3}+C\textsubscript{{oF}2})$
and
$\omega^{\prime\prime}\textsubscript{{p}2}=1/(R\textsubscript{{o}4}||R\textsubscript{{oF}3})(C\textsubscript{{o}4}+C\textsubscript{{oF}3})$,
respectively. In Eq. (7) obtained using the single-pole approximation model,
the zero of the four-stage feedforward OTA is given by

$\displaystyle\omega^{\prime\prime}\textsubscript{{z}1}$
$\displaystyle=\omega^{\prime\prime}\textsubscript{{p}1}\bigg{(}\frac{A\textsubscript{3-{stage}}A\textsubscript{{v}4}}{A\textsubscript{{vF}3}}+1\bigg{)}$
$\displaystyle=\omega^{\prime\prime}\textsubscript{{p}1}\bigg{[}\frac{g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})g\textsubscript{{m}3}(R\textsubscript{{o}3}||R\textsubscript{{oF}2})g\textsubscript{{m}4}}{g\textsubscript{{mF}3}}+1\bigg{]}$
(8)

Table 1 summarizes the zeros according to the number of stages. Recall that
$\omega^{\prime}\textsubscript{{z}1}$ and
$\omega^{\prime\prime}\textsubscript{{z}1}$ are obtained assuming that
$g\textsubscript{{mF}1}$ and $g\textsubscript{{mF}2}$ are large enough.
Similar to two- and three-stage OTAs, for improving the PM in the four-stage
feedforward OTA, $g\textsubscript{{mF}3}$ must also be large enough to cancel
out $\omega^{\prime\prime}\textsubscript{{p}1}$. From the equations of
$\omega\textsubscript{{z}1}$, $\omega^{\prime}\textsubscript{{z}1}$, and
$\omega^{\prime\prime}\textsubscript{{z}1}$ as summarized in Table 1, it can
be concluded that the transconductance of the feedforward path must be
sufficiently large to secure the PM. Assuming that $g\textsubscript{{mF}1}$
and $g\textsubscript{{mF}2}$ are large enough, Eq. (7) can be rewritten as

$\displaystyle\frac{V\textit{\textsubscript{out}}}{V\textit{\textsubscript{in}}}(s)\bigg{|}\textsubscript{4-{stage}}=\frac{A\textsubscript{4-{stage}}\Big{(}1+\frac{s}{\omega\textsubscript{{z}1}}\Big{)}\Big{(}1+\frac{s}{\omega^{\prime}\textsubscript{{z}1}}\Big{)}\Big{(}1+\frac{s}{\omega^{\prime\prime}\textsubscript{{z}1}}\Big{)}}{\Big{(}1+\frac{s}{\omega\textsubscript{{p}1}}\Big{)}\Big{(}1+\frac{s}{\omega^{\prime}\textsubscript{{p}1}}\Big{)}\Big{(}1+\frac{s}{\omega^{\prime\prime}\textsubscript{{p}1}}\Big{)}\Big{(}1+\frac{s}{\omega^{\prime\prime}\textsubscript{{p}2}}\Big{)}}$
(9)

where $A\textsubscript{4-{stage}}$ can be approximated by
$g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})g\textsubscript{{m}3}(R\textsubscript{{o}3}||R\textsubscript{{oF}2})g\textsubscript{{m}4}(R\textsubscript{{o}4}||R\textsubscript{{oF}3})$.
The four-stage OTA shown in Fig. 7 consists of four poles and three zeros in
Eq. (9). Ideally, by making $g\textsubscript{{mF}1}$ large enough,
$[1+(s/\omega\textsubscript{{p}1})]$ in the denominator can be canceled out by
$[1+(s/\omega\textsubscript{{z}1})]$ in the numerator. Similarly,
$[1+(s/\omega^{\prime}\textsubscript{{p}1})]$ in the denominator can be
canceled out by $[1+(s/\omega^{\prime}\textsubscript{{z}1})]$ in the numerator
as $g\textsubscript{{mF}2}$ becomes large enough. Eventually, the four-stage
OTA can be approximated by a single-pole system assuming that
$[1+(s/\omega^{\prime\prime}\textsubscript{{p}1})]$ in the denominator is
canceled out by $[1+(s/\omega^{\prime\prime}\textsubscript{{z}1})]$ in the
numerator as $g\textsubscript{{mF}3}$ becomes sufficiently large. As mentioned
above, to satisfy the condition that the poles are canceled out by the zeros,
all feedforward stages need to be designed to have sufficiently large
transconductances while sustaining $g\textsubscript{{mF}1}\ll
g\textsubscript{{mF}2}\ll g\textsubscript{{mF}3}$ as shown in Table 1.
However, as the number of stages increases, the burden on the transconductance
of the feedforward path increases for canceling out poles, which is associated
with design area and power consumption.

Table 1. Zeros according to the number of stages
---
Order |  Zero
[8pt]2-stage |  $\omega\textsubscript{{z}1}=\omega\textsubscript{{p}1}\Big{(}\frac{g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}}{g\textsubscript{{mF}1}}+1\Big{)}$
[12pt]3-stage |  $\omega^{\prime}\textsubscript{{z}1}=\omega^{\prime}\textsubscript{{p}1}\bigg{[}\frac{g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})g\textsubscript{{m}3}}{g\textsubscript{{mF}2}}+1\bigg{]}$
[12pt]4-stage |  $\omega^{\prime\prime}\textsubscript{{z}1}=\omega^{\prime\prime}\textsubscript{{p}1}\bigg{[}\frac{g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})g\textsubscript{{m}3}(R\textsubscript{{o}3}||R\textsubscript{{oF}2})g\textsubscript{{m}4}}{g\textsubscript{{mF}3}}+1\bigg{]}$

In other words, to cancel out a pole by a zero in each stage, the
transconductance ratio between the output stages of each order must be greater
than the gain of the previous stage. Therefore, the following conditions must
be satisfied: $(g\textsubscript{{mF}1}/g\textsubscript{{m}2})\gg
g\textsubscript{{m}1}R\textsubscript{{o}1}$,
$(g\textsubscript{{mF}2}/g\textsubscript{{m}3})\gg
g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})$,
and $(g\textsubscript{{mF}3}/g\textsubscript{{m}4})\gg
g\textsubscript{{m}1}R\textsubscript{{o}1}g\textsubscript{{m}2}(R\textsubscript{{o}2}||R\textsubscript{{oF}1})g\textsubscript{{m}3}(R\textsubscript{{o}3}||R\textsubscript{{oF}2})$.
However, considering DC bias conditions, power consumption, and design area,
it becomes more difficult to satisfy the above conditions as the number of
stages increases. For these reasons, it becomes difficult to cancel out a pole
that is generated as the number of stages increases, ultimately leading to the
degradation of the PM.

Figure 8: (a) Bode plots of the four-stage feedforward OTA without the Miller
capacitors and (b) Bode plots employing the Miller capacitors.

Fig. 8(a) shows the simulated Bode plots obtained using the four-stage
feedforward OTA without Miller compensation. Employing the feedforward path
with $g\textsubscript{{mF}3}$ and without $C\textsubscript{{d}}$ alleviates
the phase shift at high frequencies compared to the design without the
feedforward path (blue and red lines in Fig. 8(a)). However, the zero is
located at a higher frequency than the pole, and the zero does not completely
cancel out the pole, leading to the PM degradation. As shown in the green line
in Fig. 8(a), by isolating the output feedforward path using
$C\textsubscript{{d}}$, the voltage gain is improved from
$G^{\prime\prime}\textsubscript{{m}}R^{\prime\prime}\textsubscript{{o}}g\textsubscript{{m}4}(R\textsubscript{{o}4}||R\textsubscript{{oF}3})$
to
$G^{\prime\prime}\textsubscript{{m}}R^{\prime\prime}\textsubscript{{o}}g\textsubscript{{m}4}R\textsubscript{{o}4}$
at low frequencies. Also, the dominant pole
$\omega^{\prime\prime}\textsubscript{{p}2}$ changes from
$1/(R\textsubscript{{o}4}||R\textsubscript{{oF}3})(C\textsubscript{{o}4}+C\textsubscript{{oF}3})$
to $1/(R\textsubscript{{o}4}C\textsubscript{{o}4})$ as $C\textsubscript{{d}}$
is employed. The other poles and zeros located at high frequencies are almost
unaffected by $C\textsubscript{{d}}$ since $C\textsubscript{{d}}$ becomes
almost a short circuit. As shown in Fig. 8(b), the PM can be improved by
employing the Miller capacitors $C\textsubscript{{m}1,2,3}$ shown in Fig. 7.
Similar to Miller compensation in the two-stage design, the poles and zeros
can be located closely together through pole splitting [4]. Therefore, the
zero can effectively cancel out the pole that shifts toward high frequencies
by pole splitting, resulting in an improved PM. However, the pole that moves
toward the origin through pole splitting reduces the bandwidth. In this
analysis, $C\textsubscript{{m}1}$ and $C\textsubscript{{m}3}$ are set to 10
fF, and $C\textsubscript{{m}2}$ is set to 200 fF. Also, the load impedance is
set to 10 M$\Omega$||2 pF on $V\textsubscript{{out}}$ for this analysis. Note
that the output load impedance is set to 10 M$\Omega$||2 pF in analyzing all
multistage feedforward OTAs throughout this work, and all output stages are
designed using large device size to achieve a large transconductance.
Accordingly, when analyzing the OTAs without Miller capacitors, it is assumed
that $\omega\textsubscript{{p}2}$, $\omega^{\prime}\textsubscript{{p}2}$, and
$\omega^{\prime\prime}\textsubscript{{p}2}$ are the dominant poles.

Fig. 9 shows the fully differential feedforward OTA used to obtain the results
shown in Fig. 8. In Fig. 9, the bias block and common-mode feedbacks
$CM\textsubscript{{n=}1,2,3}$ are the same as the circuits used in two- and
three-stage OTAs. The common-mode voltages of $V\textsubscript{{in}+/--}$ and
$V\textsubscript{{out}+/--}$ are set to 0.6 V. $C\textsubscript{{d}}$ is set
to 2 pF. The current consumption of the four-stage feedforward OTA is as
follows: $g\textsubscript{{m}1}$ (273 $\mu$A), $g\textsubscript{{m}2}$ (273
$\mu$A), $g\textsubscript{{mF}1}$ (820 $\mu$A), $g\textsubscript{{m}3}$ (1.61
mA), $g\textsubscript{{mF}2}$ (1.64 mA), $g\textsubscript{{m}4}$ (1.63 mA),
$g\textsubscript{{mF}3}$ (3.28 mA), and $CM\textsubscript{{n=}1,2,3}$ (104
$\mu$A). The OTA is driven using a 1.2 V supply voltage.

Figure 9: Fully differential four-stage feedforward OTA.

Table 2 summarizes the overall transconductances used in this analysis
according to the number of stages. Note that the transconductance ratio is
defined as $g\textsubscript{{mF}(k)}/g\textsubscript{{m}(k+1)}\textsubscript{,
k=1,2,3}$ in this analysis. As discussed above,
$g\textsubscript{{mF}(k)}/g\textsubscript{{m}(k+1)}$ must be designed as large
as possible to secure sufficient phase margin in a multistage feedforward OTA.
However, as the OTA order increases, achieving sufficient
$g\textsubscript{{mF}(k)}/g\textsubscript{{m}(k+1)}$ is a challenging task and
becomes even more difficult when considering DC bias conditions, power
consumption, and design area. Therefore, the analysis is carried out by
setting $g\textsubscript{{mF}(k)}/g\textsubscript{{m}(k+1)}$ to be larger than
2 in this work, as summarized in Table 2.

Table 2. Overall transconductances
---
Order |  Transconductance |  |
[3pt]2-stage |  $g\textsubscript{{m}1}$ (1.99 mS) |  $g\textsubscript{{m}2}$ (22.12 mS) |  $g\textsubscript{{mF}1}$ (44.97 mS)
[3pt]3-stage |  $g\textsubscript{{m}1}$ (1.99 mS) |  $g\textsubscript{{m}2}$ (2.27 mS) |  $g\textsubscript{{mF}1}$ (5.97 mS)
|  |  $g\textsubscript{{m}3}$ (28.04 mS) |  $g\textsubscript{{mF}2}$ (56.29 mS)
[3pt]4-stage |  $g\textsubscript{{m}1}$ (1.99 mS) |  $g\textsubscript{{m}2}$ (0.79 mS) |  $g\textsubscript{{mF}1}$ (5.96 mS)
|  |  $g\textsubscript{{m}3}$ (4.76 mS) |  $g\textsubscript{{mF}2}$ (11.95 mS)
|  |  $g\textsubscript{{m}4}$ (28.05 mS) |  $g\textsubscript{{mF}3}$ (56.28 mS)

##  IV. CONCLUSION

In this work, the analysis of multistage feedforward OTAs is conducted by
employing the single-pole approximation model. In each multistage feedforward
OTA, the first two-stage OTA is modeled as a single-pole system by assuming
that the first feedforward transconductance is large enough. Then, the next
stage is again approximated identically to the approximation method in the
previous stage by leveraging the previous single-pole system and making the
second feedforward transconductance sufficiently large. Although the single-
pole approximation model becomes difficult to apply as the number of stages
increases in actual circuit design, it provides an intuitive way to analyze
and design multistage feedforward OTAs.

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