A Triad of Circles Associated with a Triangle

Sangaku Journal of Mathematics (SJM) ©SJM
ISSN 2534-9562
Volume 7 (2023), pp. 54–70
Received 18 July 2023 Published on-line 3 August 2023
web: http://www.sangaku-journal.eu/
©The Author(s) This article is published with open access.111This article is
distributed under the terms of the Creative Commons Attribution License which
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A Triad of Circles Associated with a Triangle

Ercole Suppaa and Stanley Rabinowitzb

a Via B. Croce 54, 64100 Teramo, Italia

e-mail<EMAIL_ADDRESS>
web: http://www.esuppa.it/

b 545 Elm St Unit 1, Milford, New Hampshire 03055, USA

e-mail<EMAIL_ADDRESS>author

web: http://www.StanleyRabinowitz.com/

Abstract. We study some properties of a triad of circles associated with a
triangle. Each circle is inside the triangle, tangent to two sides of the
triangle, and externally tangent to the circle on the third side as diameter.
In particular, we find a nice relation involving the radii of the inner and
outer Apollonius circles of the three circles in the triad.

Keywords. Paasche point, Apollonius circle, barycentric coordinates,
Mathematica.

Mathematics Subject Classification (2020). 51-02, 51M04.

##  1\. Introduction

Notation. Throughout this paper, we will use the following notation, where
$\triangle ABC$ is a fixed acute triangle in the plane. We let $a=BC$, $b=CA$,
$c=AB$, $r$ is the inradius of $\triangle ABC$, $R$ is the circumradius of
$\triangle ABC$, $p=\frac{a+b+c}{2}$, $\Delta=[ABC]$ is the area of the
triangle, and $S=2\Delta$. We also let $I$ denote the incenter of $\triangle
ABC$.

The semicircle erected inwardly on side $BC$ will be named $\omega_{a}$ as
shown in Figure 1 (left). Semicircles $\omega_{b}$ and $\omega_{c}$ are
defined similarly. The circle inside $\triangle ABC$, tangent to sides $AB$
and $AC$, and externally tangent to semicircle $\omega_{a}$ will be named
$\gamma_{a}$. Circles $\gamma_{b}$ and $\gamma_{c}$ are defined similarly. The
radii of circles $\gamma_{a}$, $\gamma_{b}$, and $\gamma_{c}$ are denoted by
$\rho_{a}$, $\rho_{b}$, and $\rho_{c}$, respectively. The centers of these
circles are named $D$, $E$, and $F$, respectively, as shown in Figure 1
(right).

Figure 1.

For purposes of this paper, these three circles will be called the _triad of
circles_ associated with $\triangle ABC$.

This triad of circles appears in a Sangaku described in [4] and reprinted in
[7, problem 6]. The statement in the Sangaku is given as Theorem 1.1.

######  Theorem 1.1.

For the triad of circles associated with $\triangle ABC$, we have

$r=\frac{1}{2}\left(\rho_{a}+\rho_{b}+\rho_{c}+\sqrt{\rho_{a}^{2}+\rho_{b}^{2}+\rho_{c}^{2}}\right).$

A proof of this result can be found in [11]. A variant on this result when the
triangle is not acute can also be found in [11].

It is the purpose of this paper to give other properties of such a triad of
circles.

##  2\. Known Results

Before giving new results, we summarize some of the properties already known
about the triad of circles. The following five theorems come from [11].

######  Theorem 2.1.

For the triad of circles associated with $\triangle ABC$, we have

$\rho_{a}=r\left(1-\tan\frac{A}{2}\right).$

Similar formulas hold for $\rho_{b}$ and $\rho_{c}$.

######  Theorem 2.2.

Let $P$ and $Q$ be the feet of the perpendiculars from $D$ and $I$ to side
$AC$, respectively. Then $PQ=IQ$. (Figure 2)

Figure 2. red lengths are equal

######  Theorem 2.3.

The lengths of the common external tangents between any two circles of the
triad are equal. The common length is $2r$. (Figure 3)

Figure 3. blue lengths are equal

######  Theorem 2.4.

The six points of contact of the triad of circles associated with $\triangle
ABC$ lie on a circle with center $I$ and radius $r\sqrt{2}$. (Figure 4)

Figure 4.

This circle will be called the _contact circle_.

The following corollary follows immediately from Theorem 2.4.

######  Corollary 2.1.

In Figure 5 showing the contact circle and the incircle, the green area is
equal to the blue area.

Figure 5. green area = blue area

######  Theorem 2.5.

For the triad of circles associated with $\triangle ABC$, we have

$\rho_{a}^{2}+\rho_{b}^{2}+\rho_{c}^{2}=\frac{r^{2}(p-4R-r)^{2}}{p^{2}}.$ (1)

The following result comes from [5] where it is stated that the result is due
to Tomasz Cieśla.

######  Theorem 2.6.

Let $T_{a}$, $T_{b}$, and $T_{c}$ be the touch points of the circles in the
triad with their corresponding semicircles as shown in Figure 6. Then
$AT_{a}$, $BT_{b}$, and $CT_{c}$ are concurrent.

Figure 6.

The point of concurrence is catalogued as point $X_{1123}$ in the Encyclopedia
of Triangle Centers [5]. Since reference [5] does not include a proof of this
result, we will give our own proof later in Section 5 of this paper.

The point $X_{1123}$ is known as the _Paasche point_ of the triangle because
Paasche proved the following result in [8].

######  Theorem 2.7.

Congruent circles with centers $A_{1}$ and $A_{2}$ touch each other externally
at point $A^{\prime}$ outside $\triangle ABC$. Circle $(A_{1})$ is tangent to
$AB$ and $BC$. Circle $(A_{2})$ is tangent to $AC$ and $BC$. Points
$B^{\prime}$ and $C^{\prime}$ are defined similarly (Figure 7). Then
$AA^{\prime}$, $BB^{\prime}$, and $CC^{\prime}$ are concurrent.

Figure 7.

Remark. The same result is true if the pairs of congruent circles are inside
the triangle instead of outside. Figure 8 illustrates this. (Only the two
congruent circles tangent to side $BC$ are shown.) This result comes from [12,
Art. 3.5.4, ex. 4c].

Figure 8.

The Paasche point can also be characterized as follows according to [3].

######  Theorem 2.8.

In $\triangle ABC$, let $\omega_{a}$, $\omega_{b}$, and $\omega_{c}$ be the
circles constructed using sides $BC$, $CA$, and $AB$, respectively, as
diameters. Let $\Omega$ be the circle internally tangent to $\omega_{a}$,
$\omega_{b}$, and $\omega_{c}$. Let $A^{\prime}$ be the touch point between
$\omega_{a}$ and $\Omega$. Points $B^{\prime}$ and $C^{\prime}$ are defined
similarly (Figure 9). Then $AA^{\prime}$, $BB^{\prime}$, and $CC^{\prime}$ are
concurrent at $X_{1123}$, the Paasche point of $\triangle ABC$.

Figure 9.

A circle that is tangent to three given circles is called an _Apollonius
circle_ of those three circles.

If all three circles lie inside an Apollonius circle, then the Apollonius
circle is called the _outer Apollonius circle_ of the three circles. The outer
Apollonius circle surrounds the three circles and is internally tangent to all
three.

If all three circles lie outside an Apollonius circle, then the Apollonius
circle is called the _inner Apollonius circle_ of the three circles. The inner
Apollonius circle will either be internally tangent to the three given circles
or it will be externally tangent to all the circles. The following theorem
comes from [6].

######  Theorem 2.9.

For the triad of circles associated with $\triangle ABC$, the inner Apollonius
circle of $\gamma_{a}$, $\gamma_{b}$, $\gamma_{c}$, is internally tangent to
the inner Apollonius circle of $\omega_{a}$, $\omega_{b}$, $\omega_{c}$
(Figure 10).

Remark. The inner Apollonius circle of $\gamma_{a}$, $\gamma_{b}$,
$\gamma_{c}$ is known as the 1st Miyamoto-Moses-Apollonius circle and the
outer Apollonius circle of $\gamma_{a}$, $\gamma_{b}$, $\gamma_{c}$ is known
as the 2nd Miyamoto-Moses-Apollonius circle (see [6]).

Figure 10.

##  3\. Metric Relationships involving $\rho_{a}$, $\rho_{b}$, $\rho_{c}$

In addition to Theorem 1.1 and Theorem 2.5, the following symmetric
relationship involving $\rho_{a}$, $\rho_{b}$, and $\rho_{c}$ holds.

######  Theorem 3.1.

For the triad of circles associated with $\triangle ABC$, we have

$\rho_{a}\rho_{b}+\rho_{b}\rho_{c}+\rho_{c}\rho_{a}-2r(\rho_{a}+\rho_{b}+\rho_{c})+2r^{2}=0.$

###### Proof.

From Theorem 2.1, we have $\rho_{a}=r(1-\tan\frac{A}{2})$,
$\rho_{b}=r(1-\tan\frac{B}{2})$, and $\rho_{c}=r(1-\tan\frac{C}{2})$.
Substituting these values into the expression

$\rho_{a}\rho_{b}+\rho_{b}\rho_{c}+\rho_{c}\rho_{a}-2r(\rho_{a}+\rho_{b}+\rho_{c})+2r^{2}$

and simplifying (using Mathematica), shows that the expression is equal to

$-r^{2}\cos\left(\frac{A+B+C}{2}\right)\sec\frac{A}{2}\sec\frac{B}{2}\sec\frac{C}{2}.$

Since $A+B+C=\pi$, this expression is equal to 0. ∎

######  Theorem 3.2.

For the triad of circles associated with $\triangle ABC$, we have

$p(r-\rho_{a})(r-\rho_{b})(r-\rho_{c})=r^{4}.$

###### Proof.

This result follows from Theorem 2.1 and the trigonometric identity

$\tan\frac{A}{2}\tan\frac{B}{2}\tan\frac{C}{2}=\frac{r}{p}$

which comes from [2, p. 358]. ∎

######  Theorem 3.3.

For the triad of circles associated with $\triangle ABC$, we have

$\rho_{a}=\frac{\Delta-(p-b)(p-c)}{p}.$ (2)

Similar formulas hold for $\rho_{b}$ and $\rho_{c}$.

###### Proof.

Using Theorem 2.1 and the well-known identities

$\tan\frac{A}{2}=\frac{r}{p-a}\qquad\hbox{and}\qquad r=\frac{\Delta}{p},$

we get

$\displaystyle\rho_{a}$
$\displaystyle=r\left(1-\frac{r}{p-a}\right)=r-\frac{r^{2}}{p-a}=\frac{\Delta}{p}-\frac{\Delta^{2}}{p^{2}(p-a)}$
(3)
$\displaystyle=\frac{\Delta}{p}-\frac{p(p-a)(p-b)(p-c)}{p^{2}(p-a)}=\frac{\Delta-(p-b)(p-c)}{p}.$
(4)

This complete the proof. ∎

##  4\. Barycentric coordinates of centers of $\gamma_{a}$, $\gamma_{b}$,
$\gamma_{c}$

######  Theorem 4.1.

The barycentric coordinates of the center of $\gamma_{a}$ are

$D=aS+2(p-b)(p-c)(b+c):bS-2b(p-b)(p-c):cS-2c(p-b)(p-c).$

###### Proof.

Let $y$ be the distance between $D$ and the sideline $BC$. Summing the areas
of triangles $DBC$, $DCA$ and $DAB$ we obtain

$ay+b\rho_{a}+c\rho_{a}=2\Delta.$ (5)

Plugging (2) into (5) we get

$\displaystyle ay+(b+c)\cdot\frac{\Delta-(p-b)(p-c)}{p}=S$ $\displaystyle
ay+(b+c)\cdot\frac{S-2(p-b)(p-c)}{2p}=S$ $\displaystyle
2pay+(b+c)S-2(p-b)(p-c)(b+c)=(a+b+c)S$ $\displaystyle
2pay=2(p-b)(p-c)(b+c)+aS$ $\displaystyle ay=\frac{aS+2(p-b)(p-c)(b+c)}{2p}$
(6)

By using (2) and (6), we obtain

$\displaystyle D$ $\displaystyle=\Delta DBC:\Delta
DCA:DAB=ay:b\rho_{a}:c\rho_{a}$
$\displaystyle=aS+2(p-b)(p-c)(b+c):bS-2b(p-b)(p-c):cS-2c(p-b)(p-c)$

which are the desired barycentric coordinates. ∎

######  Theorem 4.2.

The radical center of circles $\gamma_{a}$, $\gamma_{b}$ and $\gamma_{c}$ is
the Gergonne point of $\triangle ABC$.

###### Proof.

Using Mathematica and the package baricentricas.m333The package
baricentricas.m written by F.J.G.Capitan can be freely downloaded from
http://garciacapitan.epizy.com/baricentricas/, it can be proved that

  * (a)

the radical axis of $\gamma_{a}$ and $\gamma_{b}$ is $(p-a)x-(p-b)y=0$;

  * (b)

the radical axis of $\gamma_{b}$ and $\gamma_{c}$ is $(p-b)y-(p-c)z=0$;

  * (c)

the radical axis of $\gamma_{c}$ and $\gamma_{a}$ is $(p-a)x-(p-c)z=0$.

An easy verification shows that these radical axes concur at

$G_{e}=(p-b)(p-c):(p-c)(p-a):(p-a)(p-b),$

which is the Gergonne point of $\triangle ABC$. ∎

We can also give a purely geometric proof.

###### Proof.

Let the incircle touch the sides of $\triangle ABC$ at $Q_{a}$, $Q_{b}$, and
$Q_{c}$ as shown in Figure 11. From Theorem 2.2, $Q_{a}E_{a}=Q_{a}F_{a}=r$.
Thus, the tangents from $Q_{a}$ to $\gamma_{b}$ and $\gamma_{c}$ are equal.
Since $AD_{c}=AD_{b}$ and $D_{c}E_{c}=D_{b}F_{b}$ (Theorem 2.3), this means
$AE_{c}=AF_{b}$. Hence the tangents from $A$ to $\gamma_{b}$ and $\gamma_{c}$
are equal. The radical axis of circles $\gamma_{b}$ and $\gamma_{c}$ is the
locus of points such that the lengths of the tangents to the two circles from
that point are equal. The radical axis of two circles is a straight line.
Therefore, the radical axis of circles $\gamma_{b}$ and $\gamma_{c}$ is
$AQ_{a}$, the Gergonne cevian from $A$.

Similarly, the radical axis of circles $\gamma_{a}$ and $\gamma_{c}$ is the
Gergonne cevian from $B$ and the radical axis of circles $\gamma_{a}$ and
$\gamma_{b}$ is the Gergonne cevian from $C$. Hence, the radical center of the
triad of circles is the intersection point of the three Gergonne cevians,
namely, the Gergonne point of $\triangle ABC$. ∎

Figure 11.

##  5\. A concurrence at the Paasche point.

In [5] the following result is stated.

######  Theorem 5.1.

Suppose that $ABC$ is an acute triangle. Let $\gamma_{a}$ be the circle
touching $CA$ and $AB$ from inside of $ABC$ and also externally tangent to the
semicircle of diameter $BC$, in point $T_{a}$. Define $T_{b}$ and $T_{c}$
cyclically (Figure 12). Then $ABC$ is perspective to $T_{a}T_{b}T_{c}$, and
the perspector is $X_{1123}$, the Paasche point of $\triangle ABC$.

Figure 12.

###### Proof.

We use homogeneous barycentric coordinates with respect to the triangle $ABC$.
Let $M_{a}$ be the midpoint of $BC$. The point $T_{a}$ divides the segment
joining the centers of the circles $\gamma_{a}$ and $\omega_{a}$ in the ratio
$\rho_{a}:\frac{a}{2}$. Using theorem 4.1 we have that the sum of coordinates
of $D$ is

$\displaystyle aS+2(p-b)(p-c)(b+c)+bS-2b(p-b)(p-c)+cS-2c(p-b)(p-c)$
$\displaystyle=$ $\displaystyle(a+b+c)S+2(p-b)(p-c)(b+c)-2(b+c)(p-b)(p-c)$
$\displaystyle=$ $\displaystyle(a+b+c)S=2pS.$

Therefore, by writing the coordinates of $M_{a}$ in the form $M_{a}=0:pS:pS$,
we get

$T_{a}=\frac{a}{2}\cdot D+\rho_{a}\cdot M_{a}.$

It follows that the first coordinate of $T_{a}=x_{a}:y_{a}:z_{a}$ is given by

$\displaystyle x_{a}$
$\displaystyle=\frac{a}{2}\left(aS+2(p-b)(p-c)(b+c)\right)+\rho_{a}\cdot 0$
$\displaystyle=\frac{a}{2}\left(aS+2(p-b)(p-c)(b+c)\right).$

In a similar way we find that

$\displaystyle y_{a}$
$\displaystyle=\frac{a}{2}\left(bS-2b(p-b)(p-c)\right)+\rho_{a}pS$
$\displaystyle=\frac{1}{2}\left(S-2(p-b)(p-c)\right)(ab+S)$

and

$\displaystyle z_{a}$
$\displaystyle=\frac{a}{2}\left(cS-2c(p-b)(p-c)\right)+\rho_{a}pS$
$\displaystyle=\frac{1}{2}\left(S-2(p-b)(p-c)\right)(ac+S).$

Hence

$\displaystyle T_{a}=$ $\displaystyle a^{2}S+2a(p-b)(p-c)(b+c):$
$\displaystyle\left(S-2(p-b)(p-c)\right)(ab+S):$
$\displaystyle\left(S-2(p-b)(p-c)\right)(ac+S).$

The equation of line $AT_{a}$ is $z_{a}y+y_{a}z=0$, i.e.

$AT_{a}:\quad(ac+S)y-(ab+S)z=0.$

The cyclic substitution $a\to b$, $b\to c$, $c\to a$ gives

$\displaystyle BT_{b}:\quad(bc+S)x-(ab+S)z=0,$ $\displaystyle
CT_{c}:\quad(bc+S)x-(ac+S)y=0.$

A direct verification shows that $AT_{a}$, $BT_{b}$, and $CT_{c}$ concur at
the Paasche point

$X_{1123}=(ab+S)(ac+S):(ab+S)(bc+S):(ac+S)(bc+S).\qed$

##  6\. Apollonius circles of $\gamma_{a}$, $\gamma_{b}$, $\gamma_{c}$

In order to find the radii of the inner and outer Apollonius circles tangent
to $\gamma_{a}$, $\gamma_{b}$, and $\gamma_{c}$ we will use the method
explained in [10] and some preliminary lemmas. The more complicated
calculations are performed with Mathematica.

######  Lemma 6.1.

If $u=EF$, $v=DF$, $w=DE$ are the distances between the centers of the circles
$\gamma_{a}$, $\gamma_{b}$, and $\gamma_{c}$, we have

$\displaystyle u^{2}$
$\displaystyle=\frac{a(b+c-a)\left(a^{2}+ab+ac-2b^{2}+4bc-2c^{2}\right)}{(a+b+c)^{2}},$
$\displaystyle v^{2}$
$\displaystyle=\frac{b(a-b+c)\left(-2a^{2}+ab+4ac+b^{2}+bc-2c^{2}\right)}{(a+b+c)^{2}},$
$\displaystyle w^{2}$
$\displaystyle=\frac{c(a+b-c)\left(-2a^{2}+4ab+ac-2b^{2}+bc+c^{2}\right)}{(a+b+c)^{2}}.$

###### Proof.

From Theorem 3.3, we have

$\rho_{a}=\frac{\Delta-(p-b)(p-c)}{p},$

and similarly

$\rho_{b}=\frac{\Delta-(p-a)(p-c)}{p},\qquad\rho_{c}=\frac{\Delta-(p-a)(p-b)}{p}.$

Figure 13.

Assume, without loss of generality, that $\rho_{b}<\rho_{c}$. Let $E^{\prime}$
be the foot of the perpendicular from $E$ to $FF_{a}$. Applying the
Pythagorean Theorem to triangle $EFE^{\prime}$ (see figure 13), taking into
account that $EE^{\prime}=E_{a}F_{a}=2r$ and $FE^{\prime}=\rho_{c}-\rho_{b}$,
we obtain

$\displaystyle u^{2}$
$\displaystyle=EF^{2}=E_{a}F_{a}^{2}+(FF_{a}-EE_{a})^{2}=(2r)^{2}+(\rho_{c}-\rho_{b})^{2}$
$\displaystyle=4r^{2}+\left(\frac{(p-a)(p-c)-(p-a)(p-b)}{p}\right)^{2}$
$\displaystyle=4\cdot\frac{\Delta^{2}}{p^{2}}+\frac{(p-a)^{2}(b-c)^{2}}{p^{2}}$
$\displaystyle=\frac{4p(p-a)(p-b)(p-c)+(p-a)^{2}(b-c)^{2}}{p^{2}}$
$\displaystyle=\frac{a(b+c-a)\left(a^{2}+ab+ac-2b^{2}+4bc-2c^{2}\right)}{(a+b+c)^{2}}.$

The formulas relating to $v^{2}$ and $w^{2}$ can be proved in a similar way. ∎

######  Lemma 6.2.

If $\theta$, $\varphi$, $\psi$ are three positive real numbers such that
$\theta+\varphi+\psi=360^{\circ}$, then we have

$\cos^{2}\theta+\cos^{2}\varphi+\cos^{2}\psi-2\cos\theta\cos\varphi\cos\psi=1.$

###### Proof.

Using the addition formulas we have

$\displaystyle\cos^{2}\theta+\cos^{2}\varphi+\cos^{2}\psi-2\cos\theta\cos\varphi\cos\psi$
$\displaystyle=$
$\displaystyle\cos^{2}\theta+\cos^{2}\varphi+\cos^{2}(360^{\circ}-\theta-\varphi)-2\cos\theta\cos\varphi\cos(360^{\circ}-\theta-\varphi)$
$\displaystyle=$
$\displaystyle\cos^{2}\theta+\cos^{2}\varphi+\cos^{2}(\theta+\varphi)-2\cos\theta\cos\varphi\cos(\theta+\varphi)$
$\displaystyle=$
$\displaystyle\cos^{2}\theta+\cos^{2}\varphi+\left(\cos\theta\cos\varphi-\sin\theta\sin\varphi\right)^{2}-2\cos\theta\cos\varphi\left(\cos\theta\cos\varphi-\sin\theta\sin\varphi\right)$
$\displaystyle=$
$\displaystyle\cos^{2}\theta+\cos^{2}\varphi+\cos^{2}\theta\cos^{2}\varphi+\sin^{2}\theta\sin^{2}\varphi-2\cos^{2}\theta\cos^{2}\varphi$
$\displaystyle=$
$\displaystyle\cos^{2}\theta+\cos^{2}\varphi+\sin^{2}\theta\sin^{2}\varphi-\cos^{2}\theta\cos^{2}\varphi$
$\displaystyle=$
$\displaystyle\cos^{2}\theta(1-\cos^{2}\varphi)+\cos^{2}\varphi+\sin^{2}\theta\sin^{2}\varphi$
$\displaystyle=$
$\displaystyle\sin^{2}\varphi(\cos^{2}\theta+\sin^{2}\theta)+\cos^{2}\varphi$
$\displaystyle=$ $\displaystyle\sin^{2}\varphi+\cos^{2}\varphi=1.\qed$

######  Theorem 6.1.

Let $\rho_{i}$ be the radius of the inner Apollonius circle externally tangent
to $\gamma_{a}$, $\gamma_{b}$, and $\gamma_{c}$ (see figure 14). Then

$\rho_{i}^{2}=\rho_{a}^{2}+\rho_{b}^{2}+\rho_{c}^{2}.$

Figure 14.

###### Proof.

Let $U$ be the center of the inner Apollonius circle and let $x=\rho_{i}$. Let
us consider the angles $\varphi_{a}=\angle EUF$, $\varphi_{b}=\angle FUD$, and
$\varphi_{c}=\angle DUE$.

Since $\varphi_{a}+\varphi_{b}+\varphi_{c}=360^{\circ}$ by Lemma 6.2 we have

$\cos^{2}\varphi_{a}+\cos^{2}\varphi_{b}+\cos^{2}\varphi_{c}-2\cos\varphi_{a}\cos\varphi_{b}\cos\varphi_{c}=1.$
(7)

If we substitute

$t_{a}=\sin^{2}\frac{\varphi_{a}}{2},\qquad
t_{b}=\sin^{2}\frac{\varphi_{b}}{2},\qquad
t_{c}=\sin^{2}\frac{\varphi_{c}}{2}$ (8)

in (7), we obtain

$t_{a}^{2}+t_{b}^{2}+t_{c}^{2}-2\left(t_{a}t_{b}+t_{b}t_{c}+t_{c}t_{a}\right)+4t_{a}t_{2}t_{c}=0.$
(9)

Since $UD=x+\rho_{a}$, $UE=x+\rho_{b}$, $UF=x+\rho_{c}$, the Law of Cosines
yields

$\displaystyle\cos\varphi_{a}=\frac{\left(x+\rho_{b}\right)^{2}+\left(x+\rho_{c}\right)^{2}-u^{2}}{2(x+\rho_{b})(x+\rho_{c})}\quad\Rightarrow$
$\displaystyle\quad
t_{a}=\frac{1-\cos\varphi_{1}}{2}=\frac{u^{2}-\left(\rho_{b}-\rho_{c}\right)^{2}}{4(x+\rho_{b})(x+\rho_{c})},$
(10)

and analogously

$t_{b}=\frac{v^{2}-\left(\rho_{c}-\rho_{a}\right)^{2}}{4(x+\rho_{c})(x+\rho_{a})},\qquad
t_{c}=\frac{w^{2}-\left(\rho_{a}-\rho_{b}\right)^{2}}{4(x+\rho_{a})(x+\rho_{b})}.$
(11)

Plugging (10) and (11) in (9) and using Lemma 6.1, after a straightforward
calculation, we get an equation of the form $f(x)g(x)=0$, where

$f(x)=6(a+b+c)x+2ab+2bc+2ac-a^{2}-b^{2}-c^{2}+12\Delta$ (12)

and

$g(x)=2(a+b+c)x+a^{2}+b^{2}+c^{2}-2ab-2ac-2bc+4\Delta.$ (13)

The root of (12) is

$x=\frac{a^{2}+b^{2}+c^{2}-2ab-2bc-2ca-12\Delta}{6(a+b+c)}=-\frac{2r(4R+r)+6\Delta}{6p}<0.$

The root of (13) is

$x=\frac{-a^{2}-b^{2}-c^{2}+2ba+2bc+2ca-4\Delta}{2(a+b+c)}=\frac{r(4R+r-p)}{p}>0.$

Therefore, discarding the negative root, we have
$\rho_{i}=\frac{r(4R+r-p)}{p}$. Hence, taking into account equation (1), we
get

$\rho_{i}^{2}=\rho_{a}^{2}+\rho_{b}^{2}+\rho_{c}^{2}.\qed$

######  Corollary 6.1.

In Figure 15 showing the triad of circles associated with $\triangle ABC$ and
the inner Apollonius circle externally tangent to each circle in the triad, we
have that the sum of the yellow areas is equal to the green area.

Figure 15. yellow area = green area

######  Theorem 6.2.

Let $\rho_{o}$ be the radius of the outer Apollonius circle internally tangent
to $\gamma_{a}$, $\gamma_{b}$, $\gamma_{c}$ (see Figure 16). We have

$\rho_{o}=\frac{2}{3}\left(\rho_{a}+\rho_{b}+\rho_{c}\right)+\sqrt{\rho_{a}^{2}+\rho_{b}^{2}+\rho_{c}^{2}}.\\\
$

Figure 16.

###### Proof.

The proof is similar to that of Theorem 6.1. ∎

Combining Theorem 1.1, Theorem 6.1, and Theorem 6.2, we get the following nice
results.

######  Corollary 6.2.

The inradius $r$ and the radii $\rho_{i}$, $\rho_{o}$ of the Apollonius
circles satisfy the relation

$3\rho_{o}=\rho_{i}+4r.$

######  Corollary 6.3.

The radii $\rho_{i}$, $\rho_{o}$ of the Apollonius circles satisfy the
relation

$3\rho_{o}=2(\rho_{a}+\rho_{b}+\rho_{c})+3\rho_{i}.$

Remark. The centers $U$ and $V$ of the inner and outer Apollonius circles of
$\gamma_{a}$, $\gamma_{b}$, and $\gamma_{c}$ are known ETC centers, namely
$U=X(52805)$ and $V=X(52806)$.

######  Theorem 6.3.

Let $\omega_{i}$ be the inner Apollonius circle, externally tangent to
$\gamma_{a}$, $\gamma_{b}$, and $\gamma_{c}$. Let $U_{a}$ be the touch point
between $\omega_{i}$ and $\gamma_{a}$. Define $U_{b}$ and $U_{c}$ cyclically.
Let $\omega_{o}$ be the outer Apollonius circle, internally tangent to
$\gamma_{a}$, $\gamma_{b}$, and $\gamma_{c}$. Let $V_{a}$ be the touch point
between $\omega_{o}$ and $\gamma_{a}$. Define $V_{b}$ and $V_{c}$ cyclically.
Let $G_{e}$ be the Gergonne point of $\triangle ABC$. Then the points $A$,
$V_{a}$, $U_{a}$, and $G_{e}$ are collinear. Similarly, the points $B$,
$V_{b}$, $U_{b}$, and $G_{e}$ are collinear; and the points $C$, $V_{c}$,
$U_{c}$, and $G_{e}$ are collinear (Figure 17).

Figure 17.

###### Proof.

Clearly, by symmetry, it is enough to prove that $A$, $V_{a}$, $U_{a}$, and
$G_{e}$ are collinear. From Theorem 4.2, we know that $G_{e}$ is the radical
center of $\gamma_{a}$, $\gamma_{b}$, and $\gamma_{c}$. Hence, from the
Gergonne construction of Apollonius circles, it follows that $U_{a}$, $V_{a}$,
and $G_{e}$ are collinear. Therefore, it remains to prove that $A$, $U_{a}$,
and $G_{e}$ are collinear. To this end we use barycentric coordinates. We have
$A=1:0:0$ and $G_{e}=\frac{1}{p-a}:\frac{1}{p-b}:\frac{1}{p-c}$. The point
$U_{a}$ divides the segment $DU$ joining the centers of the circles
$\gamma_{a}$ and $\omega_{i}$ in the ratio $\rho_{a}:\rho_{i}$. By using
Mathematica, we find that

$U_{a}=\frac{-a}{(p-b)(p-c)-\Delta}:\frac{1}{p-b}:\frac{1}{p-c}.$

The points $A$, $U_{a}$, and $G_{e}$ are collinear because

$\left|{\begin{array}[]{*{20}{c}}1&0&0\\\ \\\
\frac{-a}{(p-b)(p-c)-\Delta}&\frac{1}{p-b}&\frac{1}{p-c}\\\ \\\
\frac{1}{p-a}&\frac{1}{p-b}&\frac{1}{p-c}\end{array}}\right|=0.$

This completes the proof. ∎

Remark. We could also show that the lines $AU_{a}$, $BU_{b}$, and $CU_{c}$ are
concurrent by using Theorem 2 of [9]. That theorem also shows that the point
of concurrence, $G_{e}$, is the internal center of similitude of the incircle
of $\triangle ABC$ and the circle $\omega_{i}$.

######  Corollary 6.4.

Let $U_{a}$ be the touch point between $\gamma_{a}$ and $\omega_{i}$ (Figure
18). Then the tangents from $U_{a}$ to $\gamma_{b}$ and $\gamma_{c}$ are
equal.

Figure 18. red tangents are equal

###### Proof.

From Theorem 6.3, $AU_{a}$ is the Gergonne cevian from vertex $A$. But from
the proof of Theorem 4.2, this Gergonne cevian is the radical axis of circles
$\gamma_{b}$ and $\gamma_{c}$. Thus the two tangents have the same length. ∎

######  Theorem 6.4.

Let $\omega_{i}=(U,\rho_{i})$, $\omega_{o}=(V,\rho_{o})$ be the inner and
outer Apollonius circles externally and internally tangent to $\gamma_{a}$,
$\gamma_{b}$, and $\gamma_{c}$, respectively. Let $U_{a}$, $V_{a}$ be the
touch point of $\gamma_{a}$ with $\omega_{i}$ and $\omega_{o}$ respectively.
Define $U_{b}$, $U_{c}$, $V_{b}$, $V_{c}$ cyclically. Let $I=X_{1}$,
$G_{e}=X_{7}$ be the incenter and the Gergonne points of $\triangle ABC$
respectively (Figure 19). Then $U$ and $V$ lie on the Soddy line $IG_{e}$ and
$UI:IV=3$.

Figure 19.

###### Proof.

This follows directly from the barycentric coordinates for $U$ and $V$. ∎

##  7\. Other properties

######  Theorem 7.1.

For the triad of circles associated with $\triangle ABC$, let $x=\rho_{a}$,
$y=\rho_{b}$, and $z=\rho_{c}$. Let $u,v,w$ be the radii of the greatest
circles inscribed in the circular segments shown in Figure 20. Then

$xw+yu+zv=xv+zu+yw.$

Figure 20.

###### Proof.

From Theorem 2.1, we have

$x=r\left(1-\tan\frac{A}{2}\right),\quad
y=r\left(1-\tan\frac{B}{2}\right),\quad z=r\left(1-\tan\frac{C}{2}\right).$

Figure 21.

On the other hand, we have (see Figure 21)

$u=\frac{1}{2}MN=\frac{1}{2}BM\cdot\tan\frac{A}{2}=\frac{a}{4}\tan\frac{A}{2}$

and similarly

$v=\frac{b}{4}\tan\frac{B}{2},\quad w=\frac{c}{4}\tan\frac{C}{2}.$

Observe that

$\tan\frac{A}{2}=\frac{r}{s-a}=\frac{\Delta}{s(s-a)},\quad\tan\frac{B}{2}=\frac{\Delta}{s(s-b)},\quad\tan\frac{C}{2}=\frac{\Delta}{s(s-c)}$

so, using the Heron formula $\Delta^{2}=s(s-a)(s-b)(s-c)$, we get

$\displaystyle u(y-z)$
$\displaystyle=\frac{a}{4}\tan\frac{A}{2}\left(r\left(1-\tan\frac{B}{2}\right)-r\left(1-\tan\frac{C}{2}\right)\right)$
$\displaystyle=\frac{ar}{4}\tan\frac{A}{2}\left(\tan\frac{C}{2}-\tan\frac{B}{2}\right)$
$\displaystyle=\frac{ar}{4}\left(\tan\frac{A}{2}\tan\frac{C}{2}-\tan\frac{A}{2}\tan\frac{B}{2}\right)$
$\displaystyle=\frac{ar}{4}\left(\frac{\Delta}{s(s-a)}\frac{\Delta}{s(s-c)}-\frac{\Delta}{s(s-a)}\frac{\Delta}{s(s-b)}\right)$
$\displaystyle=\frac{ar}{4}\left(\frac{s-b}{s}-\frac{s-c}{s}\right)$
$\displaystyle=\frac{r}{4}\cdot\frac{ac-ab}{s}.$

Similarly, we have

$v(z-x)=\frac{r}{4}\cdot\frac{ba-bc}{s},\qquad
w(x-y)=\frac{r}{4}\cdot\frac{cb-ca}{s}.$

Therefore

$\displaystyle xw+yu+zv-(v+zu+yw)=$ $\displaystyle u(y-z)+v(z-x)+w(x-y)$
$\displaystyle=$ $\displaystyle\frac{r}{4}\cdot\frac{ac-
ab}{s}+\frac{r}{4}\cdot\frac{ba-bc}{s}+\frac{r}{4}\cdot\frac{cb-ca}{s}$
$\displaystyle=$ $\displaystyle\frac{r}{4}\cdot\frac{ac-ab+ba-bc+cb-ca}{s}=0.$

This completes the proof. ∎

## References

  * [1]
  * [2] Titu Andreescu and Oleg Mushkarov, Topics in Geometric Inequalities, XYZ Press, 2019.
  * [3] Francisco Javier Garcia Capitán, Problem 2428, Romantics of Geometry Facebook Group, October 2018.
https://www.facebook.com/groups/parmenides52/posts/1926266464153717/

  * [4] Honma, ed., Zoku Kanji Sampō, Tohoku University Digital Collection, 1849.
  * [5] Clark Kimberling, Encyclopedia of Triangle Centers, entry for X(1123), the Paasche Point.
https://faculty.evansville.edu/ck6/encyclopedia/ETCPart2.html#X1123

  * [6] Clark Kimberling, Encyclopedia of Triangle Centers, preamble to entry X(52805), Miyamoto-Moses Points.
https://faculty.evansville.edu/ck6/encyclopedia/ETCPart27.html#X52805

  * [7] Hiroshi Okumura, Problems 2023–1, Sangaku Journal of Mathematics, 7(2023)9–12.
http://www.sangaku-journal.eu/2023/SJM_2023_9-12_problems_2023-1.pdf

  * [8] Ivan Paasche, Aufgabe P933: Ankreispaare, Praxis der Mathematik 1(1990)40.
  * [9] Stanley Rabinowitz, Pseudo-Incircles, Forum Geometricorum, 6(2006)107–115.
https://forumgeom.fau.edu/FG2006volume6/FG200612.pdf

  * [10] Milorad R. Stevanović, Predrag B. Petrović, and Marina M. Stevanović, Radii of Circles in Apollonius’s Problem, Forum Geometricorum, 17(2017)359–372.
https://forumgeom.fau.edu/FG2017volume17/FG201735.pdf

  * [11] Ercole Suppa and Marian Cucoanes, Solution of Problem 2023–1–6, Sangaku Journal of Mathematics, 7(2023)21–28.
http://www.sangaku-journal.eu/2023/SJM_2023_21-28_Suppa,Cucoanes.pdf

  * [12] Paul Yiu, Introduction to the Geometry of the Triangle, December 2012.
http://math.fau.edu/Yiu/YIUIntroductionToTriangleGeometry121226.pdf

Errata. On page 66, in equation (9), “$t_{2}$” should be “$t_{b}$” and in
equation (10), “$\cos\varphi_{1}$” should be “$\cos\varphi_{a}$”.


# Detection of collective modes in unconventional superconductors using
tunneling spectroscopy

Patrick A. Lee  Department of Physics, Massachusetts Institute of Technology,
Cambridge, MA, USA  Jacob F. Steiner  Department of Physics and Institute for
Quantum Information and Matter, California Institute of Technology, Pasadena,
CA 91125, USA

###### Abstract

We propose using tunneling spectroscopy with a superconducting electrode to
probe the collective modes of unconventional superconductors. The modes are
predicted to appear as peaks in $dI/dV$ at voltages given by $eV=\omega_{i}/2$
where $\omega_{i}$ denotes the mode frequencies. This may prove to be a
powerful tool to investigate the pairing symmetry of unconventional
superconductors. The peaks associated with the collective modes appear at
fourth order in the single particle tunneling matrix element. At the same
fourth order, multiple Andreev reflection (MAR) leads to peaks at voltage
equal to the energy gaps, which, in BCS superconductors, coincides with the
expected position of the amplitude (Higgs) mode. The peaks stemming from the
collective modes of unconventional superconductors do not suffer from this
coincidence. For scanning tunneling microscopes (STM), we estimate that the
magnitude of the collective mode contribution is smaller than the MAR
contribution by the ratio of the energy gap to the Fermi energy. Moreover,
there is no access to the mode dispersion. Conversely, for planar tunnel
junctions the collective mode peak is expected to dominate over the MAR peak,
and the mode dispersion can be measured. We discuss systems where the search
for such collective modes is promising.

##  I Introduction

In the past three decades, many examples of unconventional superconductors
(SC) have been discovered. Many of these have multiple order parameters,
either due to pairing in several disconnected Fermi surfaces, or due to
pairing that is intrinsically multi-component. In the latter case, the order
parameters may be members of a particular irreducible representation, prime
examples being MgB2 [1] and the iron based superconductors [2]. Alternatively,
they are of mixed symmetry due to the breaking of lattice or time reversal
symmetry. While there are numerous examples of mixed symmetry pairing, very
often it is difficult to identify the precise order parameter symmetry in
these materials. In an interesting recent paper Poniatowski et al. [3] pointed
out that, since these systems exhibit collective modes beyond the familiar
phase and amplitude (Higgs) modes, the detection of these modes may serve as
signature of the order parameter symmetry. They investigated several examples
and showed that commonly these modes lie below the quasi-particle gap
$2\Delta$ and hence form well defined excitations. Some of these collective
modes are analogs of the Leggett mode [4], or of the “clapping” mode, familiar
from the He3 literature [5, 6]. While progress in the detection of such modes
has been made using nonlinear optical spectroscopy [7], they are often charge
neutral and thus evade detection using conventional tools. Motivated by this,
we study the question whether the collective modes of unconventional
superconductors may be detected using tunneling spectroscopy. We investigate
point contact tunneling such as scanning tunneling microscopy (STM) as well as
planar tunneling and compare their respective advantages and disadvantages. We
will also discuss examples where such experiments may be feasible.

This paper is structured as follows. In Sec. II we demonstrate within linear
response theory how collective modes of the pairing give rise to features in
the tunneling current between two SCs. In Sec. III we discuss processes that
give rise to features in the tunneling current at the same order in
perturbation theory which are commonly observed, namely the Josephson peak and
multiple Andreev reflections (MAR). In Sec. IV we present a microscopic
treatment of the current due to collective modes, which includes the
possibility of multi-component order parameters. In Sec. V we discuss the
magnitude of the features in the collective mode relative to the Josephson
peak and the leading order MAR contribution. We proceed in Sec. VI by applying
our results to several experimentally relevant multi-component SCs,
highlighting examples where it is promising to look for the current due to
collective modes. Finally, in we conclude Sec. VII with a discussion of the
experimental challenges associated with measuring the current due to
collective modes.

Figure 1:  Schematic drawing of the STM tunneling conductance $dI/dV$ with a
SC tip (with energy gap $\Delta_{L}$) showing the expected subgap features up
to fourth order in the tunneling amplitude. The standard quasi-particle peak
starting at $\Delta_{R}+\Delta_{L}$ has been reduced by $g=(h/e^{2})/R_{N}$,
the dimensionless normal state conductance. Below this energy we find the
multiple Andreev reflection (MAR) peaks at $\Delta_{R}$ and $\Delta_{L}$ which
overlap the respective amplitude (Higgs) modes. Shown in red is the
contribution from a collective mode for an unconventional SC on the $R$ side
at frequency $\omega_{i}$. It consists of a peak at $eV=\omega_{i}/2$ and a
tail towards higher voltage. Its height has been multiplied by
$E_{F}/\Delta_{R}$. Shown in blue is the Josephson current that has been
broadened by dissipation. The lineshape is given by Eq. (4) for $kT_{0}\gg
E_{J}$ which is the typical situation [8] and is much narrower in the opposite
limit [9]. The collective mode is the new feature discussed in this paper.

##  II Collective modes in the tunneling spectrum

The idea of using tunneling to detect pair fluctuations goes back to the
seminal papers by Ferrell [10] and Scalapino [11]. They were interested in
pair fluctuations above the critical temperature $T_{c}$, and pointed out that
the pair fluctuations appear in linear response to an external pairing order
parameter, just like magnetization fluctuations appear as the linear
susceptibility to an external magnetic field. More specifically, they
considered a tunnel junction with voltage bias $V$ between two SCs $L$ (left)
and $R$ (right) with different $T_{c,j}$ and energy gaps $\Delta_{j}$,
$j\in\\{L,R\\}$, where the left SC is assumed to have a higher $T_{c}$. The
pair tunneling Hamiltonian is obtained by expanding the Josephson energy
$E_{J}$ of a junction with area $A$ to linear order in $\Delta_{R}$, which is
then replaced by the pair destruction operator
$\hat{\Delta}_{R}({\boldsymbol{r}})=|g_{0}|\
\psi_{R,\downarrow}({\boldsymbol{r}})\psi_{R,\uparrow}({\boldsymbol{r}})$,
$g_{0}$ being the BCS coupling. This gives the coupling Hamiltonian

$H_{\mathrm{pair}}=\int d{\boldsymbol{r}}\,Ce^{-i2eVt}\
\hat{\Delta}_{R}({\boldsymbol{r}})+\textrm{h.c.},$ (1)

which oscillates at the Josephson frequency $2eV$ (setting $\hbar=1$). Here,
we defined the coupling strength $C=\partial(E_{J}/A)/\partial\Delta_{R}$ as
well as the Josephson energy
$E_{J}=(g/4\pi)\Delta_{R}K(\sqrt{1-\Delta_{R}^{2}/\Delta_{L}^{2}})$ in terms
of the elliptic function $K$. Moreover, we introduced the dimensionless
conductance $g=(h/e^{2})/R_{N}$, where $R_{N}$ is the junction resistance in
the normal state [12]. In the limit $\Delta_{L}\gg\Delta_{R}$,
$E_{J}=(g/2\pi)\Delta_{R}\ln(4|\Delta_{L}/\Delta_{R}|)$ and we recover the
expressions given in Ref. [11]. Standard linear response theory gives the
current as

$I_{\mathrm{pair}}(V,H)=4eC^{2}A\
\mathrm{Im}\,\chi_{R}({\boldsymbol{q}}=2{\boldsymbol{q}}_{H},\omega=2eV),$ (2)

where $\chi_{R}({\boldsymbol{q}},\omega)$ is the Fourier transform of the pair
susceptibility

$\chi_{R}({\boldsymbol{r}},t)=-i\langle[\hat{\Delta}_{R}({\boldsymbol{r}},t),\,\hat{\Delta}_{R}(0)^{\dagger}]\rangle\theta(t),$
(3)

and $2{\boldsymbol{q}}_{H}$ is the pair momentum induced by a magnetic field
$H$ parallel to the junction [11]. (We have assumed that $R$ is a two
dimensional SC. Otherwise, the current will depend on the thickness of $R$
provided it is less than the coherence length, c.f. Ref. [11].)

The pair fluctuations were successfully measured very close to the transition
temperature $T_{c,R}$ [13]. In principle, there is no reason why the same
arguments cannot be applied to low temperatures. In that case, the collective
mode dispersions corresponds to poles in $\chi_{R}({\boldsymbol{q}},\omega)$
and should manifest as peaks in the tunneling current. In fact, this has been
proposed as a way to measure the Higgs mode [14]. In practice, peaks
corresponding to putative collective modes have never been observed in
tunneling experiments. A purpose of this paper is to explain this absence, and
to point out the conditions under which such observations may become
successful in the future.

##  III Josephson peak and multiple Andreev reflections

We begin by noting that the current $I_{\textrm{pair}}$ is proportional to
$g^{2}$ and thus fourth order in the tunneling matrix element. We first
consider the STM case and consider other terms to the same order in the
tunneling current. STM spectra exhibit sub-gap structures Stemming from
processes commonly known as multiple Andreev reflections (MAR) [8, 15]. They
may be calculated in an expansion in powers of the tunneling matrix element
[16]. At fourth order, the first set of MAR peaks appears at $eV=\Delta_{L}$
and $\Delta_{R}$ in $dI/dV$. They correspond to processes where a pair tunnels
across the junction and gains an energy $2eV$. For $2eV>2\Delta_{R}$ this
energy can go into exciting a pair of quasi-particles on the $R$ side. This
gives rise to a step threshold in the current $I(V)$ and consequently a peak
in $dI/dV$ at $eV=\Delta_{R}$. A similar argument produces a step at
$\Delta_{L}$. MAR peaks are commonly seen in STM when the tip is brought close
to the surface, increasing $g$ [8, 15]. The ratio of the lowest order MAR peak
in $dI/dV$ to the conductance above the coherence peak threshold is simply of
order $g$ (see Fig. 1). We note that $g$ of order 0.01 or even unity can be
achieved [17, 15]. Nevertheless, collective modes such as the Higgs mode have
not been reported in STM experiments. One reason lies in the fact that, in
conventional SC, the phase mode is pushed up to the plasma frequency and the
only remaining collective mode is the Higgs mode which has energy
2$\Delta_{R}$. This gives rise to a peak at $eV=\Delta_{R}$ which happens to
coincide with the lowest order MAR peak. Furthermore, as shown below, in STM
the magnitude of the collective mode contribution is reduced from the MAR
magnitude by a factor $\Delta_{R}/E_{F}$ where $E_{F}$ is the Fermi energy of
the $R$ SC. This reduction stems from point tunneling: in this case, the
current involves a convolution over the momentum of the mode. The latter
disperses rapidly on the scale of the inverse coherence length $\xi^{-1}$,
giving rise to this suppression. We conclude that the collective mode may be
visible in STM only for strongly correlated materials where the factor
$\Delta_{R}/E_{F}$ is not too small.

Another contribution to the same order in $g^{2}$ commonly seen in STM is the
Josephson current broadened by thermal noise. Thermal fluctuations dephase the
junction and convert the Josephson current from a delta function to a peak
structure at low but finite bias. The theory has been given by Ivanchenko and
Zilberman [18]. The result depends on the relative size of the Josephson
energy $E_{J}$ of the junction to the noise temperature $k_{B}T_{0}$. (Note
that $T_{0}$ is in general different from and larger than the sample
temperature $T$.) In the limit $E_{J}\ll k_{B}T_{0}$ the current is given by

$I_{J}(V)=e\frac{E_{J}^{2}}{k_{B}T_{0}}\frac{2eV\Gamma_{0}}{(2eV)^{2}+\Gamma_{0}^{2}},$
(4)

where the width is given by $\Gamma_{0}=k_{B}T_{0}R_{0}(2e)^{2}$ with the
dissipation parametrized by an effective resistance $R_{0}$. More
specifically, the external circuit is modelled by a series resistance $R_{0}$
(not to be confused with the normal state junction resistance $R_{N}$) which
gives rise to voltage fluctuations across the junction characterized by
$\langle\delta V(t)\delta
V(t^{\prime})\rangle=2k_{B}T_{0}R_{0}\delta(t-t^{\prime})$. Note that in Eq.
(4) the maximum current is proportional to $E_{J}^{2}/T_{0}$ which is
proportional to $g^{2}$. In fact, STM data are usually in this limit: the
lineshape predicted by Eq. (4) is often seen as a peak in $dI/dV$ whose height
is comparable to and scales in the same way as the MAR peak at $eV=\Delta_{R}$
with changing tip height [8]. On the other hand, planar junctions are in the
opposite limit $E_{J}\gg k_{B}T_{0}$ because $E_{J}$ scales with the area. In
this case, the peak is very narrow and steep [9]. A useful physical picture is
that of an overdamped particle moving in a tilted “washboard potential”. In
the limit $E_{J}\ll k_{B}T_{0}$ thermal fluctuations lead to rapid jumps over
the washboard barrier and give rise to phase slips, resulting in Eq. (4).

In the case of the Josephson current, the voltage bias is small and the
washboard is relatively flat. In the case of the collective mode we are in a
large voltage regime where the phase is running rapidly down the washboard and
subject to weak modulation due to $E_{J}$. In this case the phase across the
junction is given to a good approximation by $\theta(t)\approx
2e[V_{\textrm{ext}}t+\int_{0}^{t}dt’\delta V(t’)]$. Hence, the fluctuating
part of the phase correlation is given by
$\langle(\delta\theta(t)-\delta\theta(0))^{2}\rangle=(2e)^{2}2k_{B}T_{0}R_{0}t$.
Inserting this into Eq. (3), we find that the effect of thermal noise is to
introduce an additional Lorentzian convolution to the response function, with
a width given by $\Gamma_{0}$. Similar arguments show that the width of the
MAR peak is also given by $\Gamma_{0}$ (see Appendix B). Thus, the minimal
width of all the sub-gap structures shown in Fig. 1 is set by the width of the
Josephson peak in Eq. (4) which can be readily measured. It follows that a
condition for the visibility of the collective mode is simply that its width
given by $\Gamma_{0}$ is not so large that it will overlap other features such
as the Josephson peak or the MAR peak. Note that the width of the collective
mode and the MAR peak are the same whether $k_{B}T_{0}$ is large or small
compared with $E_{J}$. Only the Josephson peak is affected by this condition.

Figure 2:  Diagrams that contribute to the STM tunneling current to fourth
order in the tunneling matrix element $t_{{\boldsymbol{k}},{\boldsymbol{p}}}$
represented by the solid dot. The diagrams that couple to the collective mode
is shown in (c) where the double line represents the pair propagator of the
$R$ SC which is related to the pair susceptibility
$\chi_{\alpha,\beta}({\boldsymbol{q}},\omega=2eV)$ by analytic continuation.
(a) and (b) show the two diagrams $M_{1}$ and $M_{2}$ that contribute to the
triangle on the right side of (c). Two similar diagrams contribute to the left
triangle related to the right triangle by complex conjugation. The anomalous
Green function of the $L$ SC is shown in red. In (a) the $R$ SC Green function
is the regular $G$ while in (b) it is the anomalous $F$ function. The latter
are shown in (d). Note that frequency and momentum reverse sign on opposite
ends of $F$. (e) shows the BCS coupling in separable form for each channel
$\alpha$.

##  IV Microscopic treatment of the current due to collective modes

Next, we turn to a microscopic treatment of the problem, including the multi-
component SCs mentioned in the introduction. We will derive an extension of
the pair tunneling Hamiltonian Eq. (1) by calculating the in-gap current to
fourth order in $t_{{\boldsymbol{k}},{\boldsymbol{p}}}$, following earlier
work by Takayama [19]. The voltage drop across the junction can be absorbed
into a time dependent tunneling matrix element
$t_{{\boldsymbol{k}},{\boldsymbol{p}}}e^{ieVt}$, rendering the SC leads at
equilibrium. This allows a treatment within the conventional Matsubara
formalism, and the more elaborate Keldysh treatment [16] is not necessary. The
ingap current due to collective modes of the $R$ SC is then given by the
diagram Fig. 2(c). Details of the calculation are presented in Appendix A.
Below, we discuss the main results.

In the STM case, tunneling occurs at a single point and does not resolve the
momentum of the pair response function as in Eq. (2). Instead, the current
involves an integral over the momentum ${\boldsymbol{q}}$:

$I_{\textrm{STM}}(V)=4e\sum_{\alpha,\beta}\int d{\boldsymbol{q}}\
M_{\alpha}^{*}({\boldsymbol{q}},V)M_{\beta}({\boldsymbol{q}},V)\\\
\times\mathrm{Im}\,\chi_{\alpha,\beta}({\boldsymbol{q}},\omega=2eV).$ (5)

Here, we have introduced multiple pairing order parameters
$\Delta_{\boldsymbol{k}}^{(\alpha)}$ for the $R$ SC. We will drop the $R$
label from now on. The label $\alpha$ may refer to pairing in different bands,
or to members of a irreducible representation, or to superposition of
different pairing symmetries when time reversal or crystalline symmetry is
spontaneously broken. Following Ref. [3], we assume a separable form for the
attractive interaction
$U_{{\boldsymbol{k}},{\boldsymbol{k}^{\prime}}}=-\sum_{\alpha}g_{\alpha}\zeta_{\boldsymbol{k}}^{(\alpha)}\zeta_{\boldsymbol{k}^{\prime}}^{(\alpha)}$,
where the $\zeta_{\boldsymbol{k}}^{(\alpha)}$ are orthonormal form factors and
the $g_{\alpha}$ are the coupling constants in the corresponding channel. We
neglect the dependence on ${\boldsymbol{q}}$, the center of mass momentum of
the Cooper pair. This vertex is shown in Fig. 2(e). The pair destruction
operator is generalized to
$\hat{\Delta}_{\boldsymbol{k}}^{(\alpha)}({\boldsymbol{q}})=g_{\alpha}\sum_{\boldsymbol{k}}\zeta_{\boldsymbol{k}}^{(\alpha)}c^{(\alpha)}_{-{\boldsymbol{k}},\downarrow}c^{(\alpha)}_{{\boldsymbol{k}}+{\boldsymbol{q}},\uparrow}$.
For simplicity of notation, we assume singlet pairing, but the calculation can
be straightforwardly extended to general pairing symmetry. The pair
susceptibility can be generalized from Eq. (3) in a natural way as

$\chi_{\alpha,\beta}({\boldsymbol{q}},t)=-i\langle[\hat{\Delta}_{{\boldsymbol{k}}}^{(\alpha)}({\boldsymbol{q}},t),\hat{\Delta}_{\boldsymbol{k}}^{(\beta)}({\boldsymbol{q},}0)^{\dagger}]\rangle\theta(t).$
(6)

The matrix element $M_{\alpha}({\boldsymbol{q}},V)$ is given by the sum of
$M_{1}$ and $M_{2}$, which are defined by the two triangle diagrams shown in
Fig. 2(a) and (b), respectively. It is

$M_{\alpha}({\boldsymbol{q}},V)=T\sum_{\omega_{m}}\int
d{\boldsymbol{k}}\,d{\boldsymbol{p}}\,|\tilde{t}|^{2}\zeta_{\boldsymbol{p}}^{(\alpha)}F_{L}({\boldsymbol{k}},\omega_{m})\\\
\times[G_{R}^{(\alpha)}({\boldsymbol{p}}+{\boldsymbol{q}},\omega_{m}+ieV)G_{R}^{(\alpha)}(-{\boldsymbol{p}},-\omega_{m}+ieV)\\\
-F_{R}^{(\alpha)}({\boldsymbol{p}}+{\boldsymbol{q}},\omega_{m}+ieV)F_{R}^{(\alpha)}(-{\boldsymbol{p}},-\omega_{m}+ieV)].$
(7)

Here, as appropriate for STM tunneling, we have neglected the momentum
dependence of the tunneling matrix element
$t_{{\boldsymbol{k}},{\boldsymbol{p}}}$, replacing it by $\tilde{t}$. We have
further used the fact that the anomalous electron Green function satisfies
$F_{\downarrow\uparrow}({\boldsymbol{k}},\omega_{m})=-F_{\uparrow\downarrow}({\boldsymbol{k}},\omega_{m})$
which accounts for the negative sign in the second term. Note that one factor
of $\zeta_{\boldsymbol{k}}^{(\alpha)}$ enters the matrix element in Eq. (7)
and one factor enters the pair susceptibility in Eq. (5). It is easy to see
that the left triangle in Fig. 2(c) is the complex conjugate of the right
triangle. The product
$\zeta_{\boldsymbol{p}}^{(\alpha)}F_{L}({\boldsymbol{k}},\omega_{m})$ in Eq.
(7) determines which components of the pair fluctuation can be probed. For
example, if $L$ is a conventional s-wave SC, only collective modes with a
component $\alpha$ corresponding to s-wave will couple, as we shall illustrate
by an example below.

Based on the form Eq. (5), we suggest a nonlocal generalization of the pair
tunneling Hamiltonian,

$\tilde{H}_{\mathrm{pair}}=\sum_{\alpha}\int
d{\boldsymbol{r}}^{\prime}\,\tilde{C}_{\alpha}({\boldsymbol{r}}-{\boldsymbol{r}^{\prime}},V)e^{-i2eVt}\
\hat{\Delta}^{(\alpha)}_{R}({\boldsymbol{r}^{\prime}})+\textrm{h.c.},$ (8)

for STM tunneling at position ${\boldsymbol{r}}$. (Integration over
${\boldsymbol{r}}$ in Eq. (8) gives the generalization of Eq. (1) for planar
junctions.) It is clear that linear response based on Eq. (8) leads to Eq. (5)
if we identify the Fourier transform of
$\tilde{C}_{\alpha}({\boldsymbol{r}},V)$ with
$M_{\alpha}({\boldsymbol{q}},V)$. As shown in Appendix A, $M$ has a smooth $V$
dependence which can usually be ignored. More importantly, Eq. (5) involves a
convolution in momentum space between the pair susceptibility and the product
of the matrix element. We find that $M_{\alpha}({\boldsymbol{q}},V)$ goes to a
constant for small $q$ and falls off with $q$ on a scale given by the inverse
of the coherence length $\xi_{R}$ when $\Delta_{L}\gtrsim\Delta_{R}$ (see
Appendix for details). The physical origin of the nonlocality in Eq.(8) and
the convolution over ${\boldsymbol{q}}$ in Eq. (5) is that the Cooper pair is
injected from the $L$ SC one electron at a time by the single particle
tunneling matrix element. Consequently quasi-particles exists virtually over a
distance of order $\xi_{R}$ before recombining to form a Cooper pair on the
$R$ SC. We note that for the single order parameter case the simple pair
tunneling Hamiltonian Eq. (1) can be readily derived from Eq. (8). Focusing on
the planar case, in the absence of a magnetic field, and for $V\ll\Delta_{R}$,
it is $\tilde{C}\simeq
M({\boldsymbol{q}}=0,V=0)/A=\partial(E_{J}/A)/\partial\Delta_{R}$ (see
Appendix A), fully consistent with Eq. (1).

##  V Visibility

Next, we estimate the magnitude of the collective mode contribution to the
current. For simplicity we discuss the single order parameter case. In this
case, the inverse pair propagator generically takes the form
$\chi^{-1}({\boldsymbol{q}},\omega_{n})=N(0)[1+(\omega_{n}^{2}+b_{i}v_{F}^{2}q^{2})/(a_{i}\Delta^{2})]$,
where $a_{i}$ and $b_{i}$ are numbers of order unity [3]. The zeros of this
function give the collective mode dispersion $\omega_{i}(q)$. The pair
susceptibility thus takes the form

$\mathrm{Im}\
\chi({\boldsymbol{q}},\omega)=\frac{1}{N(0)}\frac{\pi\omega_{i0}^{2}}{2\omega_{i}({\boldsymbol{q}})}\delta(\omega-\omega_{i}({\boldsymbol{q}})),$
(9)

where $\omega^{2}_{i}({\boldsymbol{q}})=\omega_{i0}^{2}+b_{i}v_{F}^{2}q^{2}$
and $\omega_{i0}=\sqrt{a_{i}}\Delta$. Note that the dispersion is very steep:
$\omega_{i}({\boldsymbol{q}})$ roughly doubles in value when $q$ is of order
the inverse of the coherence length $\xi=v_{F}/\pi\Delta$. In planar junctions
this form leads to delta functions in the current $I(V)$ at
$2eV=\omega_{i}({\boldsymbol{q}})$. Thus, planar junction tunneling allows
access to the dispersion of the mode. This is not the case for which requires
an additional integration over ${\boldsymbol{q}}$. Instead of a delta
function, the $I(V)$ now features a step at $2eV=\omega_{i0}$ followed by a
smooth drop off on a scale set by $M({\boldsymbol{q}},V)$. The step function
gives rise to a delta function in $dI/dV$,

$\displaystyle\frac{dI_{\textrm{STM}}}{dV}=e^{2}\frac{8\pi^{2}a_{i}\Delta^{2}}{N(0)b_{i}v_{F}^{2}}\left|M(0,V)\right|^{2}\delta(2eV-\omega_{i0}),$
(10)

which is followed by a negative tail towards larger voltages as sketched in
Fig. 1. To estimate $M$ in Eq. (10), it suffices to consider its $V=0$ limit.
Focusing on a symmetric junction $\Delta_{R}=\Delta_{L}=\Delta$, it is
$M=\partial E_{J}/\partial\Delta=g/8$ [12]. With this, we may compare the peak
in $dI/dV$ due to the collective mode to the step in the MAR current which is
given by $eg^{2}\Delta$ [16]. (See also Appendix B.) Using $N(0)=m/2\pi$ and
$E_{F}=mv_{F}^{2}/2$ we find the relative magnitude of the collective mode and
MAR steps to be approximately $\Delta/E_{F}$ as stated earlier. For
conventional SCs this ratio is very small which makes the detection of the
collective mode infeasible. Nonetheless, there are now examples of strongly
correlated SCs where this ratio is not very small. It is worthwhile to look
for the collective mode contribution in STM in such systems.

The situation is more promising for planar junctions. (See Appendix A for
details.) It is useful to consider the ratio of the collective mode current
$I_{\textrm{planar}}$ to the current in the normal state at $eV=2\Delta_{R}$
which is given by $I_{N}=2\Delta/(eR_{N})$. For simplicity, we again consider
the symmetric case $\Delta_{R}=\Delta_{L}=\Delta$. We find the ratio

$\frac{I_{\textrm{planar}}}{I_{N}}=\frac{\pi^{3}}{32}\frac{a_{i}g}{Ak_{F}^{2}}\frac{\Delta
E_{F}}{\omega_{i0}}\frac{\Gamma}{(2eV-\omega_{i0})^{2}+\Gamma^{2}}.$ (11)

Here, we have replaced the delta function in Eq. (9) by a Lorentzian with
width $\Gamma$ which is given by $\Gamma_{0}$ plus other sources of broadening
such as inhomogeneity. A similar ratio for the planar MAR current is obtained
in Appendix B, where it is found to have parametrically the same prefactor up
to numerical constants, but with the Lorentzian in Eq. (11) replaced by
$(eV/\Delta)[(eV)^{2}-\Delta^{2}]^{-1/2}$. The latter should also be broadened
by voltage noise and inhomogeneity. Since this form is less singular than the
Lorentzian, the collective mode contribution should dominate over the MAR peak
in planar junctions, in contrast to the situation in STM. More precisely, with
the same broadening $\Gamma$, the peak currents due to the collective mode and
the MAR have a ratio of $\sqrt{\Delta/\Gamma}$ which is greater than one.

We now address the size of the signal from the collective mode contribution
given by Eq. (11). We interpret $Ak_{F}^{2}$ as the number of tunneling
channels in a planar junction and use the Landauer formula to define the ratio
$\mathcal{T}_{\textrm{eff}}=g/(Ak_{F}^{2})$ as the effective tunneling
probability per channel. $\mathcal{T}_{\textrm{eff}}$ gives the intrinsic
transparency of a tunnel junction and is generally a very small number.
Conversely, the peak value of the Lorentzian is $1/\Gamma_{i}$, and the ratio
$E_{F}/\Gamma_{i}$ is a very large number. For a typical planar junction,
$\mathcal{T}_{\textrm{eff}}$ is so small that the product is still too small
to be observable for reasonable $\Gamma_{i}$. This may be the reason why
neither MAR nor collective modes have been observed in planar junctions.
However, as we shall see, the numbers are not too far off, and there may be
reasons for optimism. To see this we estimate that for the typical oxide
tunnel barrier used in Ref. [13], the transparency is
$\mathcal{T}_{\textrm{eff}}\approx 10^{-8}$ (assuming $A\sim 100^{2}$nm2,
$k_{F}\sim 1$Å-1, and $R_{N}\sim 2\Omega$). In this experiment a fluctuating
pair tunneling peak with width of about $1\mu$eV was readily observed. We
conclude that a collective mode with width of order $1\mu$eV should be
observable in a conventional oxide planar junction. because the signal in Eq.
(11) is proportional to the ratio $\mathcal{T}_{\textrm{eff}}/\Gamma_{i}$. The
minimal contribution to the width comes from voltage fluctuations and the
corresponding $\Gamma_{0}$ can be made very small [9]. In practice, in many of
the strongly correlated SC of interest, such as cuprates or iron based SC,
local inhomogeneity may lead to significant broadening of the collective mode
in planar junctions. Their detection may require higher tunneling transparency
$\mathcal{T}_{\textrm{eff}}$ and possibly smaller area junctions. The latter
requirement will reduce the current, making the experiment more challenging.

Apart from the larger signal compared with STM, we note that the current
itself is predicted to show a narrow peak, so that the $dI/dV$ signal is the
derivative of a Lorentzian which has a distinctive lineshape with a large
negative part. Furthermore, the planar junction has the advantage that the
dispersion of the collective mode may be probed by applying an in-plane
magnetic field. These distinctive features will provide strong evidence that a
collective mode is being observed.

##  VI Promising experimental platforms

Now, we discuss several multi-component examples where the collective modes
may be detected in $I(V)$ or $dI/dV$ as sharp peaks. We note that
generalization of Eqs. (10) and (11) to the case of multiple order parameters
is straightforward. Indeed, one may assume that the pair susceptibility takes
the form of a weighted sum of delta functions, $\mathrm{Im}\
\chi_{\alpha,\beta}({\boldsymbol{q}},\omega)=\sum_{i}w_{\alpha,\beta}^{i}({\boldsymbol{q}})\delta(\omega-\omega_{i}({\boldsymbol{q}}))$.
This translates directly into a sum of peaks at $2eV=\omega_{i0}$ and weighted
by $w_{\alpha,\beta}^{i}$ in the $dI/dV$ [$I(V)$] for STM [planar] tunneling.
We expect the magnitude estimates, which were obtained above in the single
order parameter case, to apply also in the more general case. For simplicity,
we assume that the SC being probed is inversion symmetric. If the $L$ SC is
conventional, only s-wave Cooper pairs can be injected from the $L$ electrode
and only the $\alpha$-components corresponding to s-wave pairing survive.
Consider first multi-band SCs, where s-wave pairing occurs in two different
Fermi surfaces $\alpha=1,2$, as, e.g., in MgB2 and in iron based
superconductors. In the latter case the two s-wave components are out of
phase, referred to as $s_{\pm}$ state. We expect a collective mode (the
Leggett mode) corresponding to the out of phase oscillation of the two order
parameters $\Delta^{(1)}$ and $\Delta^{(2)}$. This mode will manifest as a
pole in $\chi_{1,2}$. The Leggett mode has been observed by Raman scattering
in MgB2 at a relatively high energy of 9.2 meV which lies between the two
doubled energy gaps [20]. Hence, this mode is damped, but it is nonetheless
interesting to search for this peak within tunneling spectroscopy. We note
that MgB2 planar tunneling junctions have been successfully fabricated [21].
For the Fe based SCs, the situation is not so clear. We note that recently an
observation of the Leggett mode was reported in single layer NbSe2 in an
experiment using a normal STM tip [22]. Here, the Leggett mode is interpreted
as giving an excitation above the gap at energy $\Delta+\omega_{i}$.
Interestingly, the experiment found that $\omega_{i}/2\approx 0.7\Delta$ which
places the peak well inside the gap and well below the MAR structure. Note
that NbSe2 features a small ratio of $\Delta/E_{F}$ so that the STM signal of
the Leggett mode is expected to be very small, but perhaps a planar junction
experiment can be attempted.

As a second application we consider the case of a time reversal breaking SC,
more specifically of the type $s+id$, i. e. an admixture of s- and d-wave
pairing. This case is treated in detail in Ref. [3] and here we only summarize
the salient features. The important point is that the presence of the s wave
component allows us to couple to novel collective modes such as clapping
modes. We define the s- and d-wave order parameter components as
$(\Delta^{(0)},\Delta^{(2)})=(\eta_{0},-i\eta_{2})\Delta_{0}e^{i\theta}$,
where $\theta$ is the overall pair phase, and $\eta_{0}$ and $\eta_{2}$ are
real numbers satisfying $\eta_{0}^{2}+\eta_{2}^{2}=1$. It is convenient to
introduce $\Delta^{\pm}=\Delta^{(0)}/\eta_{0}\pm\Delta^{(2)}/(i\eta_{2})$. The
saddle point solution occurs at $\Delta^{+}=\Delta_{0}$ and $\Delta^{-}=0$.
Expanding around the saddle point we find the coordinates of the collective
modes as

$\Delta^{+}({\boldsymbol{r}},t)=e^{i\theta}[\Delta_{0}+h({\boldsymbol{r}},t)],$
(12)
$\Delta^{-}({\boldsymbol{r}},t)=e^{i\theta}[a({\boldsymbol{r}},t)+ib({\boldsymbol{r}},t)].$
(13)

Here, $h$ denotes the amplitude or Higgs mode, while $a$ and $b$ denote two
new modes which are generalizations of the clapping mode in $p+ip$ SCs.
Poniatowski et al. [3] show that these modes lie at approximately
$\sqrt{2}\Delta_{0}$. We will now show that they appear in the s-wave pair
fluctuation channel in Eq. (10). To this end, we expand

$\Delta^{(0)}({\boldsymbol{r}},t)=e^{i\theta}[\eta_{0}\Delta_{0}+\tilde{\Delta}^{(0)}({\boldsymbol{r}},t)].$
(14)

It is easy to see that the fluctuating part is given by

$\tilde{\Delta}^{(0)}({\boldsymbol{r}},t)=\eta_{0}[h({\boldsymbol{r}},t)+a({\boldsymbol{r}},t)+ib({\boldsymbol{r}},t)].$
(15)

Hence, all three modes will appear in the $\alpha=0$ pair fluctuation
component in Eq. (10). In particular, the generalized clapping modes will show
up as peaks in the vicinity of $eV=\Delta_{0}/\sqrt{2}$, well separated from
the MAR peak at $\Delta_{0}$. This is shown schematically in Fig. 1. In the
iron based SC Ba1-xKxFe2As2, a time reversal breaking SC state appears for $x$
between 0.7 and 0.8 and is suspected to be an $s+id$ SC [23]. This is an
excellent candidate to search for these collective modes.

We consider a third example of triplet pairing where the time reversal
breaking state may be of the $p+ip$ or $p+if$ type. UTe2 may be an example
[24]. This rather complicated structure preserves inversion in the bulk, but
it is possible, indeed likely, that the top layer breaks inversion due to some
local structural relaxation. In this case the s-wave pair injected by the $L$
SC is admixed with the p- and f-wave order parameter in the first layer of the
$R$ SC, and the matrix element $M_{\alpha}$ in Eq. (5) is nonzero for non
s-wave components. In this way, collective modes may couple to the current.
Indeed, a recent STM experiment using a Nb tip found a subgap peak near the
expected energy gap for UTe2 and a peak in $dI/dV$ near zero voltage
suggestive of a broadened Josephson current [24]. The latter observation
points to an admixture of s-wave pairing in the top layer, as we need.
Unfortunately, the ratio $\Delta/E_{F}$ may be too small for STM to observe
the collective mode in this system.

##  VII Discussion

We end by discussing the feasibility of probing collective modes of
unconventional SC using either planar or STM tunnel junctions. It is generally
considered difficult to make planar tunnel junctions in these systems, but
with modern fabrication techniques it may be possible to create nanoscale
tunnel junctions with high transparency that are free of pin-holes. Another
approach may involve stacks of van der Waals materials such as transition
metal dichalcogenides (TMD). A variety of insulating TMD may be used to create
monolayer barriers between SC layers. In cuprates, a single layer of
insulating parent state may be used as tunneling barrier [25]. On the STM
front, it has been demonstrated that it is possible to pick up a piece of
layered SC with an STM tip, which then serves as the SC electrode [26]. This
holds promise for the present application: e.g., a cuprate SC tip would allow
the collective modes of systems with $d$-wave [27] or $d+id$ pairing symmetry
to be probed. As noted above, the detection of collective modes in STM relies
on a relatively large ratio of $\Delta/E_{F}$. Recently, a range of strongly
correlated SC have been discovered, which represent promising candidates for
the present proposal. Notable examples are twisted bilayer and trilayer
graphene, where the ratio is so large that the BEC limit may be reached [28].
Another example are the iron based topological SC, which have very small Fermi
energy [17]. An example that is not well understood is the superconductivity
observed in YPtBi which involves doping of a quadratic touching band with a
very small Fermi energy and a short coherence length [29]. We conclude that,
while they are challenging, tunneling experiments probing the signatures of
the collective modes in unconventional SC are within reach.

## Acknowledgements

We thank Shuqiu Wang and Seamus Davis for sharing their data on UTe2 which
stimulated this investigation. We thank Nicholas Poniatowski, Leonid Glazman
and Iliya Esin for helpful discussions. PL acknowledges support by DOE (USA)
office of Basic Sciences Grant No. DE-FG02-03ER46076. JFS acknowledges support
by the Air Force Office of Scientific Research under award number
FA9550-22-1-0339.

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##  Appendix A Derivation of tunneling current due to collective modes

We model the junction by the single-electron tunneling Hamiltonian

$H_{\textrm{tun}}=\sum_{{\boldsymbol{k}},{\boldsymbol{p}},\sigma}t_{{\boldsymbol{k}},{\boldsymbol{p}}}e^{ieVt}c_{L,{\boldsymbol{k}},\sigma}^{\dagger}c_{R,{\boldsymbol{p}},\sigma}+\textrm{h.c.},$
(16)

where we have included the voltage $V$ across the junction through time-
dependent tunneling. In this framework, one may assume the electrons of both
sides to be in equilibrium and standard Matsubara diagrammatic techniques may
be applied. The DC current is given by averaging the current operator related
to $H_{\textrm{tun}}$ over time,

$I=\frac{1}{\beta}\int_{0}^{\infty}d\tau
I(\tau)=-\frac{2e}{\beta}\sum_{kp\sigma}\textrm{Im}\int_{0}^{\infty}d\tau\langle
T_{\tau}t_{{\boldsymbol{k}},{\boldsymbol{p}}}e^{i(ieV)\tau}c_{Lk\sigma}^{\dagger}(\tau)c_{Rp\sigma}(\tau)U(\beta)\rangle,$
(17)

where we defined the imaginary time evolution operator

$U(\beta)=T_{\tau}\exp{[-\int_{0}^{\beta}d\tau^{\prime}H_{\textrm{tun}}(\tau^{\prime})]}.$
(18)

(See Appendix B for a more careful treatment of the time-dependent factor in
imaginary time.) We can now expand Eq. (17) to fourth order in
$t_{{\boldsymbol{k}},{\boldsymbol{p}}}$ and proceed with the usual decoupling
into products of Green functions. We are interested in diagrams that involve
the imaginary time ordered pair propagator for the $R$ SC,

$\chi_{\alpha,\beta}({\boldsymbol{q}},\tau)=-\langle
T_{\tau}[\hat{\Delta}_{{\boldsymbol{k}}}^{(\alpha)}({\boldsymbol{q}},\tau)\
\hat{\Delta}_{\boldsymbol{k}}^{(\beta)}({\boldsymbol{q}},0)^{\dagger}]\rangle.$
(19)

Upon Fourier transform and analytic continuation,
$\chi_{\alpha,\beta}({\boldsymbol{q}},i\omega_{n}\rightarrow\omega+i\eta)$,
$\eta$ positive infinitesimal, becomes the retarded pair susceptibility
defined in Eq. (6). The relevant diagrams are shown in Fig. 2(c). They consist
of the propagator connected to a triangular graph on each side. The sides of
the triangle that connect to $\chi$ must refer to the $R$ SC, which means the
remaining side refers to the $L$ SC. Furthermore, this side must couple to the
anomalous propagator $F_{L}$ because we want to inject a pair from $L$ to $R$.
There are two versions of the triangles, depending on whether the $L$ Green
function is the normal $G$ or the anomalous $F$. We call the ones on the right
of the pair propagator $M_{1}$ and $M_{2}$. It is easy to see that the ones on
the left are the respective complex conjugates. In this way, we arrive at Eq.
(7) where $M=M_{1}+M_{2}$. After analytic continuation, we obtain Eq. (5).
Note that the voltage $eV$ is injected at each tunneling vertex and is passed
directly to the pair propagator. This has the important consequence that the
voltage gives a direct measure of the collective mode frequency via
$2eV=\omega_{i}$.

###  A.1 Collective mode: STM junction

The momentum label on the triangles depends on the nature of the tunnel
junction. We first discuss the STM or point contact tunneling case. The
corresponding diagrams are shown in Fig. 2(a) and (b). To simplify notation,
we consider a single s-wave component and suppress the $\alpha$ label. The
tunneling occurs at a single point, which means that
$t_{{\boldsymbol{k}},{\boldsymbol{p}}}$ is independent of momentum and can be
set to a constant $\tilde{t}$. In contrast to frequency, momentum is not
conserved at the vertex for tunneling at a point. The integration over $k$ for
the $L$ SC can be done separately, which simply gives the anomalous Green
function at ${\boldsymbol{r}}=0$, i.e.
$F_{L}({\boldsymbol{r}}=0,\omega_{n})=\pi
N_{L}(0)\Delta_{L}/\sqrt{\omega_{m}^{2}+\Delta_{L}^{2}}$ where $N(0)$ is the
density of states at the Fermi level. Furthermore, a finite momentum $q$ is
passed on to the propagator, which needs to be integrated over. This gives
rise to the integral over ${\boldsymbol{q}}$ shown in Eq. (4). This is the
main difference between STM and planar tunneling, as shall be discussed below.

The full expression for $M({\boldsymbol{q}},V)$ requires numerical
integration. Here we consider some simple limits. First we set
${\boldsymbol{q}}=0$ and then consider the expansion to quadratic order in
${\boldsymbol{q}}$. Eq. (7) becomes

$\displaystyle M({\boldsymbol{q}}=0,V=0)=$ $\displaystyle\
T\sum_{m}\tilde{t}^{2}F_{L}({\boldsymbol{r}}=0,\omega_{m})\int
d{\boldsymbol{p}}\,[G_{R}(\omega_{m},{\boldsymbol{p}})G_{R}(-\omega_{m},-{\boldsymbol{p}})-F_{R}(\omega_{m},{\boldsymbol{p}})F_{R}(-\omega_{m},-{\boldsymbol{p}})]$
(20) $\displaystyle=$ $\displaystyle\
T\sum_{m}\tilde{t}^{2}\frac{\Delta_{L}}{\sqrt{\omega_{m}^{2}+\Delta_{L}^{2}}}N_{L}(0)N_{R}(0)\int
d\xi_{\boldsymbol{p}}\left[\left(\frac{i\omega_{m}+\xi_{\boldsymbol{p}}}{\omega_{m}^{2}+E_{\boldsymbol{p}}^{2}}\right)\left(\frac{-i\omega_{m}+\xi_{\boldsymbol{p}}}{\omega_{m}^{2}+E_{\boldsymbol{p}}^{2}}\right)-\frac{\Delta_{R}^{2}}{(\omega_{m}^{2}+E_{\boldsymbol{p}}^{2})^{2}}\right]$
(21)

where $E_{\boldsymbol{p}}=\sqrt{\Delta^{2}+\xi_{\boldsymbol{p}}^{2}}$ and
$\xi_{\boldsymbol{p}}=\epsilon({\boldsymbol{p}})-\mu$. We have set $V$=0 in
this expression because its dependence on $V$ is expected to be smooth for the
following reason. As seen from the diagrams in Fig. 2, the effect of $V$ is to
shift $\omega_{m}$ to $\omega_{m}+ieV$ and $\omega_{m}-ieV$ in the two normal
Green functions $G$. The sum over $\omega_{m}$ in Eq. (20) can be converted to
an integral over the real variable $\omega_{m}\rightarrow\omega$ in the
$T\rightarrow 0$ limit. It is clear that adding an imaginary part will affect
the integral in a smooth way.

The factor in square brackets in Eq. (20) can be written as

$\frac{1}{\omega_{m}^{2}+\xi_{\boldsymbol{p}}^{2}+\Delta_{R}^{2}}-\frac{2\Delta_{R}}{(\omega_{m}^{2}+\xi_{\boldsymbol{p}}^{2}+\Delta_{R}^{2})^{2}}=\frac{d}{d\Delta_{R}}\left[\frac{\Delta_{R}}{\omega_{m}^{2}+\xi_{\boldsymbol{p}}^{2}+\Delta_{R}^{2}}\right].$
(22)

On the other hand, the Josephson energy is given by [12]

$E_{J}=T\sum_{m}\tilde{t}^{2}\int
d{\boldsymbol{k}}d{\boldsymbol{p}}\,F_{L}(\omega_{m},{\boldsymbol{k}})F_{R}(\omega_{m},{\boldsymbol{p}})=T\sum_{m}\tilde{t}^{2}N_{L}(0)N_{R}(0)\int
d\xi_{\boldsymbol{p}}\frac{\Delta_{L}}{\sqrt{\omega_{m}^{2}+\Delta_{L}^{2}}}\frac{\Delta_{R}}{\omega_{m}^{2}+\xi_{\boldsymbol{p}}^{2}+\Delta_{R}^{2}}.$
(23)

Hence, we find that

$M({\boldsymbol{q}}=0,V=0)=\frac{\partial E_{J}}{\partial\Delta_{R}}$ (24)

as quoted in the text.

Next we calculate the leading $q^{2}$ term in $M({\boldsymbol{q}},V)$. We
expand
$\xi_{{\boldsymbol{p}}+{\boldsymbol{q}}}\approx\xi_{\boldsymbol{p}}+v_{F}q\cos{\theta}$
where $\theta$ is the angle between ${\boldsymbol{p}}$ and ${\boldsymbol{q}}$.
We keep only the linear in $q$ term in this expansion. This amounts to
assuming a constant Fermi velocity $v_{F}$. After a straightforward but
laborious calculation, we arrive at

$\displaystyle M({\boldsymbol{q}},V=0)-M({\boldsymbol{q}}=0,V=0)\approx$
$\displaystyle\
-v_{F}^{2}q^{2}\frac{\pi}{4}\tilde{t}^{2}N_{L}(0)N_{R}(0)\int\frac{d\omega}{2\pi}\frac{\Delta_{L}}{\sqrt{\omega^{2}+\Delta_{L}^{2}}}\frac{\Delta_{R}^{2}-\omega^{2}}{(\Delta_{R}^{2}+\omega^{2})^{5/2}}$
(25) $\displaystyle=$ $\displaystyle\
-q^{2}\xi_{R}^{2}\frac{\pi^{2}}{8}\tilde{t}^{2}N_{L}(0)N_{R}(0)f(y=\Delta_{L}/\Delta_{R})$
(26)

where we defined the BCS coherence length of the $R$ SC
$\xi_{R}=v_{F}/\pi\Delta_{R}$, as well as the dimensionless function

$f(y)=y\int dw\frac{1}{\sqrt{w^{2}+y^{2}}}\frac{1-w^{2}}{(1+w^{2})^{5/2}}.$
(27)

For $y\gg 1$, $f\rightarrow 2/3$ while for $y\ll 1$ it behaves as $f\sim
y\ln{1/y}$. At $\Delta_{L}=\Delta_{R}$, $f(1)=\pi/4$. Thus, for
$\Delta_{L}\gg\Delta_{R}$ and $\Delta_{L}\sim\Delta_{R}$,
$M({\boldsymbol{q}},V=0)$ decreases with $q$ on the scale of the inverse of
the coherence length $\xi_{R}$. The physical picture is that the injected
quasi-particle pair can travel a distance $\xi_{R}$ in the $R$ SC before
forming a Cooper pair. In the $\Delta_{L}\ll\Delta_{R}$ limit the scale
depends on $\Delta_{L}$ as well.

Figure 3:  Diagrams that contribute to $M_{1}$ in planar tunnel junctions. (a)
momentum parallel to the plane is conserved. (b,c) diffusive case. Dashed line
represents averaging over the random tunneling matrix elements. (b) is in
position space. Note the the averaging forces both ends of the red line to be
at the same ${\boldsymbol{r}}$. (c) The momentum space version of (b) obtained
by Fourier transform.

###  A.2 Collective mode: Planar junction

We now turn to the case of a planar tunnel junction. We consider two tunneling
models. First, we assume that momentum parallel to the plane is conserved
during the tunneling process. The momentum labels in the triangle graph are
shown in Fig. 3(a). Note the difference to the STM case shown in Fig. 2. Now
there is only one momentum integration. Importantly, the pair propagator
carries momentum $q=0$. This will be changed to $q_{H}$ in the presence of a
parallel magnetic field [11]. In contrast to the STM case, there is no
convolution over ${\boldsymbol{q}}$. We thus recover the prediction from
linear response to the pair tunneling Hamiltonian Eq. (1). Second, we consider
diffusive scattering at the junction interface. We follow Takayama [19] and
assume local tunneling of the form

$H_{\textrm{local tun}}=\int
d{\boldsymbol{r}}\,\hat{t}({\boldsymbol{r}})\sum_{\sigma}\psi_{L,\sigma}^{\dagger}({\boldsymbol{r}})\psi_{R,\sigma}({\boldsymbol{r}})+\textrm{h.c.},$
(28)

where ${\boldsymbol{r}}$ is the spatial coordinate in the plane, and $\hat{t}$
is a random variable such that
$\langle\hat{t}(\vec{r})\hat{t}(\vec{r^{\prime}})\rangle=t_{0}^{2}\delta({\boldsymbol{r}}-{\boldsymbol{r}^{\prime}})$.
This model has the desirable feature that the current, when computed to fourth
order in $t_{0}$, scales correctly with the junction area. A more realistic
model would have a nonzero average $\langle\hat{t}(\vec{r})\rangle$. This
latter part conserves parallel momentum and thus may be included by simply
adding the diagrams of Fig. 3(a) and (c). In Fig. 3(b) we show the triangle
diagram in real space where the dashed line represents averaging over the
random variable. Upon Fourier transform we obtain the diagram in momentum
space as shown in Fig. 3(c). The momentum transfer to the pair propagator is
zero in this case. In a parallel magnetic field, momentum
${\boldsymbol{q}}_{H}$ is injected at the tunneling vertices, leading to
${\boldsymbol{q}}_{H}$ dependence of $M$ and excitation of the collective mode
at $2{\boldsymbol{q}}_{H}$. We conclude that in both cases, the planar
tunneling case can be described by a pair tunneling Hamiltonian which is
analogous to Eq. (1), except that $C$ should be replaced by a slowly varying
function of $V$ which is nonlocal in space. Indeed, this function is readily
obtained from its Fourier transform
$\tilde{C}({\boldsymbol{q}},V)=M({\boldsymbol{q}},V)$, as described in the
main text below Eq. (8).

##  Appendix B Calculation of leading MAR subgap structures

In this section we employ the Matsubara method to treat the leading MAR
structure which is fourth order in the tunneling matrix element. The
equivalent calculation was done by Cuevas et al. [16] for the STM case using
Keldysh technique. We first present results for the planar tunneling case,
including also finite temperature effects. We were not able to find this case
treated in the literature, and we find the Matsubara method to be less
laborious than the Keldysh method, even though some subtlety is involved.

As noted in the text, the voltage drop across the junction can be absorbed
into a time dependent tunneling matrix element
$t_{{\boldsymbol{k}},{\boldsymbol{p}}}e^{ieVt}$. After this step the SC in the
leads are at equilibrium so that conventional Matsubara method can be used.
This was done in calculating the matrix element $M$ that couples to the
collective mode in the last section. The proper procedure was given by Tsuzuki
[30] and used by Takayama [19]. The idea is to introduce $\Omega_{n}=2\pi nT$,
where $n$ is a positive integer, and replace
$t_{{\boldsymbol{k}},{\boldsymbol{p}}}e^{ieVt}$ by
$t_{{\boldsymbol{k}},{\boldsymbol{p}}}e^{i\Omega_{n}\tau}$, where $\tau=it$.
In this way the evolution in $\tau$ is unitary. At the end of the calculation,
after internal Matsubara frequencies have been summed, we replace
$i\Omega_{n}$ by $eV$. In a slight departure from Tsuzuki, we keep the
adiabatic turning on term $e^{\eta t}$ as $e^{-i\eta\tau}$ so that
$i\Omega_{n}\rightarrow eV+i\eta$. We will see that failure to follow this
procedure will result in erroneous thermal factors. This subtlety did not
arise in the calculation of the matrix element $M$ in the previous section
which involves only virtual excitations of quasi-particles. In contrast MAR
involves the real excitation of quasi-particles and their thermal occupation
appears in a crucial way.

###  B.1 MAR: Planar junction

To illustrate this point, we give some details for the case of planar
junctions with diffusive scattering using the random tunneling model described
in the last section [19]. A representative diagram for the current that shows
the flow of momentum and frequency is shown in 4(a). There are other diagrams
where each solid line can be either $G$ or $F$ functions. In other words, the
diagram should be viewed as the trace of the matrix product of four Green
function in Nambu space. In addition, the dashed line representing averaging
over random tunneling matrix elements can connect the ends of the solid lines
on the top and bottom instead of left and right as shown in Fig.4a. However,
its contribution can be shown to be less singular. Here, we show the
evaluation of this particular diagram which we label by $i$. The sum over
${\boldsymbol{k}}$ and ${\boldsymbol{k}^{\prime}}$ converts the anomalous
Green function $F_{L}$ to the local form, and we have a single sum over
${\boldsymbol{p}}$. We write the contribution of this diagram to the MAR
current for the diffusive planar junction as

$I^{\textrm{MAR}}_{\textrm{planar},i}=4e|t_{0}|^{4}\textrm{Im}\
J_{i}(i\Omega_{n}\rightarrow eV+i\eta),$ (29)

where

$\displaystyle J_{i}(i\Omega_{n})=$ $\displaystyle\
T\sum_{m}|F_{L}(x=0,\omega_{m})|^{2}A\int
d{\boldsymbol{p}}\,[G_{R}(\omega_{m}+\Omega_{n},{\boldsymbol{p}})G_{R}(-\omega_{m}+\Omega_{n},-{\boldsymbol{p}})$
(30) $\displaystyle=$ $\displaystyle\
T\sum_{m}\frac{\pi^{2}\Delta_{L}^{2}}{\omega_{m}^{2}+\Delta_{L}^{2}}N^{2}_{L}(0)N_{R}(0)A\int
d\xi_{\boldsymbol{p}}\,\left[\frac{i\omega_{m}+i\Omega_{n}+\xi_{\boldsymbol{p}}}{(i\omega_{m}+i\Omega_{n})^{2}-E_{\boldsymbol{p}}^{2}}\right]\left[\frac{-i\omega_{m}+i\Omega_{n}+\xi_{\boldsymbol{p}}}{(-i\omega_{m}+i\Omega_{n})^{2}-E_{\boldsymbol{p}}^{2}}\right].$
(31)

Here $A$ is the junction area. The sum over $m$ is done in the standard way by
extending $i\omega_{m}$ to the complex $z$ and converting $\int d\xi$ to $\int
dE\ E/\sqrt{E^{2}-\Delta^{2}}$ where $E=\sqrt{\xi^{2}+\Delta^{2}}$. There are
6 poles $z_{i}$ in Eq. (30). The sum over $n$ leads to a sum over the product
of the residues of these poles and $n_{F}(z_{i})$ where $n_{F}$ is the Fermi
function. For $eV>0$ the important poles are at $z_{3}=i\Omega_{n}-E$ and
$z_{2}=-z_{3}$. The Fermi function becomes
$n_{F}(z_{3})=n_{F}(i\Omega_{n}-E)=n_{F}(-E)$. The last step is crucial. Had
we not replaced $eV$ by $i\Omega_{n}$, we would have gotten $n_{F}(eV-E)$
which is the wrong distribution and will give a thermal smearing for $eV$ near
the gap $\Delta$.

After this step we can set $i\Omega_{n}\rightarrow eV-i\eta$. A pole appears
at $E=eV+i\eta$. Taking the imaginary part of $J_{i}(i\Omega_{n}\rightarrow
eV+i\eta)$ gives a delta function $\delta(2eV-2E)$. This is the expression of
energy conservation: the tunneling of a Cooper pair gains an energy $2eV$
which are used to excite two quasi-particles each at energy $E$. This is
because momentum is conserved on average after averaging over the random
tunneling amplitudes and a pair of quasi-particles with opposite momenta
${\boldsymbol{p}}$ and $-{\boldsymbol{p}}$ and equal energy $E$ are created.
For $eV>\Delta$ this opens up a new threshold for conduction. The contribution
of this diagram is given by

$I^{\textrm{MAR}}_{\textrm{planar},i}=e\frac{\pi^{3}}{2}|t_{0}|^{4}N^{2}_{L}(0)N_{R}(0)A\frac{eV}{\sqrt{(eV)^{2}-\Delta^{2}}}[n_{F}(-eV)-n_{F}(eV)].$
(32)

Inclusion of the other diagrams only changes the numerical prefactor. The
important thing to note is that the current appears as an inverse square root
divergence above the threshold. This just reflects the density of states of
finding the quasi-particle and is very different from the step singularity in
the well known STM case. As we shall see below, this is because in that case a
pair of quasi-particles at energy $E_{p}$ and $E_{p^{\prime}}$ are excited and
a convolution over two quasi-particle density of states appear. Also note that
the thermal factor $n_{F}(-eV)-n_{F}(eV)$ comes from the poles at $z_{3}$ and
$z_{2}$. This factor equals unity up to corrections of order $e^{-\Delta/T}$
which is negligible. Therefore there is no thermal smearing of the square root
divergence near threshold.

Figure 4:  Representative diagrams that contribute to the MAR currents. (a)
Planar junction, diffusive case. Dashed line represents averaging over the
random tunneling matrix elements. (b) Planar junction, clean case. Momentum
parallel to the plane is conserved resulting in a single momentum integral
over $k$. (c) STM case. The tunneling matrix element represented by the solid
dots add or remove Matsubara frequency $\Omega_{n}$ which is analytically
continued to $\Omega_{n}\rightarrow eV+i\eta$ after the sum over $\omega_{m}$
is performed. In addition to the diagrams shown, each solid line can be either
$G$ or $F$ functions.

It is useful to express the results in terms of the conductance $1/R_{N}$ of
the tunnel junction, which, in the diffusive case, is given by

$\frac{1}{R_{N}}=4\pi\frac{e^{2}}{\hbar}|t_{0}|^{2}N_{L}(0)N_{R}(0)A,$ (33)

and consider the ratio of the MAR current to the normal state current at
$eV=2\Delta$, $I_{N}=\frac{2\Delta}{eR_{N}}$. We also set
$N_{R}(0)=\frac{m}{2\pi}=\frac{k_{F}^{2}}{4\pi E_{F}}$ where $k_{F}$ and
$E_{F}$ are the Fermi momentum and Fermi energy of the $R$ SC. We find

$\frac{I^{\textrm{MAR}}_{\textrm{planar}}}{I_{N}}\propto\frac{g}{Ak_{F}^{2}}\frac{E_{F}}{\Delta}\frac{eV}{\sqrt{(eV)^{2}-\Delta^{2}}}.$
(34)

We can interpret $Ak_{F}$ as the number of tunneling channels and
$\mathcal{T}_{\textrm{eff}}=\frac{g}{Ak_{F}^{2}}$ as the tunneling probability
per channel. $\mathcal{T}_{\textrm{eff}}$ describes the intrinsic transparency
of a planar junction.

We can repeat the calculation for the smooth planar junction where momentum
parallel to the interface is conserved. A representative diagram is shown in
Fig. 4(b). Now there is only a single momentum sum and the results when
expressed in terms of the conductance is essentially the same as Eq. (34).

###  B.2 MAR: STM junction

We have also reproduced the result for the STM case. A representative diagram
is shown in Fig. 4(c). Now there are two momentum sums over ${\boldsymbol{p}}$
and ${\boldsymbol{p}^{\prime}}$, resulting in two energy integrals. Taking the
imaginary part results in a delta function $\delta(E+E^{\prime}=2eV)$. This
expresses the fact that the energy gained by tunneling a pair is used to
excite a pair of quasi-particles with momenta ${\boldsymbol{p}}$ and
${\boldsymbol{p}^{\prime}}$ and energy $E$ and $E^{\prime}$. Summing over all
diagrams and taking $\Delta_{L}=\Delta_{R}=\Delta$, we find for $eV>\Delta$

$I_{\textrm{STM}}^{\textrm{MAR}}=2e\pi^{3}|\tilde{t}|^{4}N^{2}_{L}(0)N^{2}_{R}(0)\Delta^{2}\int_{\Delta}^{2eV-\Delta}\
dE\
\frac{n_{F}(-E)-n_{F}(E)}{\sqrt{E^{2}-\Delta^{2}}\sqrt{(2eV-E)^{2}-\Delta^{2}}},$
(35)

leading to a step function at $eV=\Delta$. This agrees with, and extends Ref.
[16] to finite temperature, demonstrating that the finite temperature
correction is exponentially small in $\Delta/T$. Its ratio to the
corresponding normal current $I_{N}$ is given by

$\frac{I_{\textrm{STM}}^{\textrm{MAR}}}{I_{N}}=\frac{\pi}{16}g\
\theta(eV-\Delta)$ (36)

Note that unlike the planar case, the factor $N_{R}(0)$ in the denominator,
which led to the Fermi energy $E_{F}$ appearing in the numerator of Eq. (34),
does not arise.

We emphasize that there is no thermal smearing of the MAR peak until a
temperature comparable to the energy gap $\Delta$ is reached. On the other
hand, thermal noise gives rise to voltage fluctuations as discussed in the
main text. The MAR structure can also be viewed as linear response to the
order parameter in the $R$ SC except that quasi-particles are excited instead
of a collective mode. Therefore the voltage fluctuation will broaden the MAR
structure in the same way as the collective mode is broadened. Thus the
minimum broadening is given by $\Gamma_{0}$. In practice, in planar junctions
there are other sources of broadening such as local inhomogeneity. The latter
is particularly important for many strongly correlated SCs such as cuprates
and iron based SCs.