# Thermoelectric response across the semiconductor-semimetal transition
in black phosphorus

Yuna Nakajima  Department of Physics, Gakushuin University, Tokyo 171-8588,
Japan.  Yuichi Akahama  Graduate School of Material Science, University of
Hyogo, Kamigori 678-1297, Japan.  Yo Machida<EMAIL_ADDRESS>Department of Physics, Gakushuin University, Tokyo 171-8588, Japan.

###### Abstract

In spite of intensive studies on thermoelectricity in metals, little is known
about thermoelectric response in semiconductors at low temperature. An even
more fascinating and unanswered question is what happens to the Seebeck
coefficient when the semiconductor turns to a metal. By precisely tuning the
ground state of black phosphorus with pressure from the semiconducting to
semimetallic state, we track a systematic evolution of the Seebeck
coefficient. Thanks to a manifest correlation between the Seebeck coefficient
and resistivity, the Seebeck response in each conduction regime, i.e.,
intrinsic, saturation, extrinsic, and variable range hopping (VRH) regimes, is
identified. In the former two regimes, the Seebeck coefficient behaves in
accordance with the present theories, whereas in the later two regimes
available theories do not give a satisfactory account for its response.
However, by eliminating the extrinsic sample dependence in the resistivity
$\rho$ and Seebeck coefficient $S$, the Peltier conductivity $\alpha=S/\rho$
allows to unveil the intrinsic thermoelectric response, revealing vanishing
fate for $\alpha$ in the VRH regime. The emerged ionized impurity scattering
on entry to the semimetallic state is easily surpassed by electron-electron
scattering due to squeezing of screening length accompanied by an increase of
carrier density with pressure. Each carrier scattering participates an
enhancement of the phonon drag contribution to the Seebeck effect, but creates
the phonon drag peak with opposite sign at distinct temperature. In the low
temperature limit, a small number of carriers enhances a prefactor of
$T$-linear Seebeck coefficient as large as what is observed in prototypical
semimetals. A crucial but largely ignored role of carrier scattering in
determining the magnitude and sign of the Seebeck coefficient is indicated by
the observation that a sign reversal of the $T$-linear prefactor is
concomitant with a change in dominant scattering mechanism for carriers.

## I. INTRODUCTION

When a solid is subject to temperature gradient, thermal diffusion of charge
carriers develops electric field along the trajectory of carriers. This is a
phenomenon know as the Seebeck effect. The magnitude of this effect is
quantified by Seebeck coefficient $S$, which is defined as a ratio of
temperature difference $\Delta T$ to thermoelectric voltage $V_{\rm th}$,
$S=-V_{\rm th}/\Delta T$. Let us begin by showing that depending on what
statistics electrons obey the Seebeck coefficient behaves differently between
metals and semiconductors. In the Boltzmann picture with relaxation time
approximation, the Seebeck coefficient is expressed by

$S=-\cfrac{1}{eT}\left\\{\dfrac{\bigintss_{0}^{\infty}\dfrac{\partial
f_{0}}{\partial\epsilon}D(\epsilon)\epsilon(\epsilon-\mu)\tau(\epsilon)d\epsilon}{\bigintss_{0}^{\infty}\cfrac{\partial
f_{0}}{\partial\epsilon}D(\epsilon)\epsilon\tau(\epsilon)d\epsilon}\right\\},$
(1)

where $f_{0}$ is quilibrium distribution function and the density of state is
$D(\epsilon)=(2m)^{3/2}\epsilon^{1/2}/2\pi^{2}\hbar^{3}$. The scattering time
is assumed to be $\tau(\epsilon)=\tau_{0}\epsilon^{r}$. By using the Fermi
integral $F_{n}(\eta)=\int^{\infty}_{0}f_{0}(\zeta,\eta)\zeta^{n}d\zeta$, Eq.
(1) is rewritten as ioffe

$S=\cfrac{k_{\rm
B}}{e}\left\\{\eta-\cfrac{\left(r+\cfrac{5}{2}\right)F_{r+3/2}(\eta)}{\left(r+\cfrac{3}{2}\right)F_{r+1/2}(\eta)}\right\\},$
(2)

where $\zeta=\epsilon/k_{\rm B}T$ and $\eta=\mu/k_{\rm B}T$ ($\mu$ being the
chemical potential).

For metals the Fermi-Drac distribution function applies to $f_{0}$. Employing
the Sommerfeld expansion for large positive values of $\mu/k_{\rm B}T$, the
Seebeck coefficient for electrons in metals is given by

$S=-\frac{\pi^{2}}{3}\frac{k_{\rm B}}{e}\frac{k_{\rm B}T}{\epsilon_{\rm
F}}\biggl{(}r+\frac{3}{2}\biggr{)}.$ (3)

Here, $\epsilon_{\rm F}$ is the Fermi energy and we assume $\mu=\epsilon_{\rm
F}$. From Eq. (3) one finds that the magnitude of Seebeck coefficient is set
by the Fermi energy $\epsilon_{\rm F}$ and the exponent of energy dependence
of scattering time $r$, which takes different values depending on the
scattering mechanism: $r=-1/2$ for phonon scattering, $r=0$ for neutral
impurity scattering, and $r=3/2$ for ionized impurity scattering ioffe ;
seeger . $S$ linearly decreases with temperature and vanishes at zero
temperature. Validity of this expression is documented by a universal relation
between $S/T$ and the Sommerfled value $\gamma$ as well as $S/T$ and the
prefactor of $T^{2}$ resistivity for a wide variety of metals behnia ; okazaki
.

For insulators the Maxwell-Boltzmann distribution function applies to $f_{0}$,
then the Seebeck coefficient is obtained to be

$S_{\rm e,h}=\mp\frac{k_{\rm B}}{e}\biggl{\\{}\frac{\Delta}{2k_{\rm
B}T}+\biggl{(}r+\frac{5}{2}\biggr{)}\biggr{\\}},$ (4)

where $\Delta$ is a gap between the valence and conduction band. The suffixes
e and h denote electron and hole, respectively. One can see that the magnitude
of Seebeck coefficient is set by $\Delta$ and $r$, and it diverges on cooling
inversely proportional to temperature. This expression well describes the
Seebeck coefficient of semiconductors in the intrinsic regime as reported in
Ref. johnson .

Here several questions arise. Q1 What is the fate of Seebeck coefficient in
semiconductors even below the intrinsic regime? Surprisingly, after 200 years
of discovery of Seebeck effect there is no consensus about what is going on in
this fundamental transport quantity when a semiconductor is cooled down to low
temperatures. The limited number of studies have reported the contradicting
observations; it diverges in some cases liuSi ; machidaPF  and vanishes as
$T\rightarrow 0$ in other cases buhannic ; demishev ; ishida ; berthebaud . Q2
What does happen in the Seebeck coefficient at the metal-insulator (M-I)
transition? In other words, how do the contrasting Seebeck responses between
metals and insulators converge at the transition? The thermoelectricity near
the M-I transition has been barely explored lakner ; ishida . The pioneer work
has done by H. v. Lohneysen et. al. lakner  on the doped Si, who found that
the Kondo effect emerges close to the critical doping. Because of this
extrinsic contribution induced by impurities, the systematic evolution of the
intrinsic diffusive contribution to Seebeck coefficient near the M-I
transition is hardly observable. Q3 How does the electron scattering affect
the Seebeck coefficient? Since the magnitude of Seebeck coefficient is
primaliry detrmined either by $\epsilon_{\rm F}$ or $\Delta$, in most cases
much attension has not given to the scattering parameter $r$ in Eqs. (3) and
(4).

Black phosphorus (BP), an elemental semiconductor ling , provides a rare
opportunity to address the above issues. Prior transport studies have revealed
the resistivity behavior like the doped (p-type) semiconductor despite that BP
is undoped akahama . The atomic vacancy is attributed to the hole doping
riffle . The low density of carriers ($\sim$1015 cm-3) leads to a heat
transport dominated by phonons, which yield a variaty of fascinating phenomena
including phonon hydrodynamics machidabp  and sizable phonon Hall effect li .
Application of hydrostatic pressure exceeding 1 GPa is enough to close the
band gap of $\sim$0.3 eV and the semimetallic state is stabilized akiba1 ;
xiang ; akiba2 . It should be emphasize that the semiconductor-semimetal
transition is isostructural, thus it is the Lifshitz transition xiang . This
is in contrast with the pressure-induced M-I transition in Si which is
accompanied by a stractural change minomura . The ability to use the pressure
to tune the ground state is advantage of BP over such as Si lakner  because an
introduction of impurities can be avoided. The Dirac nature of mobile carries
created in the semimetallic state makes this elemental semimetal more
attractive xiang ; gong ; fujii .

Our focus is to unveil the thermoelectric property of semiconductor and its
evolution on entering the semimetallic state across the semiconductor-
semimetal (S-S) transition utilizing BP as a representative semiconductor. Our
precise transport measurements under pressure up to 1.7 GPa and in the
temperature range between 2 K and 300 K allow to track the systematic
evolution of resistivity and Seebeck coefficient as a function of temperature
and pressure. We show that thanks to the clear correspondence to the
resistivity which can be separated into four conduction regimes, i.e.,
intrinsic, saturation, extrinsic, and variable range hopping (VRH) regimes,
the characteristic Seebeck response in each regime is resolved. In particular,
the vanishing fate for Seebeck coefficient is uncovered in the VRH regime.
While the Seebeck behavior in the intrinsic and saturation regimes is
satisfactory accounted by the present theories, the responses in the extrinsic
and VRH regimes are not in agreement with any available theories. However, we
show that being free from extrinsic sample dependence, the Peltier
conductivity arrows to access the intrinsic thermoelectric response at low
temperature.

Concomitant with an entry to the semimetallic state, ionized impurity
scattering becomes dominant scattering process affecting electrons, but it is
overwhelmed by electron-electron scattering due to shrinkage of screening
length of impurity potential upon increasing of density of conduction
electrons with pressure. We show that both scattering could be sources to
enhance the phonon drag contribution to the Seebeck coefficient. A direct link
between the change in hierarchy of dominant scattering and a sign reversal of
the $T$-linear prefactor of Seebeck coefficient indicates that a crucial role
is played by carrier scattering in determining the magnitude and sign of the
Seebeck coefficient, although its importance has been largely ignored.

## II. METHODS

Single crystals of black phosphorus were synthesized under high pressure endo
. Seebeck coefficient under pressure was measured by the differential DC
method using chromel-constantan thermocouples as probes of thermoelectric
voltage. Schematic image of Seebeck coefficient measurement setup is shown in
Fig. 1. The junctions of the thermocouples were made by spot welding, and
glued directly to the sample by using silver paste. A temperature gradient
across the sample was generated by a chip resistor mounted on the thin glass-
epoxy flame. The resistor was connected to the hot end of the sample by a gold
wire which was also used as a current electrode for the resistivity
measurements. The cold end of the sample was thermally anchored at a large
copper plate with silver paste to establish steady heat flow from the sample.
The absolute Seebeck coefficient $S$ of the sample is evaluated from the
following relation,

$S=\frac{S_{\rm ch}-(V_{\rm ch}/V_{\rm co})S_{\rm co}}{1-(V_{\rm ch}/V_{\rm
co})},$ (5)

where $V_{\rm ch}$ ($V_{\rm co}$) is voltage across the circuit consisting of
the sample and chromel (constantan), and $S_{\rm ch}$ ($S_{\rm co}$) is an
absolute Seebeck coefficient of chromel (constantan). $S_{\rm co}$ was
evaluated from the differential Seebeck coefficient of the type E thermocouple
with independently measured $S_{\rm ch}$ slack ; zrudsky ; chaussy . Effect of
pressure on the type E thermocouple was found to be negligible (see APPENDIX).

External pressure was applied by using a NiCrAl-BeCu hybrid piston cylinder.
Daphne oil 7373 was used as a pressure transmitting medium. The applied
pressure was estimated by monitoring the superconducting transition
temperature of Pb. The resistivity was also measured under the same pressure
environment utilizing the voltage electrodes for the thermoelectric
measurements.

Figure 1: (a) Schematic of Seebeck coefficient measurement setup. (b)A photo
of the setup.

## II. RESULTS

### Electrical resistivity

Figure 2 shows temperature dependence of electrical resistivity $\rho(T)$
measured under various pressures. $\rho(T)$ at ambient pressure can be
separated into four conduction regimes, i.e., intrinsic, saturation,
extrinsic, and variable range hopping (VRH) regimes. In the intrinsic regime
between 250 K and 300 K, $\rho$ shows an activation behavior. By fitting the
data to the Arrhenius equation $\rho=\exp(\Delta_{\rm vc}/2k_{\rm B}T)$ (Fig.
3(a)), we obtained an activation energy of $\Delta_{\rm vc}\sim$ 0.3 eV which
corresponds to an energy gap between valence and conduction bands. The
estimated $\Delta_{\rm vc}$ is in agreement with what was reported in the
previous study keyes . $\rho(T)$ decreases with temperature in the saturation
regime between 50 K and 250 K, reflecting the facts that the carrier density
is insensitive to the temperature and that a scattering rate of electrons by
phonons is reduced on cooling. In the extrinsic regime between 15 K and 50 K,
$\rho(T)$ again exhibits the activation behavior (Fig. 3(b)) with an energy
gap of $\Delta_{\rm va}\sim$16 meV between the valence band and acceptor
level. Below 15 K, an evolution of $\rho$ becomes weak due to an entrance in
the VRH regime where the conduction is governed by a hopping of electrons
between impurity sites. In this regime, $\rho$ varies following the expression
$\rho\propto\exp[(T/T_{\rm 0})^{-1/(d+1)}]$ ($d$ is a space dimension). As
seen in Fig. 3(c)-(e), our limited data does not allow to determine the
hopping dimensionality.

Figure 2: Temperature dependence of resistivity $\rho$ under various pressure
in a logarithmic scale. Figure 3: Arrhenius plots in the intrinsic and
extrinsic regimes are shown in the panel (a) and (b), respectively.
Resistivity in the VRH regime is plotted against (c) $T^{-1/2}$, (d)
$T^{-1/3}$, and (e) $T^{-1/4}$.

By increasing the pressure, a rise of $\rho$ in the intrinsic and extrinsic
regimes becomes weak due to the gap closing. Indeed, the gap estimated from
the Arrhenius plot (Figs. 3(a) and (b)) decrease with pressure as shown in
Fig. 5(a). A linear extrapolation suggests that the gaps vanish around $P_{\rm
c}\sim$ 1.1 GPa. This conjecture is supported by the observation that the
semiconducting behavior in $\rho(T)$ becomes unclear at $P$ =1.07 GPa (Fig.
2(a)).

Figure 4: (a) Temperature dependence of resistivity $\rho$ under the pressure
of 1.7 GPa. At low temperature, $\rho$ follows $T^{2}$ as shown in the upper
inset. Lower inset: exponent $\gamma$ of temperature dependence of resistivity
($\rho=\rho_{0}+A_{\gamma}T^{\gamma}$) which is quantified by taking the
logarithmic derivative after subtracting residual resistivity:
$\gamma=\\{d\ln(\rho-\rho_{0})/d\ln T\\}$ is shown as a function of
temperature.

Above 1.16 GPa, a metallic behavior eventually shows up in $\rho(T)$ at high
temperature, but below 100 K $\rho(T)$ subsequently rises. Such an upturn is
observed in gapless semiconductors and semimetals blatt ; carvalho ; goswami ;
jaouiInAs , and is attributed to the ionized impurity scattering. For such a
scattering process, screening of the impurity potential by conduction
electrons is irrelevant. A comparable magnitude of the Thomas-Fermi screening
length $r_{\rm TF}\propto\sqrt{a_{\rm B}/n^{1/3}}\sim$ 15 nm and the mean-
free-path $l$ = 24 nm calculated from resistivity
$\rho=3\pi^{2}\hbar/e^{2}k_{\rm F}^{2}l$ gunnarsson  implies the ionized
impurity scattering is marginal in BP.

In fact, by further increasing the pressure, the upturn entirely suppressed
and $\rho$ keeps decreasing all the way down to the lowest temperature at the
pressure exceeding $P^{*}\sim$ 1.35 GPa. This is caused by the squeezing of
$r_{\rm TF}$ upon increasing the number of conduction electrons, and thus the
scattering from ionized impurity becomes ineffective. In Figure 4, a plot of
$\rho$ vs $T$ at $P$ = 1.7 GPa is shown as representative temperature
dependence of $\rho(T)$ in the semimetallic state. As seen in the upper inset
of Fig. 4, at low temperature $\rho(T)$ exhibits $T^{2}$ dependence arising
from electron-electron scattering. A linear fit to the data gives a $T^{2}$
prefactor of 15.2 n$\Omega$cm/K2, which is comparable to the one find in Bi
hartman . On warming, electron-phonon scattering becomes dominant and as a
result, an exponent of $T$ dependence of $\rho$ increases from two to three as
demonstrated in the lower inset of Fig. 4 where the logarithmic derivative
after subtracting residual resistivity: dln$(\rho-\rho_{0})$/dln$T$ is plotted
as a function of temperature. By further increasing temperature, the exponent
drops to approach around one. Similar variation of dln$(\rho-\rho_{0})$/dln$T$
has been observed in Sb jaoui

Figure 5: (a) The gaps between the top of valence band and the bottom of
conduction band $\Delta_{\rm vc}$ (left axis) determined by resistivity and
Peltier coefficient decrease linearly with pressure, pointing that the gap
vanishes around 1.1 GPa as denoted by a dashed line. The gap between the top
of valence band and the acceptor level $\Delta_{\rm va}$ (right axis) follows
the same trend. Contour plot of temperature derivative of resistivity and the
Seebeck coefficient in the temperature-pressure plane are shown in the panels
(b) and (c), respectively.

One can see how $\rho$ evolves under pressure from a contour plot of
temperature derivative of $\rho$, d$\rho$/d$T$, in the temperature-pressure
plane, which is shown in Fig. 5(b). In the plot, warm/cold color represents a
region where $\rho$ decrease/increase on cooling. It is clearly seen that the
intrinsic, saturation, and extrinsic/VRH regions colored by purple, red, and
purple, respectively, are squeezed upon increasing the pressure and the former
two regions are terminated at the S-S transition ($P_{\rm c}\sim$ 1.1 GPa)
denoted by a dashed line. While the extrinsic/VRH region also disappears at
the same pressure, the emergence of ionized impurity scattering above $P_{\rm
c}$ extends the cold-color region (d$\rho$/d$T<0$) to the semimetallic state.
As can be seen by a change of color from clod to hot across $P^{*}\sim$ 1.35
GPa (a dashed-and-dotted line), the dominant scattering mechanism switch from
the ionized impurity scattering to electron-electron scattering. As we will
show below, this change in scattering mechanism inside the semimetallic state
provides a great impact on the Seebeck effect.

### Seebeck coefficient

#### semiconducting state

Figure 6 depicts the temperature dependence of Seebeck coefficient $S(T)$
under various pressures. A sign of $S$ is positive in the semiconducting
state, consistent with that black phosphorus is the p-type semiconductor. At
ambient pressure, $S(T)$ increases with decreasing temperature and attains a
maximum value, 800 $\mu$V/K around 200 K. Up to the critical pressure ($P_{\rm
c}\sim$ 1.1 GPa), the maximal $S$ is diminished by a factor of five. The
magnitude of $S$ keeps decreasing even above $P_{\rm c}$ as displayed in Fig.
6(b), and $S$ takes place a sign change at $P^{*}\sim$ 1.35 GPa (the inset of
Fig. 6(b)) at low temperature. The contour plot of $S$ displayed in Fig. 5(c)
clearly shows that the sign change in $S$ is concomitant with the
disappearance of ionized impurity scattering.

Figure 6: (a) Temperature dependence of Seebeck coefficient $S$ under various
pressure. Corresponding zooms above 1.07 GPa and 1.30 GPa are shown in the
panel (b) and its inset, respectively. Figure 7: Temperature dependence of
Seebeck coefficient (left axis) together with resistivity (right axis) under
each pressure. Panels (a)-(d) and (e)-(o) correspond to the data in the
semiconducting and semimetallic states, respectively. In the panels (a)-(d),
four conduction regimes in the semiconducting state are highlighted by
different colors.

One of the main outcome of this study is a manifest correlation between
resistivity and Seebeck coefficient. Figure 7 shows the temperature dependence
of Seebeck coefficient (left axis) together with resistivity (right axis) at
each pressure. In the semiconducting state, Seebeck coefficient shows
characteristic behaviors in four conduction regimes: $S$ sharply rises in the
intrinsic regime, then becomes nearly temperature independent in the
saturation regime, and subsequently drops rapidly in the extrinsic regime.
Finally, $S$ again decreases slowly in the VRH regime. An application of the
pressure makes the intrinsic and extrinsic regime wider and the saturation and
VRH regime narrower, creating a prominent peak of $S$ in the saturation
regime. On entering the semimetallic state at $P>P_{\rm c}$, the peak in $S$
becomes less pronounced.

From Eq. (4), the diffusion thermopower is expected to evolves with
$\Delta_{\rm vc}/T$ in the intrinsic regime. Indeed, this is seen in our data
as shown in Fig. 8(a) where $S$ is plotted against $1/T$. An extraction of
$\Delta_{\rm vc}$ from the slope is not straightforward due to the presence of
thermally excited hole and electron carriers. The Seebeck coefficient of two
carrier system is expressed as

$S=\frac{\sigma_{\rm e}S_{\rm e}+\sigma_{\rm h}S_{\rm h}}{\sigma_{\rm
e}+\sigma_{\rm h}},$ (6)

where $\sigma_{\rm e}$ ($\sigma_{\rm h}$) and $S_{\rm e}$ ($S_{\rm h}$) are
conductivity and Seebeck coefficient of electrons (holes), respectively. By
substituting Eq. (4) to Eq. (6), we get

$S=\cfrac{k_{\rm B}}{e}\cfrac{1-(\sigma_{\rm e}/\sigma_{\rm
h})}{1+(\sigma_{\rm e}/\sigma_{\rm h})}\biggl{\\{}\frac{\Delta}{2k_{\rm
B}T}+\biggl{(}r+\frac{5}{2}\biggr{)}\biggr{\\}}.$ (7)

Suppose that $\sigma_{\rm e}/\sigma_{\rm h}\sim 1$, the Seebeck coefficient
will be very small when the chemical potential is located close to middle of
the energy gap. As the chemical potential moves toward one of the bands (the
valence band for the case of BP) on cooling, $\sigma_{\rm e}/\sigma_{\rm h}$
becomes small and the Seebeck coefficient is dominated by the contribution
from holes. This is what occurs in our sample when it is cooled from room
temperature. By assuming that the Seebeck coefficient is dominated only by
holes, $\Delta_{\rm vc}$ is obtained by fitting the data to Eq. (4). As seen
in Fig. 5(a), the extracted $\Delta_{\rm vc}$ is about twice as large as the
one from resistivity. This discrepancy may be due to the additional
contribution by electrons and/or the presence of two kinds of holes with
distinct mobilities as suggested by the Hall conductivity measurements akiba2
. This complexity also makes difficult to find implication of the scattering
parameter $r$ that evolves from -13 to -4 on approaching $P_{\rm c}$.

Figure 8: (a) Seebeck coefficient exhibits 1/$T$ dependence in the intrinsic
regime as expected from the theory. Peltier coefficient also follows the
expected $T$ dependence in the saturation regime, which is shaded by orange in
the panels (b)-(f). The panel (g) demonstrates that the Seebeck coefficient in
the saturation regime shaded by orange is well described by the Pisarenko
formula (Eq. (9)). The calculated results are shown by a dashed line. Figure
9: Temperature dependence of resistivity (a), Seebeck coefficient (b), and
Peltier conductivity (c) for four different BP samples form the same batch
together with the literature data hu . While $\rho$ and $S$ depend largely on
the samples, $\alpha$ falls on an almost the same curve, which arrows us to
access an intrinsic thermoelectric behavior of semiconductors.

An alternative estimation of $\Delta_{\rm vc}$ is provided by the Peltier
coefficient $\Pi=ST$ (extracted using the Kelvin relation) which quantifies
average thermal energy transported by carriers and it corresponds to the
activation energy $\Delta_{\rm vc}$. Figures 8(b)-(f) depict temperature
dependence of $\Pi$ at $P<P_{\rm c}$ (the saturation regime is shaded by
orange). $\Pi$ attains a maximum value at the lower temperature end of the
intrinsic regime. $\Delta_{\rm vc}$ extracted from the maximal $\Pi_{\rm max}$
as $e\Pi_{\rm max}=(\Delta_{\rm vc}/2T)T=\Delta_{\rm vc}/2$ is in good
agreement with the one from resistivity, as seen in Fig. 5(a). We note that
this way of the gap estimation is compatible with the empirical method using
the relation of $E_{\rm g}=2e|S_{\rm max}|T_{\rm max}$ gibbs , where $E_{\rm
g}$, $S_{\rm max}$, and $T_{\rm max}$ are the energy gap, the maximum Seebeck
coefficient, and the temperature at which $S_{\rm max}$ occurs, respectively.
This yields $E_{\rm g}\sim$ 0.32 eV at ambient pressure, in good agreement
with $\Delta_{\rm vc}\sim$ 0.3-0.4 eV extracted from resistivity and $\Pi$.

According to the Wilson-Sommerfeld treatment geballe , combining the equation
of carrier density $n=A\exp(-\Delta_{\rm vc}/2k_{\rm B}T)$ ($A$ being
constant) with $\Pi\propto\Delta_{\rm vc}$ gives the qualitative description
of Peltier coefficient as,

$\Pi\propto 2k_{\rm B}T(\ln A-\ln n).$ (8)

Given that $n$ is nearly constant in the saturation regime, $\Pi$ is expected
to vary proportional to temperature. We show in figure 8(b)-(f) that $\Pi$ is
indeed in $T$-linear in the saturation regime shaded by orange. A quantitative
account of the magnitude of Seebeck coefficient in the saturation regime is
given by the Pisarenko formula johnson ; collignon  which is valid when the
number of carriers is set by extrinsic dopants,

$S=\cfrac{k_{\rm
B}}{e}\left[2+r+\ln\left\\{\cfrac{2}{n}\left(\cfrac{m^{*}k_{\rm
B}T}{2\pi\hbar^{2}}\right)^{3/2}\right\\}\right],$ (9)

where $m^{*}$ is the effective carrier mass. Since the dominance of phonon
scattering between 50 K and 200 K is indicated by the Hall mobility
experiments akahama , by setting $r$ = -1/2 and using $n$ extracted from the
published data akahama , we obtained a fair agreement of the computed $S$ from
Eq. (9) to the experiments with $m^{*}$ = 0.45 $m_{0}$ in the saturation
regime (see the dashed line in Fig. 8(g)). The deduced effective mass of
$m^{*}$ = 0.45 $m_{0}$ reasonably agrees with the one extracted from the
cyclotron resonance experiments narita .

Contrary to the activation behavior in resistivity, the Seebeck coefficient
steeply drops upon entry to the extrinsic regime. The similar behavior can be
found in other semiconductors ioffe ; yoshikawa . While a satisfactory account
of this contradiction is still missing, we infer that the presence of electron
carriers whose mobility is comparable with that of hole carriers would cancel
out the $1/T$ behavior because the Seebeck coefficient of two carrier system
is expressed by Eq. (6). The presence of electron carriers is backed by the
fact that oxygen dopants induce n-doped BP pei .

In the VRH regime, $S(T)$ varies linearly with temperature as shown in the
inset of Fig. 9(b). (See the data of sample #14 from which the preceding data
is obtained.) Our observation is not agreement with what was seen in the BP
nanoribbons in which $S(T)$ follows $T^{1/2}$ in the temperature range
(between 50 K and 200 K) where resistivity obeys the VRH transport liuBP , and
is in contrast with what was found in the doped Si lakner  and the organic
insulator machidaPF  where $S(T)$ diverges on cooling. None of available
theories predicting the vanishing VRH thermopower $S\sim T^{x}$ ($x$ = 1/2
zvyagin , 3/4 yamamoto  for three dimensions) are compatible with our result.
We show here that the Peltier conductivity $\alpha=S/\rho$ rather than $S$ is
useful to probe the intrinsic thermoelectric feature of semiconductors. In
Figs. 9(a)-(c), $\rho(T)$, $S(T)$, and $\alpha(T)$ of the several BP samples
from the same batch together with the literature data hu  are compared,
respectively. Clearly, $\rho(T)$ and $S(T)$ are highly sample dependent
especially below the extrinsic regime because they are affected by the
impurities. By suppressing the extrinsic sample dependence in $\rho(T)$ and
$S(T)$, however, all $\alpha(T)$ data fall into the same curve, allowing to
address the intrinsic thermoelectric responses: $\alpha(T)$ varies with
$T^{1.5}$ and $T^{5}$ in the VRH and extrinsic regimes, respectively.

Now, let us discuss the phonon drag contribution to the Seebeck coefficient in
the semiconducting state. In general, the Seebeck coefficient is composed of
the diffusion term $S_{\rm dif}$ and the phonon drug term $S_{\rm ph}$,

$S=S_{\rm dif}+S_{\rm ph}.$ (10)

The facts that the Seebeck coefficient obeys the activation behavior in the
intrinsic regime and that the Pisarenko formula is valid in the saturation
regime (both are features of the diffusive Seebeck response) indicate that a
dominant role is played by the carrier diffusive in both regimes. This means
that in these regimes every response in the Seebeck coefficient is
predominantly controlled by $\Delta_{\rm vc}$. We do not exclude the phonon
drag contribution below the extrinsic regime, but the lacking of quantitative
explanation of the diffusive contribution prevents the further analysis.
Notably, the gigantic phonon drag effect which arrows the Seebeck coefficient
to attain as large as 6 mV/K below 50 K (the extrinsic region) is reported in
germanium geballe .

#### semimetallic state

The Seebeck effect in the semimetallic state is characterized by the presence
of positive and negative peaks centered around 10 K and 30 K, respectively
(red and blue triangles in Figs. 7(f)-(o)). The former persists up to $P^{*}$
where the dominant scatterers for carriers change from the ionized impurities
to other electrons. The latter appears as a dip above $P_{\rm c}$ and becomes
a prominent negative peak above $P^{*}$. To search for the origin of peaks,
comparison to prototypical semimetals is instructive. In semimetals, phonon
drag effect provides an enhancement of Seebeck coefficient around a
temperature where phonon scatter electrons most efficiently issi . Such a
condition is satisfied when typical phonon wave vector $q$ becomes comparable
to twice of electron wave vector, $q\sim 2k_{\rm F}$. The effective Debye
temperature $\Theta_{D}^{*}=2k_{\rm F}\bar{v}_{\rm s}\hbar/k_{\rm B}$
($\bar{v}_{\rm s}$ being the average of phonon velocity), rather than the true
Debye temperature, can be used to describe such a characteristic temperature.
As shown in Table 1, the estimated $\Theta_{D}^{*}$ for Bi, Sb, and As is of
the same order of magnitude as $T_{\rm max}$ where the peaks of Seebeck
coefficient appear uher ; jaoui2 ; uher2 . Here, $k_{\rm F}$ for each system
is evaluated from the carrier density issi . Under the assumption that the
calculated $\bar{v}_{\rm s}$ for the orthorhombic structure under ambient
pressure kaneta  is retained even under the pressure and with the knowledge of
carrier density at 1.7 GPa akiba2 , $\Theta_{D}^{*}$ for BP is estimated to be
12 K, which coincides with $T_{\rm max}\sim$ 10-30 K within experimental
margin. This provides reasonable ground to argue that the phonon drag
participates the formation of peaks.

Table 1: Comparison of prominent semimetals with BP. | Bi | Sb | As | BP
---|---|---|---|---
$n$ (cm-3) | 2.7$\times$1017 | 3.7$\times$1019 | 2.0$\times$1020 | 2.9$\times$1017
$k_{\rm F}$ (nm-1) | 0.2 | 1.0 | 1.8 | 0.2
$\bar{v}_{\rm s}$ (km/s) | 1.1 | 2.9 | 3.0 | 3.8
$\Theta_{\rm D}^{*}$ (K) | 3.3 | 44 | 82 | 12
$T_{\rm{max}}$ (K) | 3 | 10 | 30 | 10-30
Ref. | issi ; uher  | issi ; epstein ; jaoui2  | issi ; pace ; uher2  | akiba2 ; kaneta
Figure 10: Seebeck coefficient divided by temperature $S/T$ as a function of
temperature in the semiconducting state (a) and semimetallic state (b),
respectively. The inset shows a zoom of the data at $P\geq$ 1.30 GPa. While
$S/T$ is not constant at low temperature in the semiconducting state except at
ambient pressure where the VRH transport governs the carrier conduction, it
becomes constant in the semimetallic state as observed in ordinary metals.
Figure 11: A plot of a prefactor of $T^{2}$-resistivity $A$ vs. $(S/T)^{2}$
for various materials. The data is adapted from Ref. okazaki . Note that not
only prototypical semimetals of Bi and Sb, but also semimetallic BP (1.7 GPa)
obey the universal relation between $A$ and $(S/T)^{2}$. Figure 12: The
$T$-linear term of Seebeck coefficient $\alpha_{\rm T}$ as a function of
pressure. The computed $\alpha_{\rm T}$ does not reproduce the experimental
data when the scattering parameters $r_{\rm e,h}$ for electrons and holes are
set as the same value. An quantitative account of $\alpha_{\rm T}$ is possible
when $r_{\rm e,h}$ varies with pressure as denoted by the solid red line.
Pressure dependence of the Fermi energy from Ref. xiang  and mobility from
Ref. akiba2  for electrons and holes are shown in the panels (b) and (c),
respectively.

A qualitative description of phonon drag effect is provided by Herring herring
as,

$S_{\rm ph}=\pm\Lambda\frac{k_{\rm B}}{e}\frac{m^{*}\bar{v}_{\rm
s}^{2}}{k_{\rm B}T}\frac{\tau_{\rm ph}}{\tau_{\rm e}}.$ (11)

Here, $\Lambda$ is the momentum exchange rate between phonons and electrons,
and $\tau_{\rm ph}$ and $\tau_{\rm e}$ are phonon and electron scattering
times. According to this equation, a large phonon drag contribution to the
Seebeck response is possible when $\tau_{\rm ph}\gg\tau_{\rm e}$ herring ;
jaouiInAs . Coming back to the giant $S_{\rm ph}$ in germanium, it is shown
the large $\tau_{\rm ph}/\tau_{\rm e}$ ratio is a key to understand this
phenomenon geballe . For BP, the nearly ballistic heat conduction by phonons
below 30 K machidabp  enables the large $\tau_{\rm ph}$, while the ionized
impurity scattering and the electron-electron scattering are responsible for
the damping of $\tau_{\rm e}$. The opposite sign of peaks implies the sign of
carriers coupled with phonons is opposite. Given the Fermi surfaces
constituting of two electron pockets and one hole pocket akiba1 ; xiang ; gong
, combining the spacial distribution of Fermi surfaces and phonon wave vectors
of each branch would allow to pin down at which Fermi surface and at what
temperature the favorable condition for the phonon drag effect ($q\sim 2k_{\rm
F}$) is satisfied.

Upon cooling the sample down to low temperature, the phonon drag contribution
dies out and diffusive contribution becomes significant, recovering the
$T$-linear dependence in the Seebeck coefficient as expected from Eq. (3).
This is seen in figure 10(a)-(c), in which Seebeck coefficient divided by
temperature $S/T$ under various pressures are shown as a function of
temperature. While $S/T$ in the semiconducting state ($P<$ 1.1 GPa) shows a
striking temperature dependence with a concave curve and does not saturate
except for under ambient pressure (Fig. 10(a)), the one in the semimetallic
state exhibits a convex temperature dependence and eventually saturates to a
constant value at low temperature (Fig. 10(b)). As shown in the inset of Fig.
10 (b), $S/T$ becomes constant but negative above $P^{*}$. The saturation
behavior of $S/T$ suggests that the low-temperature Seebeck response is
governed by carrier diffusion. Let us prove this conjecture by examining a
universal relation between the prefactor of $T^{2}$ resistivity, $A$, and the
$T$-linear term of Seebeck coefficient okazaki . It is seen from Fig. 11 that
not only prominent semimetals of Bi and Sb, but also BP for which $A$ and
$S/T$ are evaluated under 1.7 GPa fall into the universal line, providing
evidence that the thermoelectric response is purely diffusive at low
temperature.

Having established the diffusive response in the semimetallic state, let us
scrutinize the response of the $T$-linear term of the Seebeck coefficient
$\alpha_{\rm T}$ against the pressure. As shown in Fig. 12(a), $\alpha_{\rm
T}$ falls above $P_{\rm c}$ and changes the sign across $P^{*}$. By combining
Eq. (3) and (6), one gets the expression of $\alpha_{\rm T}$ for semimetals
with equal number of electron and hole carrier density ($n_{\rm e}=n_{\rm
h}$),

$\alpha_{\rm T}=-\frac{\pi^{2}}{3}\frac{k_{\rm B}^{2}}{e(\mu_{\rm e}+\mu_{\rm
h})}\biggl{\\{}\frac{\mu_{\rm e}}{\epsilon_{\rm F}^{\rm e}}\biggl{(}r_{\rm
e}+\frac{3}{2}\biggr{)}-\frac{\mu_{\rm h}}{\epsilon_{\rm F}^{\rm
h}}\biggl{(}r_{\rm h}+\frac{3}{2}\biggr{)}\biggr{\\}},$ (12)

where $\mu_{\rm e}$ ($\mu_{\rm h}$), $\epsilon_{\rm F}^{\rm e}$
($\epsilon_{\rm F}^{\rm h}$), and $r_{\rm e}$ ($r_{\rm h}$) are mobility,
Fermi energy, and the exponent of energy dependence of scattering rate of
electrons (holes), respectively. To compute the pressure dependence of
$\alpha_{\rm T}$, one needs $\mu_{\rm e,h}$ and $\epsilon_{\rm F}^{\rm e,h}$
as a function of pressure which are taken from Ref. akiba2  and xiang , and
shown in Fig. 12(b) and (c), respectively. Since the data for $\epsilon_{\rm
F}^{\rm e,h}$ is interrupted around 1.3 GPa, $\epsilon_{\rm F}^{\rm e,h}$ is
assumed to vanish at $P_{\rm c}$ as shown by the dashed line in Fig. 12(b).
Due to the higher mobility and the lower Fermi energy of electrons, the
computed $\alpha_{\rm T}$ is always dominated by electrons ($\alpha_{\rm T}$
is always negative) and does not reproduce the experimental data as long as
$r_{\rm e,h}$ is set to be the same value and pressure-independent. The cases
of $r_{\rm e}=r_{\rm h}=-1/2$, $r_{\rm e}=r_{\rm h}=0$, and $r_{\rm e}=r_{\rm
h}=3/2$ are shown in Fig. 12(a). Nevertheless, the better agreement with the
experiment is obtained for $P>P^{*}$ with $r_{\rm e}=r_{\rm h}=-1/2$. This
corresponds to an energy-independent mean-free-path, which is a good
description for electron scattering even in the presence of electronic
correlations. An elaborate agreement including the sign change in $\alpha_{\rm
T}$ is achieved with the assumption that $r_{\rm e}$ and $r_{\rm h}$ evolves
differently with pressure. For the case where $r_{\rm h}$ and $r_{\rm e}$
start to changes from 3/2 and -2, respectively, and merge to be -1/2 at the
end (see the inset of Fig. 12(a)), not only the magnitude of $\alpha_{\rm T}$
but also the sign reversal at $P^{*}$ can be reproduced by the computations as
show by the solid red line in Fig. 12(a). Such an evolution of $r_{\rm h}$
with pressure is in good accordance with our observation in resistivity that
the dominant scattering process affecting carriers changes from ionized
impurity scattering ($r_{\rm h}=3/2$) to electron-electron scattering ($r_{\rm
h}=-1/2$) across $P^{*}$. While the justification of $r_{\rm e}$ = -2 near
$P_{\rm c}$ remains a challenge, an equal influence of ionized impurity
scattering for electrons and holes that was theoretically proposed in Ref.
blatt  would support our analysis. What we emphasize here is that in addition
to the thermodynamic quantity like $\epsilon_{\rm F}$, how and by what are
carriers scattered is a key ingredient that determines the magnitude as well
as the sign of Seebeck coefficient.

## IV. SUMMARY

In summary, our measurements on the Seebeck coefficient of black phosphorus
under pressure have provided several piece of information that promotes our
understanding of thermoelectricity in solids. We showed that the Seebeck
coefficient follows the activation behavior in the intrinsic regime and yields
the gaps which are reasonably in agreement with what was determined by other
experiments. The validity of the Pisarenko formula in the saturation regime is
documented with effective mass comparable with the one obtained by the
cyclotron resonant experiments. We resolved the vanishing fate for the Seebeck
coefficient with $S\sim T$ in the VRH regime whereas its $T$-linear slope is
highly sample dependent. The independence of the Peltier conductivity $\alpha$
on the different samples paves a way to access the intrinsic thermoelectric
properties even in the extrinsic semiconductors.

In the semimetallic state, we found that the increasing of carrier density
induces the change in dominant scattering mechanism from the ionized impurity
scattering to electron-electron scattering. Both mechanism participate the
enhancement of phonon drag effect at high temperature, but the opposite sign
of phonon drag peaks imply distinct type of carriers are dragged by phonons.
The intimate link between the change in dominant scatterers for carriers and
the sign reversal of low temperature Seebeck coefficient that is purely
dominated by carrier diffusion explicitly points to the crucial role of
carrier scattering in determining not only the magnitude but also the sign of
Seebeck coefficient.

##  I ACKNOWLEDGMENTS

We acknowledge discussions with Kamran Behnia. This work was supported by
Grants-in-Aid for Scientific Research (KAKENHI Grants Nos. JP16K05435,
JP17KK0088, and JP19H01840).

##  II APPENDIX: PRESSURE EFFECT ON TYPE E THERMOCOUPLE

Pressure effect on the type E thermocouple was investigated by using the setup
shown in Fig. 13. The junction of thermocouple was glued by silver paste
(DuPont 4922N) on one side of a Cernox thermometer (CX-1050-HT). A chip
resistor was attached on the other side of the thermometer, which serves the
heat current along the thermocouple. The other end of thermocouple was
thermally anchored to the body of pressure cell and connected to the copper
wires for the thermoelectric vlotage ($V_{\rm th}$) measurement. The
temperature gradient generated along the thermocouple was measured by using
two Cernox thermometers. One was at the thermocouple junction as noted above
and monitored the temperature ($T_{\rm in}$) at the junction (hot end of the
thermocouple) inside the pressure cell. The other one was placed on the body
of pressure cell and measured the temperature ($T_{\rm out}$) at the cold end
of thermocouple. The latter was always under ambient pressure. An external
pressure was applied by using a NiCrAl-BeCu hybrid piston cylinder. Daphne oil
7373 was used as a pressure transmitting medium. The applied pressure was
estimated by monitoring the superconducting transition temperature of Pb.

Figure 13: A photo of setup for calibration of the type E thermocouple under
pressure. Figure 14: Temperature dependence of Cernox thermometer at different
applied pressures. Upper inset demonstrates that temperature difference
between Cernox thermometers placed inside and outside of the pressure cell is
less that 0.3 % at ambient pressure. Lower inset: dimensionless sensitivity of
thermometer $(T/R)(dR/dT)$ at each pressure.

We first ensured that there is no appreciable temperature difference between
the two thermometers located inside and outside the pressure cell at ambient
pressure without injecting the heat current. The upper inset of Fig. 14 shows
that the temperature difference between the thermometers $(T_{\rm in}-T_{\rm
out})/T_{\rm in}$, which is less than 0.3 % in the whole temperature range
measured. Subsequently, the thermometer inside of the cell was calibrated
under pressure against the thermometer outside of the cell. The results are
shown in the main panel of Fig. 14. While the resistance of thermometer
slightly decreases with pressure, its dimensionless sensitivity $(T/R)dR/dT$
is not degraded by the pressure as shown in the lower inset of Fig. 14.

Figure 15: Temperature dependence of differential Seebeck coefficient $S_{\rm
dif}$ of type E thermocouple at ambient pressure. For comparison, the standard
data powell  and the literature data inyushkin  are also shown. Figure 16:
Temperature dependence of the differential Seebeck coefficient $S_{\rm dif}$
of the type E thermocouple measured under three different pressures. Inset
shows the correction on temperature at the thermocouple junction.

Figure 15 shows temperature dependence of the differential Seebeck coefficient
$S_{\rm dif}$ of the type E thermocouple at ambient pressure. For comparison,
the standard data powell  and the literature data inyushkin  are also shown.
As seen from the figure, our data well coincides the one from Ref. inyushkin
and deviates from the standard data especially at low temperature. The
discrepancy may be due to different amount of impurities contained in the
thermocouple wires made by different manufacturers; our constantan and chromel
wires are from Goodfellow, and the constantan wire from Goodfellow and the
chromel wire from Hoskins Manufacturing were used in Ref. inyushkin
(manufacturers are not specified for the standard data).

Figure 16 shows temperature dependence of $S_{\rm dif}$ measured under three
different pressures. Significant dependence of $S_{\rm dif}$ on the pressure
is observed only below 10 K. $S_{\rm dif}$ increases at most 80 % than that of
ambient pressure at 1 GPa and 2 K. As shown in the inset of Fig. 16, the
relative change of $S_{\rm dif}$ with pressure results in less than 2 % of the
correction on temperature at the thermocouple junction estimated as $(T_{\rm
true}-T_{\rm wrong})/T_{\rm true}$, where $T_{\rm true}=-V_{\rm
th}/S(P)+T_{\rm out}$ and $T_{\rm wrong}=-V_{\rm th}/S$(0 GPa) + $T_{\rm
out}$. This small correction led us to conclude that the pressure effect on
the type E thermocouple is neglectable up to 2.5 GPa.

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