Article
Methods
Symmetry analysis of (001)-oriented Nb:SrTiO 3 and Nb:TiO 2
Schottky junctions
The Nb:SrTiO 3 single crystal belongs to the point symmetry group of
mm 3 ̄ , which includes the symmetry elements of (1, 2{100}, 2{110}, 3, 4, 1 ̄, 3 ̄,
4 ̄, m{100}, m{110}). The electrical field in the Schottky junction of Nb:SrTiO 3
crystal points along the (001) direction. Owing to its vector nature, the
field shows the symmetry of ∞m, which includes two types of sym-
metry elements, that is, infinite rotation symmetry along (001) direc-
tion and infinite mirror symmetry. The ∞m symmetry can be
represented by a cone. Owing to the manifestation of the electrical
field in the Schottky junction, the depletion region will only show the
point symmetry that is the subgroup to both mm 3 ̄ and ∞m. As illus-
trated in Extended Data Fig. 1a, the symmetry elements common to
both symmetry groups are (1, 2(001), 4(001), m(100), m(010), m(110), m(1-10)). The
resultant group of symmetry elements corresponds to the point group
of 4mm, which represents polar structures, such as that of BaTiO 3 in
the tetragonal phase. Similarly, the rutile Nb:TiO 2 possesses the point
group of 4/mmm, which includes symmetry elements of (1, 2{100}, 2(1-10),
(^2) (110), 4(001),^1 ,^4 ̄, m{100}, m(1-10), m(110)). Its common subgroup with ∞m is
also the point group 4mm (Extended Data Fig. 1b).
Interface piezoelectricity at Schottky junction
If the work function of the metal exceeds that of the n-type semiconduc-
tor, a Schottky barrier forms at the interface between the metal and the
semiconductor (Fig. 1b). In the ideal case, the depletion width W is given by^1
W
χ
qN VV
kT
= q
2
(^3) −− , (3)
d bi
B
where χ 3 is the dielectric permittivity, q is the electron charge, Nd is the
density of dopant, Vbi is the built-in voltage, V is the external applied
bias, kB is the Boltzmann constant and T is the absolute temperature.
As the term kBT/q is usually much smaller than Vbi in the case of interest,
equation ( 3 ) can be simplified as
W
χ
qN
= VV
2
(^3) (−). (4)
d
bi
The potential variation in the depletion region is given as^1
Vx
qN
()=−χ Wx xΦ
1
2 −, (5)
d
3
2
B
where x is the distance away from the metal–semiconductor interface
into the depletion region and ΦB is the barrier height at the metal–semi-
conductor interface. Thus, the corresponding electric field is
Ex
V
x
qN
()= χ Wx
∂
∂ =(−). (6)
d
3
Therefore, the local strain ε(x) in the depletion region induced by the
electrostriction effect can be predicted as
εx ME M
qN
χ
()==^2 d (−Wx), (7)
3
2
2
where M is the electrostriction coefficient (in its one-dimensional form)
in the unit of m^2 V−2. The total displacement over the depletion region
is then
Lε∫ xx M
qN
χ
WM
qN
χ
Δ= ()d= VV
1
3
2
3
2
(−). (8)
W
0
d
3
2
3 d
3
bi
3
2
If an a.c. voltage with the form
VV=sa0in()ωt+,V (9)
where Va is the amplitude, V 0 is the voltage offset and ω is the angular
frequency, is applied on the junction, the corresponding displacement
would be given by
LM
qN
χ
Δ=^2 VVωt V
3
2
d[−sin( )− ]. (10)
3
bi a0
3
2
Owing to the nonlinear exponent, the Schottky junction would gener-
ate a first-order harmonic displacement (that is, strain), which can be
obtained by calculating the Fourier series of above equation. In the
first approximation the displacement is given by
LMVωt
qN
χ
Δ=sin( ) VV
2
f a d(−). (11)
3
0 bi^0
Therefore, the Schottky junction behaves similar to a classical piezo-
electric material whose strain varies linearly with applied bias. The
effective piezoelectric constant deff is
d
L
V M
qN
= χ VV
Δ
2
(−). (12)
f
eff a
d
3
bi 0
0
By substituting the electrostriction coefficient from M (m^2 V−2) with a
more fundamental parameter Q (M=Qχ 32 ) with units of m^4 C−2,
dQeff=2χq 3 Nχd 3 (−VVbi 0 ). (13)
According the developed phenomenological theory (see below), the
Schottky junction would possess simultaneously direct and converse
piezoelectricity with the same coefficient. Thus, the piezoelectric coef-
ficient of the Schottky junction can be given in the tensor form
dQijk=2jki 3 χq 3 Nχd 3 (−VVbi 0 ). (14)
In the case without any external bias, the Schottky junction shows a
piezoelectric tensor as
dQijk=2jki 3 χq 3 Nχd 3 Vbi. (15)
The depletion region in the Schottky junction behaves like an insulat-
ing polar thin layer with electric polarization pointing from the semi-
conductor bulk to the noble metal interface, as indicated by the red
arrow in the Extended Data Fig. 2a. In the equilibrium state, this positive
end of electric dipole is compensated by the electrons in the metal
interface, while the negative charge of the dipole is compensated by
the positive charge in depletion region of the semiconductor. As dem-
onstrated in the phenomenology theory, the interface piezoelectric
effect originates from the combination of the built-in electric field
and the electrostriction effect. Note that, the electrostriction effect
not only describes the electric-field-induced strain with a quadratic
dependence but also is a measure of the dependence of the dielectric
permittivity on external stress.
Once the junction is subjected to an external stress, for example, a
tensile stress perpendicular to the junction interface, the dielectric
permittivity of the semiconductor will increase due to the positive elec-
trostriction coefficient Q 11. This increased permittivity will give rise to
an enhanced electric polarization in the depletion region, which breaks
the screening equilibrium at the interface. Therefore, the increased
polarization will redistribute the charge between the metal and the
semiconductors to reach a new equilibrium state. As the Schottky bar-
rier prevents the electrons from directly flowing across the interface,