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Chemical Physics
journal homepage:www.elsevier.com/locate/chemphys
Electrical and thermal properties of strain- and electric field-induced topological crystalline insulators
Le T. Hoa
a,b, Huynh V. Phuc
c,⁎⁎, Le T.T. Phuong
d,⁎aInstitute of Research and Development, Duy Tan University, Da Nang 550000, Viet Nam
bFaculty of Natural Sciences, Duy Tan University, Da Nang 550000, Viet Nam
cDivision of Theoretical Physics, Dong Thap University, Cao Lanh 870000, Viet Nam
dCenter for Theoretical and Computational Physics, University of Education, Hue University, Hue 530000, Viet Nam
A B S T R A C T
This paper elaborates theoretically the strain and electric field effects on the electronic density of states (DOS) and electronic heat capacity (EHC) of the topological crystalline insulator SnTe (001) and related alloys. We employ Green’s function calculations for DOS and the Boltzmann approach for EHC. Schottky anomalies are found in EHC based on the entropy analysis. Strain- and electric field-induced DOS indicates the band gap opening in the system. Relative trends with respect to the strain modulus (electric field strength) show an increasing (a fluctuating) trend for the Schottky temperature. The results may lead to wide-ranging applications in thermoelectrics and tunable topological electronics/spintronics.
1. Introduction
Topological insulators (TIs) as novel quantum systems are char- acterized by the topological invariants such as 2 time-reversal sym- metry, chiral symmetry, particle-hole symmetry, and a gapped bulk state[1,2]. However, it has been shown that there are various topolo- gical classification of band structures characterizing by different sym- metries [3–11,5,12,10,13,14]. In these materials, an insulating bulk associated with a metallic surface state determine novel topological features in TIs. Among novel quantum materials, in topological crys- talline insulators (TCIs)[3–5]the crystal point-group symmetries such as mirror, rotation or glide protect the surface states in contrast to the topological invariant in TIs.
The non-trivial topology of band structure in TCIs such as lead thin salt family in IV-VI semiconductors relates to the reflection symmetry with respect to the (110) mirror plane[3,15–19,1,20–23]. SnTe,Pb1 x SnxSe (x 0.2) andPb1 xSnxTe (x 0.4) are the famous experimentally realized TCIs so far. In TCIs, depending on the surface, the metallic gapless states divide into two types; (111) and {(110), (001)}. In this first one, four Dirac cones are centered at time-reversal-invariant and Mpoints of the surface Brillouin zone (SBZ), while in the second type four Dirac cones are centered at non-time-reversal-invariantXandY points of the SBZ. It has been shown that the second type can be ob- served experimentally much easier than the first type. Thus, the number of Dirac cones on SBZ of TCIs is even, while an odd number of Dirac cones appear in trivial strong TIs[1,2], leading to fascinating features.
Following this, it has been demonstrated that the topological phase in TCIs can be tuned much easier than TIs by various external perturba- tions[24,21,25,26].
Symmetry breaking is the origin of topological phase transitions in topological materials, which can be appeared in various ways[27–31].
The strain is one of the ways to tune the topological phase in TCIs [22,32,15], which acts as an effective gauge field on the Dirac fermions and creates pseudo-Landau orbitals without breaking time-reversal symmetry. Orbital degrees of freedom can have pronounced effects on the fundamental properties of electrons in solids. Using the scanning- tunneling microscopy the effects of strain on the electronic structure of a heteroepitaxial thin film of a topological crystalline insulator, SnTe, have been investigated[32]. Their result shows that a surprising effect emerges when the strain is applied in one direction. The change in the electronic features in low-dimensional systems has also been predicted already[33]. Breaking the mirror symmetry in TCIs has been examined via the electric field as well, which can open up a continuously tunable band gap on the surface[15]. All methods open a new route to altering macroscopic electronic properties in TCIs and provide a starting point for realizing novel correlated states of matter.
In this paper new physical properties of TCI SnTe (001) surface in the presence of strain and electric field are described. In particular, the strain and perpendicular electric field effects on the electronic density of states (DOS) and the electronic heat capacity (EHC) quantities are theoretically examined. We are interested in EHC because it gives us some information about the response of the system to the temperature
https://doi.org/10.1016/j.chemphys.2020.110845
Received 19 March 2020; Received in revised form 26 April 2020; Accepted 1 May 2020
⁎Corresponding author.
⁎⁎Co Corresponding author.
E-mail addresses:[email protected](L.T. Hoa),[email protected](H.V. Phuc),[email protected](L.T.T. Phuong).
Available online 04 May 2020
0301-0104/ © 2020 Elsevier B.V. All rights reserved.
T
change. By this, we cover the fundamental electronic and thermal properties of TCI SnTe (001). The basic concepts, model Hamiltonians in the presence of electric field and strain effects are taken from the recent experimental observations. However, we apply the Green’s function and the Boltzmann methods to calculate the above-mentioned quantities.
In Section2, we focus on the theoretical background of the work in a pristine state. Then, the strain effects are discussed in Section 3.
Section4explains the perpendicular electric field effects and finally, a summary of the results is presented in Section5.
2. Theory and method
As mentioned before, four Dirac points are located on the SBZ of TCI SnTe (001) near twoXandYpoints. The SBZ in lead thin salt structures is a square lattice. So, aC4rotation symmetry [a /2rotation] is valid between these points and one may focus on one of them for simplicity.
The model Hamiltonian of the Dirac fermions in the vicinity ofXpoint is given by[17,16,18]
= + +
k v k v k n
( ) ,
X x x y y y x x y y
H0 (1)
where =1for simplicity and the Fermi velocitiesvx=1.3eV.Åand
=
vy 2.4 eV.Å are taken from the numerical ab initio computations [34,15,23]. This is a unifiedk p· Hamiltonian which confirms the ex- perimental observations of the band structure of clean SnTe (001) surface states in the absence of electric field. In the above equation,
=( , , )x y z [ =( , )x y] are the Pauli matrices in the spin [lattice]
system. Note that the real space comprises of the cations and anions on the SnTe (001) surface. To match the theoretical and experimental results, one may needn=70meV and =26meV as the intervalley scattering parameters at the lattice scale[15].
The dispersion energy of surface Dirac fermions can be calculated analytically via diagonalizing the Hamiltonian in Eq.(1):
= + + + + + +
k µ n v k v k n v k n v k
( ) 2 ( ) ,
Xµ
x x y y x x y y
, 2 2 2 2 2 2 2 2 2 2 2 2 2
E
(2) whereµ= +( )and = +( )refer to the conduction (valence) band and lower conduction band (upper valence band), respectively [see Fig. 1]. To clarify the role of intervalley scaterring parameters, we focus
on three possible cases, (i) n= =0 meV, (ii) n=70 meV and
=0meV, and (iii)n=0meV and =26meV. Ifn= =0meV, Eq.
(2)leads to the dispersionEXµ( )k =µ v kx x2 2+v ky y2 2 for which the va- lence and conduction surface states are twofold degenerate, while in the case ofn=70meV and =0meV, we deal with dispersion energy of
= + +
k µ n c k v k
( ) ( )
Xµ
x x y y
, 2 2 2 2
E where the linear dispersions form at
Xpoint with energies±n. However, interestingly, the lower band of the conduction band and the upper band of the valence band cross each other at zero energyEX( )k =0defined byn2=v kx x2 2+v ky y2 2. In the last case, i.e.n=0meV and =26meV, the surface states dispersion en- ergy is given byEXµ, ( )k =µ ( + v kx x)2+v ky y2 2 in which two copies of Dirac cones take place at zero energy located at kx= ± /vx and
= ±
ky /vy along X andY directions, respectively. These treatments analytically confirm the key role ofnand parameters in achieving the reported features of SnTe (001) surface states.
At first glance, one observes that the band structure is anisotropic along thexandydirection, as shown inFig. 1(a) and (b), respectively.
This anisotropic feature manifests itself in the electronic DOS [will be discussed later]. From the dispersion above, at k =( ,k kx y)=0, two crossed bands with energiesE= ± n2+ 2appear, while atky=0and at E=0, two momenta x= +( n2+ 2/ , 0)vx and
=( n + / , 0)v
x 2 2 x so-called the surface Dirac cones emerge, which mainly the topological features of TCI SnTe (001) originates from these two Dirac points. In addition, atkx=0andE= ± , two saddle pointsS1=(0,+n v/ )y andS2=(0, n v/ )y come up along they direction. It has been well-established that at these saddle points, the form of Fermi surface changes, i.e. a Lifshitz transition occurs.
To finish this section, we calculate and analyze the electronic DOS quantity. To do so, we use Green’s functions approach. Summing the imaginary parts of the Green’s function matrix, the non-interacting DOS is written as[35,36]
=
=
G k
( ) 1
Im[ ( , )],
k 0
1 4
SBZ
D E 0 E
(3) whereG0 ( , )k E can be each of the diagonal elements of the Green’s function matrixG k0( , )E =[E+i HX0( )]k 1[we set =0.1meV in numerical computations as the broadening factor to make the van Hove singularities sharp in DOS curve]. On the other hand, HX0( )k is
Fig. 1.The electronic band structure of pristine topological crystalline insulator SnTe (001) surface and related alloys along the (a)x-direction and (b)y-direction.
The total electronic density of states in the surface Brillouin zone of pristine SnTe (001) is represented in (c).
the matrix representation of the clean Hamiltonians in Eq.(1) [37,38].
InFig. 1(c), we print the electronic DOS of SnTe (001) in the absence of electric field. As can be seen, the electronic DOS is zero at the Fermi level and the system is semimetal. Two saddle points give rise to two van Hove singularities at energiesE= ± corresponding to the Lifshitz transition. On the other hand, two dips take place at energies
= ± n2+ 2
E corresponding to the crossed bands atk =0. In the following, we turn to the thermal properties, particularly the EHC. As mentioned before, a change in the total energy
=
E 0 D0( )E EnFD( , )E T dE 0EFD0( )E E Ed in a fermionic system consisting of N= 0 D0( )E nFD( , )E T dE fermions in the pre- sence of a temperature change T is called EHC. Obviously,D0( )E is the calculated DOS in Eq.(3)andnFD( , )E T =1/[1+exp( /E k TB )]is the Fermi–Dirac distribution function. Thus, EHC can be calculated easily using the semi-classical Boltzmann approach[39,37,38]
= =
T T
n T
T d
( ) ( )( ) ( , ) .
0
0 F FD
C E
D E E E E
E (4)
In our system, the Fermi levelEFlies at zero energy and by this, EHC can be restated as
= +
+
=
T
k k T
G k k T d
( ) 1
( )
Im[ ( , )]
1 cosh[ / ] .
B B 2 k
1 4
SBZ
2 0
B
C E E
E E
(5) Therefore, a temperature-dependent EHC for an interval thermal energy can be obtained. The thermal energy at room temperature is around 0.026 eV and for this reason, the thermal energy interval in the present work is up to 0.1 eV 1150 K.
We provide temperature-dependent EHC inFig. 2in the absence of the strain and electric field. At very low temperatures only the lowest level is occupied and the internal energy and heat capacity are small.
However, as the temperature is increased the upper level begins to be occupied and so the internal energy rapidly increases, then the gradient E versusTis large and so heat capacity increases. When both levels are almost occupied equally, which occurs at very high temperatures, no more energy can be absorbed and the gradientdE/dT effectively be- comes constant and so the heat capacity becomes small and tends to zero. Similar behavior is observed when there are a small number of
levels [more than two, i.e. in our case]. The Schottky anomaly is the general feature of these changes, appearing as a smooth peak in EHC. In our system, we obtain this anomaly at the thermal energy of around 0.038 eV 438.5 K.
3. Strain effects
In this section, we apply both compressive (positive sign) and tensile (tensile sign) strains to break the lattice symmetry down and to explore the change in the spatial orbital distribution effects on the electronic phase, DOS and EHC of SnTe (001) surface. In the theory we use, the momenta and energy shift with strain and eventually the electronic and thermal properties are enhanced significantly. In the presence of strain, the surface Dirac points suffer from a displacementu with components
= +
uij ( j iu i ju)/2. This displacement field can be expressed by a strain-induced gauge field vector potential A. In our theory, the re- lationship between the spatial displacement u and A is linear [the shear terms are neglected], given by[22,25,26]
= + +
A ( 1uxx 2uyy, 1uyy 2uxx), (6) where 1=0.3 Å−1 and 2=1.4 Å−1 [32]. Hence, the shift in the
momenta can be written as kx kx+ 1
+ + +
uxx 2uyy,ky ky 1uyy 2uxx. Although we restrict ourselves to theXpoint, the statements are similar for theYpoint.
3.1. Uniaxial strain
Depending on the strain modulusuxxanduyy, the electronic DOS and EHC of the system can be explored. As illustrated inFig. 3, we plot DOS and EHC respectively inFig. 3(a) and (b) for two arbitrary uniaxial strains±0.05and±0.1along thex-direction, i.e.uyy=0. It can be seen that in general, the DOS decreases with strain. From the changes in DOS, one can distinguish the changes made in the electronic band structure. The height of van Hove singularities decreases with strain, which means that the concavity of bands along they-direction is in- creased [the flatness is decreased] and the saddle points are not de- stroyed with strain. However, the Dirac points along thex-direction in the vicinity ofXpoint disappear with strong enough strains and a gap of
Fig. 2.The electronic heat capacity of pristine topological crystalline insulator SnTe (001) surface and related alloys. The Schottky anomaly is located at the temperature of 438.5 K.
around 0.045 eV opens atuxx= ±10%. The results are the same con- cerning the strain sign, which originates from the inherent rotation symmetry in the strain theory. Furthermore, the results show that the crossed bands at k =0 andE= ± n2+ 2 disappear atuxx= ±10%
along both directions, meaning that in addition to the opened gap around the Fermi level and Dirac points, a gap opens at this non-zero energy as well.
A plot of the EHC of TCI SnTe (001) in the presence of uniaxial strain along the x-direction is represented inFig. 3(b). Although the Schottky anomaly appears independent of the strain, its corresponding thermal energy is not the same for different strains. The Schottky temperature shifts forwardly from 438.5 K to around 484.5 K and 496 K foruxx= ±5%and±10%. This is a direct consequence of the band gap size and the slope/concavity of the Dirac points and saddle points. Since the total DOS is decreased with strain, the EHC should be decreased with strain as well. On the other hand, the low-temperature limit of EHC is zero over a tiny thermal energy interval when the gap opens in the system. However, the high-temperature limit of EHC does not show any significant effect because the quantum effects are not bold anymore and the thermal effects contribute mostly to the trends.
From the electronic band structure inFig. 1, we noticed that there is an anisotropic feature in the Dirac fermions behaviors along thex- and y-direction. This, in turn, may manifest itself in the direction of the applied strain. For this reason, as another possibility of uniaxial strain, we plotFig. 3(c) and (d) for DOS and EHC, respectively, of SnTe (001) surface in the presence of uniaxial strain along the y-direction, i.e.
=
uxx 0. In this case, the DOS decreases with strain as well as the pre- vious case but with different rates. Again, the van Hove singularities
appear for weak strains because of the saddle points, while in contrast to the strain along thex-direction, it disappears for stronguyy= ±10%. The anisotropic feature can be seen from different behaviors of DOS in the presence of uniaxialuxx anduyy. Another important thing is the band gap size, which is around 0.09 eV, 2 times larger than the band gap size ofFig. 3(a). As for the EHC in the presence ofuyy,Fig. 3(d) shows that EHC decreases withuyy as before with no change for the Schottky anomaly in the case of weak strains. Moreover, a forward shift emerges atuyy= ±10% for the Schottky anomaly taking place at the temperature 530.5 K. Also, the zero range of EHC at low temperatures is much more than the previous case due to the larger opened band gap.
3.2. Biaxial strain
Here, we focus on another possible way of applying strain; the biaxial strain. In this case, we consider the same values of strains foruxx anduyy, meaning that uniaxial strain is applied along bothx- and y- directions, so-called uwith two different values±5% and±10%, as shown inFig. 4. As expected, in the presence of strain along both di- rections, one should expect more decreasing behavior for both DOS and EHC with strainu. Foru= ±5%, our results show that the system is still in the metallic state and gapless surface states with different slope and concavity of bands play a role in determining the topological features.
However, as soon as the biaxial strain becomes stronger, a gap of around 0.3 eV appears. The inset panel inFig. 4(a) shows that neither the Dirac points nor the saddle points can be seen in the case of strong biaxial strain, meaning that the system is transited to a new gapped phase and two parabolic bands describe the Dirac fermions dispersion Fig. 3.The electronic DOS (a) and EHC (b) in the presence of uniaxial strain along thex-direction, namelyuxx= ±5%and±10%. The results are compared with the pristine state. Panels (c) and (d) refer to the presence of uniaxial strain along they-direction.
withing our limited energy interval. Of course, two extra bands must appear as well in the larger energies [out of the shown regime here] to support the four-band model.
Let us connect these behaviors of DOS to the EHC. InFig. 4(b), one understands that the EHC tends to zero as the biaxial strain is increased.
This stems from the large opened band gap in DOS. Additionally, the Schottky anomaly shifts to higher temperatures as expected to 484.5 K and 692 K respectively foru= ±5%and10%because the competition between the thermal energy k TB and the carrier energies is much stronger than when the uniaxial strain is applied. Both quantum and thermodynamic limits are affected by the biaxial strain as uniaxial one.
4. Electric field effects
In this section, we introduce the perpendicular electric field effects as the second effect which may affect the topological phase, the elec- tronic band structure, the electronic DOS and eventually EHC features.
For this reason, we use the simple formulation of the perpendicular electric field with the strengthEzin the real space, which induced to the system from the bias/gate voltage on the SnTe (001) surface
=Ez z,
HPEF (7)
If we add this new Hamiltonian to the original Hamiltonian in Eq.(1), we achieveH =HX0( )k +HPEF[40]and eventually
= + + + + + + + + +
k µ E n v k v k E n v k n E v k
( ) 2 ( ) ( ) ,
µ, z2 2 2 x x2 2 y y2 2 z2 2 2 x x2 2 2 z2 y y2 2
E
(8)
for which the electronic band structure changes withEz. Let us discuss the band gap situation in the presence of the electric field from the electronic DOS. Although it is arbitrary to choose any strengths, in the present paper, we focus on the electric field larger than the critical energy ± n2+ 2, namely ±0.1 eV and ±0.3 eV. Using the new electric field-induced Hamiltonian, we obtain new electric field-in- duced Green’s functions and eventually DOS from the relation
= = G k
( ) 1 4 1 k SBZIm[ ( , )]
D E E where G is the perturbed
Green’s function.
However, we noticed that the band gap of the system in DOS stems from the Dirac points located at xand xwithkx= ± n2+ 2/vxand
=
ky 0. Interestingly, the Dirac cones are shifted with electric field so that the new xand xpoints appear atkx= ± Ez2+n2+ 2/vxand
=
ky 0. So, we should look at the gap at these points in the presence of an electric field, not the previous ones. This is the first point that the Dirac cones do not destroy but they shift to new coordinates with the electric field. As it is shown inFig. 5(a), the system is gapless when the electric field is weak because the electronic DOS is zero at zero energy, while a gap of around 0.09 eV opens for strong enough electric fields.
As can be seen, the van Hove singularities disappear beside the opening band gap atEz= ±0.3eV. It should be pointed out that the area under curves for both the absence and presence of the electric field must be conserved due to the sum rule.
The changes made by the perpendicular electric field in DOS lead to some changes in the heat capacity of TCI SnTe (001), which are studied in Fig. 5(b). EHC decreases with the electric field and the Schottky anomaly moves to lower and higher temperatures for weak and strong Fig. 4.The electronic DOS (a) and EHC (b) in the absence and presence of biaxial strain, namelyuxx=uyy=u= ±5%and±10%.
Fig. 5.The electronic DOS (a) and EHC (b) in the presence of perpendicular electric field, namelyEz= ±0.1eV and±0.3eV. The gap opens when the electric field is strong enough.
electric fields, respectively. The Schottky temperatures are respectively 311.5 K and 507.5 K forEz= ±0.1eV and±0.3eV. Moreover, due to the opened band gap in the system in the presence of strong electric fields, one may notice zero EHC over a thermal energy range at low temperatures. Again, thermal effects are dominant at high enough temperatures and one expects a convergence independent of the electric field strength.
5. Summary
Using the Green’s function approach and the Boltzmann method, we have systematically studied the strain and electric field tuning of the electrical and thermal properties of TCI SnTe. We have found that: (i) by applying both uniaxial and biaxial strain a band gap opens in the system, which is initially gapless. (ii) The massless Dirac fermions on the SnTe (001) surface become massive with strain and strong electric field. This is due to the emergence of the band gap in the electronic DOS. (iii) The Schottky anomaly in electronic heat capacity moves to higher temperatures in the presence of both uniaxial and biaxial strain, while the Schottky temperature fluctuates with the electric field. Both the strain and electric field results provide physical insights into the practical applications in topological thermoelectrics.
CRediT authorship contribution statement
Le T. Hoa: Conceptualization, Investigation, Formal analysis.
Huynh V. Phuc:Software, Formal analysis, Conceptualization.Le T.T.
Phuong: Conceptualization, Methodology, Investigation, Formal ana- lysis, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influ- ence the work reported in this paper.
Acknowledgements
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 103.01- 2019.389.
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