Number-of-Layer, Pressure, and Temperature Resolved Bond-Phonon-Photon

Number-of-Layer, Pressure, and Temperature Resolved Bond-Phonon-Photon Cooperative Relaxation of Layered Black Phosphorus

Yonghui Liu^a , Xuexian Yang^b, Maolin Bo^c, Zhang Xi^d, Xinjuan Liu^e, Chang Q Sun^f,*, Yongli Huang^a,*

a Key Laboratory of Low-Dimensional Materials and Application Technologies (Ministry of Education), Hunan Provincial Key Laboratory of Thin Film Materials and Devices, School of Materials Science and Engineering, Xiangtan University, Hunan 411105, China

b Department of Physics, Jishou University, Jishou 416000, Hunan, China

cCollege of Mechanical and Electrical Engineering, Yangtze Normal University, Chongqing 408100,China

d Institute of Nanosurface Science and Engineering, Shenzhen University, Shenzhen 518060, China

eInstitute of Coordination Bond Metrology and Engineering, School of Materials Science and Engineering, China Jiliang University, Hangzhou 330018, China

f NOVITAS, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore

E-mail addresses: [email protected], [email protected]

Abstract

We systematically examined the effects of number-of-layer, pressure, and temperature on the bond

length and energy, Debye temperature, atomic cohesive energy, and binding energy density for

layered black phosphorus (BP) using bond-phonon-photon spectrometric methods. We clarified the

following: (i) atomic under-coordination shortens and stiffens the P-P bond, which raises the B

and

A

phonon frequency and widens the bandgap; (ii) bond thermal elongation and weakening soften

all phonon modes; and (iii) bond mechanical compression has the opposite effect of heating on

phonon frequency relaxation. The phonon and photon energy depends on the bond length and energy,

which determine the relevant elasticity and thermal stability of layered structure. More broadly, the

approaches and findings of this work provide both insight into and efficient tools for further

exploration of unusual behaviors of other two-dimensional substance.

Keywords: Bond Relaxation, Phonon and Photon, Number-of-Layer, Temperature, Pressure.

1. Introduction

Black Phosphorus (BP) differs significantly from other layered 2D semiconductor nanomaterials, such as graphene,

hexagonal boron nitride (h-BN),

and transition metal dichalcogenides (TMDs),

in its physical and chemical properties.

Layered BP has potential applications in optoelectronic and nanoelectronic devices.

For example, few-layered BP shows high carrier mobilities, up to 1000 cm

V

s

, and high current on/off ratios, up to 10

at room temperature, potentially making them excellent field-effect transistors.

Furthermore, all physical properties of the BP can be tuned by varying the number-of-layer (N), temperature (T), and pressure (P).

Understanding the dependence of material properties on the intrinsic bond relaxation dynamics and any applied stimulus is therefore critical for engineering BP properties.

Bond-phonon-photon cooperative relaxation dynamics under stimulus have the most significant effect. The bond relaxation and associated electronic behavior mediate the properties of the material.

Phonon and photon spectrometrics probe that relaxation process in detail.

For instance, heating from 77 K to 623 K softens the A

, A

and B

phonons for the few-layered BP.

Increasing

pressure from 0 to 5 GPa stiffens these three optical modes.

The variation of atomic

number-of-layer from monolayer to bulk leads to a redshift of photon and phonon modes.

The

photon energy and phonon frequency follows the relationships:

( ) ( ) ( ) ( )

( )

0

2 1

G 0

0 e d

2 0

/

A B C

( ) ( )

N d a N

E N E N N

T T T

P kP lP

ω ω

ω ω ω ω

ω ω

− −

−  −

 

−   =   − −

−   ∆ + ∆

 

−   +

(1) where ω

and E

are the phonon frequency and photon energy at the reference point; N denotes the number of atomic layers; a is the lattice constant; d, q, k, and l are adjustable parameters to match the N- and P-dependent Raman shifts; E

(N) is the bandgap of a particle; and A, B, and C are adjustable parameters. The N

and N

terms represent the potential and kinetic energies of the electron−hole pairs. These phenomenological models describe observations of unclear physical origin well.

In this communication, we show that an extension of the bond order-length-strength (BOLS)

correlation model and the local bond average (LBA)

approach can accurately reproduce the N-, T-,

and P-trends of BP Raman shifts, clarifying their physical origin and providing quantitative

predictions of bond lengths and energies. Rather than taking the more conventional approach, we

focused on the functional dependence of the Raman- and bandgap-shifts on the order, length, and

strength of the representative bond across the entire specimen, as well as their response to applied

stimuli. Agreement between modeling predictions and experimental observations validates our

insight into these relationships. In particular, we quantitatively evaluate the dependencies on the

referential wavenumber ω(1) and its bulk shift, the atomic number-of-layer N, the binding energy E

,

the binding energy density E

, the Debye temperature θ

, the compressibility β, the elastic modulus

B, the bulk modulus B

and their first-order derivatives B'

, and the effective coordination number

(CN or z) for the few-layered BP and its bond lengths d

and energies E

, which are beyond the scope

of previous approaches.

2. Principles

2.1 BOLS-LBA Notion

In the absence of phase transitions, the nature and total number of bonds are constant. This postulate applies to all specimens of interest: crystalline, non-crystalline, those with defects or impurities, and those without defects or impurities. The length and of all relevant bonds however remain obscure.

We therefore focus on the performance of a representative bond, or the average of all bonds, in our treatments.

The BOLS model suggests that bond breaking causes contraction of local bonds, an increase in bond strength, densification, and quantum entrapment of charge and energy. Hence, it modulates the local atomic cohesive energy, the binding energy density, and the Hamiltonian of the entire specimen and their relevant properties. The BOLS model can be expressed as follows:

( ) ( )

{ }

B

B

2 1 exp 12 / 8 (bond contraction ) (bond strengthening)

z z

m

z z

d d C z z

E E C

−

−

 = = +  − 

  

  =



(2) where z and B denote an atom in the zth atomic layer and in the bulk specimen as a standard, respectively. z spans from the outermost surface to the center of the solid. It can be up to three layers, as there is no bond order loss that occurs when z > 3. The bond contraction coefficient C

varies only with the effective CN (or z) of the atom of interest regardless of the nature of the bond or the solid dimensions.

Collecting statistics from a large number of bonds, the LBA provides a true spectrometer, sorting the

constituent bonds according to their vibrational frequencies. In contrast to the volume partition

approximation, which focuses on the particular value of a quantity in the partitioned volume, the

LBA approach connects the deviation of the quantity from its known bulk value under an applied external stimulus. The volume partition approximation therefore describes only the local representative atomic bonds, disregarding the manner of distribution and the number of bonds.

2.2 Bond-Phonon-Photon Correlation

2.2.1 Phonon frequency versus Bonding Identities

Usually, the Raman frequency is measured as ω ω ω =

+ ' , where ω

is the reference point and

is the Raman shift under an applied stimulus. ω

varies with the frequency of the incident radiation and substrate conditions. The responses to applied stimuli however are consistent. Expanding the interatomic potential u(r) in a Taylor series around its equilibrium and considering the effective atomic z, we can derive the phonon frequency ω of a harmonic system in the first order approximation:

( )

( ) ( )

0 2 2

3 m

d ( )

( ) !d

2

z

n

n n z

n r d

z n

z

u r u r r d

n r r d

E µω O r d

= =

≥

 

=   −

 

−  

= + +  − 

∑

(3)

The n = 0 term is the minimum binding energy E

that determines the energy shift of electrons in

various bands that can be resolved using photoelectron emission spectrometrics.

The n = 1 term is

the force [du(r)/dr = 0] at equilibrium. The n = 2 term corresponds to the harmonic vibration of a

dimer oscillator, which dominates the Raman shift.

Terms with index n ≥ 3 correspond to the

nonlinear vibrations that dominate transport dynamics, such as thermal expansion and thermal

conductivity. Setting the vibrational energy equal to the third term in the Taylor series and omitting

the higher order terms yields the following: µ ω ( ∆ ) x

≅ Ex d

. As a first-order approximation,

the lattice vibrational frequency ω corresponds to the Raman shift Δω(z, d

, E

) from the reference point ω(1, d

, E

) that depends functionally on the bond order z, length d

, energy E

, and the reduced mass μ = m

m

/(m

+m

) of the representative oscillator dimer. z and b denote an atom in the zth atomic layer and in the bulk specimen, respectively.

B B

2 1/ 2

2

( , , ) ( , , ) (1, , )

d ( ) 1

d

z

z z z z

z r d z

z d E z d E d E

E u r

r d

ω ω ω

µ

µ

∆ = −

 

= ∝  

 

(4) Increasing number-of-layer produces a redshift of the A

mode, which suggests that the A

mode is dominated by the interaction of two neighboring atoms. The D and 2D modes undergo the same redshift but the G mode blueshift when the number-of-layer is reduced in graphene.

2.2.2 Photon energy versus Bonding Identities

The nearly-free electron approximation

indicates that the bandgap E

between the conduction and the valence band depends uniquely on the first Fourier coefficient of the crystal potential V

(r)

G B

1 cry

1

( ) 2

ik r

d

V V r e

E V

r E

⋅

= ∝ 〈



 〉

 =

 ∫

(5) The bandgap energy is proportional to the mean cohesive energy per bond 〈E

〉.

is the Bloch wave function approaching nearly free electrons. The crystal potential V

(r) determines the intrinsic E

, which is the density or energy of the excitons that dictate the quantum confinement effect.

Therefore, the E

∝ E

will modify the E

intrinsically. From the perspective of the bond

relaxation approach, the perturbation to the crystal potential under the applied stimuli can be

expressed as follows: V

(N) = V

(∞)(1 + Δ

). V

(N) and V

(∞) denote the measurement of and

bulk crystal potential, respectively. Δ

is an atomic number-of-layer perturbation to the crystal potential. The relationship can also be described as:

( )

cry cry G G

cry G

3 B 3

( ) ( ) ( ) ( )

( ) ( )

1 1

N

z m

z z z

z z

V N V E N E

V E

E C

γ E γ

≤ ≤

− ∞ − ∞

∆ = =

∞ ∞

 

=  − =  −

 

∑ ∑

z 1

z z

V C N

γ

τ

(6) Where V

is the volume of the zth layer, N is the dimensionless size or the atomic number-of-layer measured along the radius of a sphere or across the thickness of the layered material or thin film, γ

is the surface-to-volume ratio of different dimensionality, and τ is the dimensionality with τ = 3, 2, 1 for a nano-dot, a nano-rod, or a nano-film, respectively. The bond index m is intrinsic to a specific material.

2.3 Photon-energy and Phonon-frequency shift 2.3.1 Number-of-layer Dependence

The number-of-layer dependence of the Raman shift and electronic bandgap are two of the most powerful tools for probing bond relaxation dynamics of two-dimensional layered materials.

The relationship between bond relaxation and average atomic coordination is:

( )

2 1

B B

G 0 3

( ) ( )

( ) 1

m z

m

z z

z

C z

z C

E N E N C C

ω

ω

τ

 

− +

−

≤

 ∆  

 =  

∆  

∆

= −



∑

(7)

Where Δω(z) and Δω(z

) denote the phonon frequency difference between the reference

wavenumber ω(1) and the measured ω(z) or the bulk ω(z

). ΔE

(N) is the difference between the measured (E

) and reference (E

) bandgaps. In particular, Eq. (7) shows that the CN-dependence of the Raman frequency and bandgap energy can be captured by the atomic average coordination without requiring any freely adjustable parameters, as opposed to Eq. (1).

2.3.2 Chemical Bond Relaxation

Bond length and energy vary according to the following relationships:

( ( ) )

( )

0 0

0 0

B

B

B

1 ( ) d d

( ) d d

1

T P

z T P

T V

T V

z

d d t t P P

t t P V V

E E

E

α β

η

 = + +

   

  + 

 =  − 

  

  



∫ ∫

∫ ∫

(8) where α(t) and β(P) are the thermal expansion and compression coefficients, respectively, and η(t) is the specific heat. These relationships indicate that heating weakens and pressure stiffens the representative bond for the entire specimen.

2.3.2 Temperature and Pressure Dependence

In order to extract the T- and P-dependence, we can use the approximation 1 + x ≅ exp(x) at x << 1.

That produces the following relationship:

( ) ( )

( ( ) ) ( )

( )

0 0

0 0

1/ 2

0

1/ 2

0

1 d exp ( ) d

( )

( ) d

1 exp ( ) d

( )

T T

T T

V P

V P

m

T t t t t

T

P V V

P P P

P E

ω η α

ω

ω β

ω

∆   −

∆  = 

∆   − 

∆   

∫ ∫

∫ ∫

(9)

The Raman shift is first calculated in the linear regime at T > θ

. Since the thermal expansion coefficient α(t) is normally of the order 10

K

, the expression simplifies to

1+

∫

α

^. It is reasonable to ignore thermal expansion under the first order approximation for T > θ

. The integration of the specific heat and pressure are:

D

0

3 3

D /

0 0 0

m m D

0

1 1

m m den

( / ) 9 e

( ) d d d

e 1

( ) d ( ) d

( ) d

T T T x

v

x

V x x

V

C T RT T x

t t t x

E E

V

p V V p x x

p x x

E E E

θ

η θ

  

 = =   −

  

  = =



∫ ∫ ∫

∫ ∫ ∫

(10)

Where R is the ideal gas constant, θ

is the Debye temperature, and C

is the specific heat. x = θ

/T is the reduced form of temperature. η(t) is the specific heat per bond, which approximately follows the Debye approximation to converge to a constant value of 3R/z at high temperature. E

and θ

are the independently adjustable parameters in calculating the specific heat. The P-induced Raman shift is calculated from the integral of the Birch-Mürnaghan (BM) equation,

the nonlinear expression x(P). Here, the BM equation has the following form: P(x) = 1.5B

(x

-x

)[1+0.75(B'

-4)(x

-1)].

x(P) is another form of the equation of states in terms of nonlinear compressibility, and its form is:

x(P) = V/V

= 1- βP +β'P

. V

denotes the initial volume. Matching the BM equation to x(P), we can

derive the nonlinear compressibility coefficients, β and β', the cohesive density E

= V

/E

, the bulk

modulus B

( βB

≌ 1), and its first-order derivative B'

These coefficients closely match the

P-dependent Raman shift. Specifically, Eqs. (9) and (10) reflect the T- and P-dependent Raman

frequency, involving only the atomic cohesive energy E

and the binding energy density E

without

requiring any freely varying parameters, in contrast to Eq. (1).

3. Calculation Procedures

We carefully measured and calculated the N-dependence of the photon energy and phonon frequency for the layered BP in order to enable verification of our predictions. We obtained m and ω(1) from the N-dependence of the Raman shift and bandgap. m is used as an initial guess, which can be further refined by carefully fitting the experimental data. The calculation is also confirmed by matching the theoretical curve to the measured T- and P-dependent Raman shift to extract the atomic cohesive energy E

, binding density E

, and Debye temperature θ

for experimental BP specimens. Further refinement of the derived E

and E

was achieved by comparing them to the experimental data over the entire temperature and pressure range with the inclusion of lattice thermal expansion, the compressibility β, and the bulk modulus B

. The accuracy of the phonon frequency and bandgap energy depend on the experimental measurements, which are exactly reproduced by the BOLS approach.

4. Results and Discussions

4.1 Layer-number Resolved Phonon and Photon Relaxation

An extension of the zone-resolved electron spectrometrics

results in the differential phonon spectrometrics (DPS) that distills clearly the characteristic phonons due to the conditioning, by differencing the spectra collected before and after conditioning, or collected at different layer number from the same specimen upon the spectral peak areas being normalized and background-corrected.

Upon the standard processes of background correction and spectral peak area normalization, the DPS purifies merely the spectral features due to conditioning. This DPS strategy can monitor the phonon relaxation both statically and dynamically with high sensitivity and accuracy without needing any approximation or assumption. The DPS distills the phonon density of states (DOS) gain as a component presenting above the horizontal axis and features the DOS loss as valley below the axis in the DPS spectrum. This process removes the commonly shared spectral feature that is out of concern.

Ideally, the resultant DPS components conserve as the spectral areas above and areas below the

lateral axis are identical. Any improper background correction or spectral normalization may asymmetrize the spectral gain and loss compared with the ideal situation. With these criteria, one can readily gain quantitative information on the local bond length, bond energy, and charge entrapment and polarization, etc. The convoluted DPS applied to both B

and A

modes for BP gives the trend of phonon relaxation direction. The valleys of differential spectra correspond to the bulk and the peaks correspond to the N-effect, demonstrating the N-induced blueshift of the characteristic peaks, as shown in Figure 1.

According to Eq. (7), the Raman frequency ω(z) and bandgap energy E

are:

[ ] ( )

( )

/ 2 1

B B

0

G 0

3

( ) (1) ( ) (1)

1

m z

m

i i

i

z z C C

E E E C C

N

ω ω ω ω

τ

− +

−

≤

 − = −

 

− = −

  ∑

(11) The number-of-layer dependence of the photon energy and phonon frequency is calculated and calibrated using Eq. (11). For N ≥ 6, z approaches the bulk BP value of 12. The empirical expression is z = 1.8N + 0.8, correlating the atomic average CN with the number-of-layer N.

Figure 2 compares theoretical and experimental results for the N-dependence of the bandgap and Raman shift. We compare Raman shift of N-dependent G mode of graphene and A

mode of BP.

Increasing number-of-layer produces a redshift of the G and A

modes, which suggests that the two

modes are dominated by the interaction of two neighboring atoms, but the collective interaction of an

atom with its z-neighbors governs the blueshift of the D and 2D modes phonon dynamics of

graphene.

Theoretical reproduction of experimental observations allowed us to evaluate the

referential wavenumber of graphene ω

= 1566.7 cm

and the bond nature index m = 2.56. The

consistency between the model predictions and experimental measurements reveal that

broken-bond-induced local strain and quantum trapping at the surface dictate the bandgap expansion

and the N-dependent Raman shifts for the A

modes. This finding evidences the importance and effectiveness of the proposed mechanisms for the lattice vibrations in BP.

4.2 Temperature-Resolved Phonon Relaxation

With increasing temperature, the Raman response proceeds gradually from the nonlinear to the linear regime. The slow decrease in Raman shift at low temperatures arises from the small

0^T

η

∫

values when the specific heat η(T) is proportional to T

at extremely low temperatures. Experimental observations show that the Raman shifts decreases linearly with temperature increasing at higher temperatures. With T >> θ

, the specific heat C

is considered to be a constant approaching 3R.

Figure 3(b) compares experimental and theoretical results for the three modes of temperature dependence, with the derived ω(1), θ

and E

given in Table 1. We observe that θ

determines the width of the curve shoulder and that E

determines the slope of the curve at high temperature. Thus, we have demonstrated that our method can reproduce experimental observations exceedingly well without involving any freely adjustable parameters, mechanisms of phonon decay, or interface interactions.

4.3 Pressure-Resolved Phonon Relaxation

The consistency between BOLS predictions and the experimental results for the pressure-dependent

Raman shift is shown in Figure 3(a). Generally, the energy increases nonlinearly with pressure

according to the BM equation. That is because the V/V

of the unit cell becomes smaller as the

specimen is compressed. Model reproduction of the P-induced shift provides quantitative evaluation

of the compressibility β = 0.32 GPa

and cohesive density E

= 0.68 eV nm

. We find that the

Raman optical modes blueshift with increasing pressure. That is caused by bond compression and

work hardening.

Figure 3(a) compares the theoretical results to the measured pressure-dependent Raman shifts of the A

, B

, and A

phonon modes for BP at room temperature. Agreement between predictions and experimental observations allows us to determine the ω(1) of the A

(360.20 cm

), B

(435.00 cm

), and A

(462.30 cm

) modes. The shift in bond energy is dependent on the ambient temperature and pressure. The measured blueshift of the Raman peaks under increasing pressure. Therefore results from the competition between thermal expansion and pressure-induced compression. Results suggest that pressure dominates the trend.

5. Conclusion

A systematic understanding of the atomic origin of unusual mechanical and optical behaviors in a nanosolid using the LBA approach and BOLS approximation have been proposed, with applications in the N-, T- and P-domains. One of the most pronounced effects observed in BP is a redshift of the Raman-active A

mode and bandgap energy with increasing number-of-layer. The unexpected redshift is shown to be due to coordination imperfections. We also showed that the P-induced blueshift of Raman optical phonons arises from the bond compression and energy storage exerted by compressive stress. We find that the thermal-softening of phonons results from bond lengthening and softening. The agreement between our calculations and experimental data, without using any empirical parameters, suggest that the current expressions accurately represent N-, P-, T-dependences.

These findings demonstrate the value of the proposed approach, particularly as a tool for extracting quantitative information on bond dynamics from photon and phonon spectroscopy.

Acknowledgment

Financial support from NSF (No.11502223) and the open project program of key laboratory of

low-dimensional materials and application technologies, ministry of education, China (KF20140202),

HPIFP(CX2015B222), and the scientific research fund of Hunan provincial education department of

China (No.15B189) are gratefully acknowledged.

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Table 1. Instrumental information derived from the reproduction of the N-, T-, P-dependence of bandgap- and Raman-shift of few-layered BP

Stimuli Quantities Values Refs

Number-of-layer

Referential frequency, ω(1) / cm^-1

1 g

2g 2 g

360.20 A 435.00 B 462.30 A







-

The bond nature, m 4.60 -

The bulk bandgap energy, E0 / eV 1.81 -

Atomic cohesive energy, Em / eV 0.42 -

Temperature Debye temperature, θD / K 400.00 ^[30]

Thermal expansion coefficient, α /10^-5K^-1 2.20 ^[34]

Binding energy density, Eden / eV nm^-3 0.68

Pressure Compressibility, β/βʹ / 10^-3GPa^-1/GPa^-2 0.32/2.20 ^[35]

Bulk modulus, B0/Bʹ 0 / GPa 84.10/4.69 ^[36]

428 432 436 440 444 448

Raman Intensity

Wavenumber / cm^-1 3-ayer 4-layer Bulk

B_2g (a)

460 464 468 472 476

Raman Intersity

Wavenumber / cm^-1

1-layer 2-layer Bulk

(b) A_2g

Figure 1. Number-of-layer resolved blueshift of the (a) B2g

[37] and (b) Ag 2[38]

phonons for the BP semiconductors.

2 4 6 8 10 0.3

0.6 0.9 1.2 1.5

Bandgap / eV

Number-of-layer

BOLS data1 data2 data3 (a)

bulk N 1 2 3 4 5 6

468 469 470 471 1582 1584 1586

Wavenumber / cm-1

Number-of-layer / N

BOLS BP-A2g mode Graphene-G mode (b)

Figure 2. Theoretical reproduction of the N-dependent (a) bandgap energy^[18,39,40] and (b) Raman shift of the BP^[17]

and graphene.^[1,41]

468 470 472

data1 data2 BOLS

Wavenumber / cm-1

A_2g

(a)

440 445 450

B_2g

0 2 4 6 8

360 370 380 390

P / GPa

A_g¹

465 468 471

Wavenumber / cm-1

BP BOLS

A_g²

435 438 441

444 B_2g

0 150 300 450 600

360 362 364

T / K

A_g¹

(b)

Figure 3. Theoretical reproduction of the T- and P-dependent Ag

1, B2g, and Ag

2 mode wavenumbers for few-layered BP.^[15,42,43]

JRS_4964_F1(a).tif

JRS_4964_F1(b).tif

JRS_4964_F2(a).tif

JRS_4964_F2(b).tif

JRS_4964_F3(a).tif

JRS_4964_F3(b).tif

Raman spectroscopy was employed to investigate bond- phonon-photon relaxation of layered black phosphorus by using the stimuli of number-of- layer, temperature and pressure.

The results demonstrate that (i) atomic under-coordination shortens and stiffens the P-P bond, which raises the B

and A

phonon frequency; (ii) bond thermal elongation and weakening soften all phonon modes; and (iii) bond mechanical compression has the opposite effect of heating on phonon frequency relaxation

Number-of-layer, pressure, and temperature resolved bond-phonon-photon

cooperative relaxation of layered black phosphorus

Y. Liu , X. Yang, M. Bo, X.

Zhang, X. Liu, C. Q. Sun, Y.

Huang