Energy and concentration nonequilibrium and nonlinear charge transport phenomena in semiconductors in a magnetic field in hot electrons approximation

International Journal of Thermal Sciences 137 (2019) 110–120

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Energy and concentration nonequilibrium and nonlinear charge transport phenomena in semiconductors in a magnetic field in hot electrons approximation

T

S. Molina-Valdovinos∗ Universidad Autónoma de Zacatecas, Unidad Académica de Física, Calzada Solidaridad esq. Paseo, La Bufa s/n, CP 98060, Zacatecas, Zac, Mexico

ARTICLE INFO

ABSTRACT

Keywords: Thermoelectric field Bipolar semiconductors Magnetoresistance Electron temperature

When external electric (E x ) and magnetic (Hy ) fields are applied to a semiconductor nonequilibrium charge carriers appear that are redistributed perpendicularly to the fields. We show that under weak recombination the nonequilibrium electron together with the hole concentration and nonequilibrium electron temperature are related by a linear function of the coordinate z. The nonequilibrium electron temperature and electron and the hole concentration have a term proportional to Hy Ex and another term proportional to Ex2 . The nonequilibrium temperature shows a dependence of the z-coordinate and includes the effects of the electron-phonon interaction as well as the absorption of heat over the surfaces of the sample. Additionally, the nonequilibrium temperature and charge carriers modified the density of current from the Ohm law. The redistribution of charge carriers originate two new contributions to the magnetoresistance, one of the contributions is linked to the diffusion of charge carriers along the z-axis, the other one is related to the thickness 2b of the sample and the heat absorption over the surfaces. Another effect that modified the density of current is associated with the absorption of heat over the surfaces of the sample, this contribution is proportional to Hy Ex and can change sign if the electric or magnetic field is inverted. In the case of a null magnetic field, the nonlinear temperature and electron (and hole) concentration are originated by the external electric field and the relaxation mechanism over the surfaces of the sample. The current-voltage characteristic deviates from the linear response due to the nonequilibrium charge carriers.

1. Introduction The Ettingshausen effect [1,2] is an effect that occurs as a manifestation of the Lorentz force on the charge carriers of the sample (metal or semiconductor) [3]. The Ettingshausen effect is linked to the Hall effect, both effects are generated for the Lorentz force. The difference is that in the Ettingshausen effect a temperature difference is measured through the surfaces of the sample in a perpendicular direction to the external electric field and the magnetic field. In the case of the Hall effect, the voltage difference (Hall voltage) is measured. The Ettingshausen effect can be used to cool a material below the ambient temperature [4–8]. When applied to an electric field E x and an orthogonal magnetic field Hy in a rectangular semiconductor, the carriers that travel with a drift velocity v are subjected to the Lorentz force e ( v × H )/c in the zdirection. In this case, the total force over an electron in the z-direction is Fz ( ) = eEz + ev ( ) H /c , where Ez is the Hall field. This force produces

∗

a deflection of the carriers in a direction perpendicular to the drift velocity v ( ) and the magnetic field. Some carriers can be deflected by the magnetic field to accumulate on either the bottom or the top face of the sample in the z-direction. The process continues until the electric field balances the Lorentz force. The Lorentz force will not be balanced across the Hall field due to the drift velocity of carriers which depends on the energy, i.e., v ( ) = µ ( ) Ex = (q ( )/m ) Ex which in turn depends on the relaxation time ( ) = 0 (T )( / T ) q which depends on the energy [9]. In this case, it is necessary to work with the average velocity of the electrons v ( ) , which give the equality in the Lorentz force, i.e., Fz = 0 , then in consequence Ez = v ( ) H / c . This process produces a voltage (Hall voltage) that can be measured through the surfaces of the sample [2,3,10]. When the system is in equilibrium, the chemical potential µ n and µ p for electrons and holes are related by the equation µ n + µ p = g [11]. When the system is placed in presence of an electric and magnetic field, the charge carriers acquire energy from the fields. As the fields increase,

Corresponding author. E-mail address: [email protected].

https://doi.org/10.1016/j.ijthermalsci.2018.11.003 Received 14 May 2018; Received in revised form 28 October 2018; Accepted 1 November 2018 1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.

International Journal of Thermal Sciences 137 (2019) 110–120

S. Molina-Valdovinos

the average energy of carriers also increases, and the system is far from in equilibrium, the relationship µ n + µ p = g is no longer satisfied. The chemical potentials for electrons and holes are out of equilibrium as a consequence of the generation of nonequilibrium charge carriers and due to the heating of the charge carriers. The electrons and holes acquire an effective temperature Tn and Tp respectively, which are higher than the lattice temperature Tph [12,13]. If the semiconductor sample is not thermally insulated at the surfaces, a mechanism for energy relaxation of carriers at the surfaces appears. This mechanism denoted by η and called surface electron heat conductivity, represents the heat flux density carriers by electrons and holes in the walls of the semiconductor [14,15]. If η is finite it means that the surfaces absorb energy. If the system is thermally insulated (adiabatic system) then = 0 . If the surfaces have a good thermal . In this case, the surfaces of the system can be conductivity then considered a heat sink, which removes the thermal energy from the surfaces of the system so that the temperature in the surfaces goes down to the equilibrium temperature T0 . In monopolar semiconductors, electrons with energies higher than the average value ( > ) are deflected and will accumulate in the region of one of the crystal surfaces, i.e, the surface will heat up. While electrons with energies lower than the average value ( < ) (cold electrons) will accumulate on the opposite surface. As a result, the surface cools. The difference of temperatures due to the carrier redistribution generates a transverse temperature gradient (or thermoelectric field). Additionally, the deflection of the electrons deviates the electrical conductivity and generates a dependence proportional to Hy2 . This effect is called classical magnetoresistance effect [2]. The carriers subjected to the fields do not move with identical velocities. The Hall voltage is set up to balance the average velocity, and carriers with velocities that differ from the average would deviate from its original trajectory. These deviations lead to increased resistance [16]. This increase depends on the magnetic field (Hy2 ) and the relaxation mechanism of the electron moment qn . In the case where the collision frequency is independent of the electron energy, i.e., qn = 0 , the classical magnetoresistance for monopolar semiconductors disappears. In bipolar semiconductors, the electrons and holes move in opposite directions in the x-direction under the presence of the external electric field. The magnetic field acts differently on each of the carriers in nondegenerate semiconductors, because each carrier has a different energy. This implies that each of the electrons and holes will have different mobilities. Thus, electrons and holes with energies higher than the average value (hot electrons) are deflected and will accumulate in the region of one of the crystal surfaces, while carriers with energies lower than the average value (cold electrons, cold holes) will accumulate on the opposite surface. As a result, this carrier redistribution generates a transverse temperature gradient T . This phenomenon is known as the Ettingshausen effect [15]. The deviations of the electrons and holes generate partial fluxes of electrons and holes along the Hall field direction; this depletion of the partial fluxes of charges increases the resistance. The increase in magnetoresistance depends on the magnetic field (Hy2 ), the relaxation mechanisms of electron momentum qn and hole momentum qp and the surface relaxation mechanisms η. Under adiabatic conditions and, considering electron-phonon interaction, if the masses of the electron and holes are equal, i.e., m n = m p , and the momentum frequency n = p (qn = qp) , the Hall field disappears ( jnz = jpz ), but the magnetoresistance and the thermoelectric field does not vanish [17]. The transverse temperature gradient T depends on the boundary conditions imposed on the system, surface electron heat conductivity and the interaction mechanisms present in the sample. If we have a system where both surfaces have good thermal conductivity ( ) then the thermoelectric field is proportional to Ex2 . If the system absorbs energy over the surfaces of the sample z = ± b, the thermoelectric field is generated by two contributions that are proportional to Hy Ex and Ex2 respectively [14,17]. A similar behavior happens with the electrical

conductivity. The electrical conductivity changes from the linear response and three new contributions appear. Each contribution is proportional to Hy2 , Hy Ex and Ex2 respectively. The longitudinal electrical conductivity of an isotropic semiconductor is dependent on the thickness b of the sample along the z-axis [18]. In a system that is bounded in the z-direction and boundless in the direction of the electric field, the current density is considered to vanish at the surface z = ± b of the sample, which is viewed as a boundary condition (i.e., jz = jzn + jzp = 0 at z = ± b) [14]. In this case a size term in the magnetoresistance appears. When the transverse dimensions of the semiconductor are very large, the size contribution to the magnetoresistance disappears recovering the classical result [19]. In a finite bipolar semiconductor the redistribution of the nonequilibrium carriers along the z-axis (thickness 2b), the generation and recombination processes assisted by traps (Shockley-Read model), and the different surface recombination velocities generate a change in the conductivity from the classical which depends on the thickness b [16]. In this work, we study the effect of external electric and magnetic fields in bipolar semiconductors. The external fields generate nonequilibrium charge carriers along the z-axis. Under weak recombination, we find a relation between nonequilibrium charge carriers and nonequilibrium electron temperature. Our analysis shows how the temperature and concentration change when we take into account electronphonon interaction, Joule heating or whether both effects are absent. Additionally, the nonequilibrium temperature and charge carriers modified the density of current from the Ohm law. The first change to the current density comes with the appearance of the magnetoresistance effect. The redistributions of carriers originate two new contributions to the magnetoresistance, one of the contributions is linked to the diffusion of charge carriers along the z-axis, the other one is related to the thickness 2b of the sample and the heat absorption over the surfaces. We study the effect over the electrical conductivity due to the absorption of different heat over the surfaces of the sample z = ± b. The absorption of heat over the surfaces generates a new contribution proportional to Hy Ex . Finally, we analyze the effect of the density of current due to the Joules of heat. We study the particular case of null magnetic field Hy = 0 and the effect on the current-voltage characteristic (CVC) that deviates from the linear response due to the Joule heating. 2. Nonequilibrium transport equations and quasineutrality approximation In this paper, we consider a bipolar semiconductor of parallelepiped form, length 2a along the x-axis and transverse length 2b along the zaxis. We assume that the semiconductor contains an impurity concentration Nt with energy t , and that the semiconductor is not degen* erate, with carrier energy given by n,p = p2 /2m n,p , where p is the * * * quasimomentum of particles, m = m n , m p are the electron and hole effective masses, ε is the carrier energy. Finally, the semiconductor is subjected to an external electric field E x along the x-axis and a magnetic field Hy along the y-axis which does not vary over time. Additionally, we considered that the holes do not obtain energy (not heated), so that the temperature of holes is equal to phonon temperature (Tp = Tph ). This can happen, because in semiconductors the effective mass of holes is greater than the effective mass of electrons (m p* > m n*) , and the electron-phonon interaction is more quasielastic. This implies that electrons acquire more energy (hot electrons), and their temperature increases above the phonon temperature, i.e., Te > Tph . By contrast, the hole-phonon interaction is less quasielastic, which implies that the hole temperature does not change. In this sense, it is only necessary to work with the energy balance for electrons. The transport phenomena in semiconductors are performed using the continuity equations for electrons (holes), Poisson equation and the energy balance equation for the nonequilibrium charge carriers [15,17,20]. We consider the static case and absence of the external 111

International Journal of Thermal Sciences 137 (2019) 110–120

S. Molina-Valdovinos

generation of carriers (by light or other mechanisms):

j =e R ,

4

E =

jz =±

= n, p

Qn + n

n

T0) = Jn E +

change with n the electron energy relaxation frequency [21], Jn E is the Joule effect, v Eg Rn is the energy due to volumetric recombination, Eg the energy gap and v is the efficiency of the thermal power density generated in the electron system due to recombination process, T = Tn, Tp are the electron and hole temperature and T0 is the temperature of the phonon system wich represents the ambient temperature due to the high thermal conductivity of phonons as compared with the electron thermal conductivity (Tph = T0 ) [15,21]. In general the electron Tn , hole Tp and phonon Tph temperatures are different [22–24]. But, let us restrict to the case where the hole and phonon temperature are equal Tph = Th = T0 . The temperature (for electrons, holes, and phonons) is expressed in terms of energy units throughout the paper. It means that temperature is multiplied by the Boltzmann constant kB . Let us write the electron temperature Tn , the distributions of electron (holes) chemical potential ( µ ) and electron and hole concentrations (β) in the following form,

µ =µ 0

0

+ µ (z ),

(5)

+

(z ),

(6)

( , T )/ T

(z ) +

0

µ

0

T0

3 2

T (z ),

x

=

0

+

1

0 xx E x

0 xz H

Hy

1 2 2 xx H Hy Ex z

T,

0 xz H

=

0 xz

=

0 zx

=

0 xz

=

0 zz

=

r=

+ 5/2) Tn

z

n (z )

n0 zz (qn

+ 5/2) Tn Ez

n0 zz (qn

+ 2)(qn + 5/2) Tn

z

Tn (z ),

4e 2

0

m*

3

4e 2

0

m*

3

4e 2

0 xz ,

(5/2 + 3q ), 0

0

m*

3

(5/2 + q ), 0

0 zz

(5/2 + 2q ), 0

=

3

4eT0 m*

0

3

4eT0 m*

0

4e

4e

=

1 xx

(5/2 + q ),

(2q + 1) (5/2 + q ), 0

0

m*

3

1 zz

(5/2 + 2q ),

0

m*

3

0 xx ,

(q + 1) (5/2 + q ), 0

1 n 0 p0 n 0 + p0

p n + + r Tn , n0 p0

(12)

n10 p0

1 n (T0 )

n

Tn

,

(13)

Tn = T0

and τ is the effective life time of electrons-hole pairs of doped semiconductors.

1

(8)

Hy Ez + Dxz0 H Hy

T,

where

It is important to note that the nonequilibrium chemical potentials are related with the appearance of nonequilibrium concentrations of charge carriers and nonequilibrium electron(hole) temperatures. For the model we are considering the presence of a unique relaxation mechanism of the electron and hole momentum, the frequency can be written as ( ) = 0 (T0 )( /T ) q , where q are the characteristics of the relaxation mechanism [9]. Additionally, we also consider a weak 1, where H = e Hy / m *c transverse magnetic field, such that H 0 are the cyclotron frequencies for electrons and holes, and 0 = 1/ 0 . The density of current of electrons and holes in presence of electric and magnetic fields are given by Ref. [14]:

j

1 xx

R=

1

T0

0 zz z

Here, (iq + k + 5/2) is the gamma function, and c is the velocity of light in vacuum. In the case of a semiconductor that contains an impurity concentration Nt , the generation and recombination processes are assisted by traps (Shockley-Read model). The recombination rate through the traps can be written as follows [13]:

Here N (T ) = 4 (m T /2 3 2)3/2 are the electron and hole densities of states. Expanding equation (7) we get

µ (z ) = ±

=

Dzz0 =

(7)

.

0 xx

Dxz0 =

where µ (z ) denotes the nonequilibrium chemical potentials for elec(z ) are electron and hole nonequilibrium controns and holes, centrations, Tn (z ) are the nonequilibrium electron temperatures, µ 0 are the equilibrium electron and hole chemical potentials, 0 = n 0 , p0 are the equilibrium electron and hole concentrations. Electron and hole concentrations in bipolar semiconductors are represented by the following expression [2],

(z ) = N (T )e µ

z

(11)

tron and hole recombination rate, Qn are the electron heat flux, n n (Tn )(Tn T0 ) describes the intensity of electron-phonon energy ex-

(4)

Dzz0

with

/ z ) , where φ is the material permittivity, E = (Ex , 0, Ez) = (Ex0, 0, electric potential, Ex0 is the fixed external electric field and Ez is the electric field (the Hall field) variable in the z direction, R is the elec-

Tn = T0 + Tn (z ),

1 2 2 zz H Hy Ez

+ 5/2) Hn Tn Hy E x +

n0 Dzz (qn

where β can take the value n for electron or p for holes, ρ is the bulk charge density, e = e, e are the charge for electrons and holes, ε is the

=

n0 xz (2qn

eQnz =

(3)

v Eg Rn ,

0 zz Ez

H = |e| 0/ m *c = |e|/ m *c 0 represents the mobility for electrons and holes per light velocity. The electron heat flux in the z-axis [2,14],

(2)

(Tn)(Tn

Hy E x +

(10)

(1)

,

0 zx H

=

n (T0 ) p (T0 ) Nt (n 0 + p0 ) + n10) + p (T0)(p +

n (T0 )(n

p10 )

.

(14)

Here, n (T0, Tn ) and p (T0, Tp ) are the electron and hole capture coefficients, n1 (p1 ) is the electron (hole) concentration when the Fermi level matches the activation energy of the impurity, n1 = n (Tn )exp [ t / Tn]; p1 = n (Tn )exp [ t g / Tn]. In general it is very difficult to solve equations (1)–(3), because it is a system of coupled nonlinear equations. To solve this system of equations we used the concept of quasineutrality [25]. Traditionally it lD2, d 2 ; where rD is the Debye radius, lD the diffusion is defined as rD2 length, d the thickness of the sample. It can be readily seen from 2 = 4 1, / , that / 0 = (rD/lD)2 . If (rD/lD)2 Poisson's equation, 0 . Under this condition the Poisson equation becomes an then algebraic equation that does not need boundary conditions. For a semiconductor with an impurity concentration Nt and from the quasie( p n nt ) 0 with the expression for neutrality condition

z

(9) 112

International Journal of Thermal Sciences 137 (2019) 110–120

S. Molina-Valdovinos

n t given in Refs. [10,13], the nonequilibrium charge carrier concentrations n , p and the temperature Tn , are related by

5

=

(15)

6

=

p=

n+

1

Tn

2

2 (5/2

3

+ qn ) T0 e 1, n0 xx

+

(5/2 + qn ) T0 e 1.

e

where 1

2

n (n 0

=

+ n10 + Nt

nt0) +

0 n (n 0 + n1 ) +

=

n (n 0

n 0 (Nt + n10) +

p (p0 +

p (p0 p10 +

nt0)

n10 n t0 0 p (p0 + p1

+ p10 )

2.1. Solutions of transport equations: the case of weak recombination

,

nt0)

n (T0 )

+ nt0)

Tn

Let us now consider that the volumetric recombination is weak. The weak recombination (R = 0 ) happens if t and kt for recombination rate through the traps. It implies that there is not energy generated by recombination processes. Under this condition, the continuity equations for electrons (and holes) and the energy balance equation transform into,

,

In this case, the recombination rate can be rewritten as:

n

R=

+

Tn

t

,

(16)

t

jn,p = 0,

where

1 t

1

1 n 0 p0 = n 0 + p0

1 n 0 p0 n 0 + p0

=

t

Qn + n

1 + 1 , n0 p0 2

1

n p

bb

(T )

1

n0 xz Hn n0 xx

+

n0 zz n0 xx

p0 xz Hp p0 xx

+

p0 2 Dzz p0 xx

+

Hy E x +

n0 Dzz

Tn , z

n0 xx

+

p0 1 Dzz p0 xx

Qnz =

1 Hy Ex

2

4 Hy Ex

5

n z

3

n z

6

Tn , z

2

=

=

3

=

4

=(

p0 n0 xx xz Hn n0 xx

+ +

n z

+

Tn (z ) = Az

B,

jnz =

1 Hy Ex

(23)

Qnz =

4 Hy Ex

(24)

2 A,

5A

Tn , z

a0

here a0 = ( 6 2 5 3)/ 2 = current along the x-axis is.

(25)

n0 xx (qn

+ 5/2) T0 e . The total density of 2

jx = jnx + jpx =

2 1 Hy

0 (1 4 Hy

+

2

n (z ) +

5 Hy

n (z ) z

Ex

Tn (z ) z

Ex

3

Tn (z )

Ex,

(26)

and 0

=

1

=

2

=

3

=

4

=

5

=

(18) (19)

,

p0 n0 n0 p0 xx zz + 2 xx Dzz , n0 p0 xx + xx

n0 xz Hn qn

(22)

where the constants A and B are determined with appropriate boundary conditions. This result shows that the nonequilibrium electron temperature appears as a consequence of the redistribution of carrier concentrations along to z-axis. By substitution of Eq. (23) into Eqs. (18) and (19), the density of current and the electron heat flux take the following form:

n0 p0 xx xz Hp , p0 xx

n0 p0 p0 n0 xx Dzz 1 + xx Dzz n0 p0 xx + xx

Tn = 0, z2

2

where 1

3

n (z ) +

(17)

Tn , z

3

bb

The first term of Ez represents the classical Hall field, the second term is a new contribution due to diffusion of the electrons and holes along the z-axis, this diffusion gives regions with low and high carrier concentrations. The third term represents the thermoelectric field generated as a consequence of the difference of energy between regions with high and low carrier concentrations. By substituting equation (17) into equations (10) and (11), we rewrite the density of current for electrons and the electron heat flux:

jnz =

2

n + z2

2

This condition gives a relationship between the nonequilibrium concentration and the nonequilibrium temperature,

0 0 0 = [ (n 0 + p0 )] 1, bb1 = . Tn bb (n 0 + p0 ) From the continuity equation (1) for electrons and holes, we obtain j = ( jn + jp ) = 0 , i.e. jnz + jpz = jz = cte . From the geometry of the Hall experiment, we can see that when there is no current flow in the z-direction, then the constant is zero at all points of the sample, i.e. jnz + jpz = jz = 0 . Using this condition, we obtain the total electric field Ez , and by considering the quasineutrality condition ( p = 1 n + 2 Tn ), we obtain

bb

(21)

T0 ) = Jn E .

2

jn = 0

If the semiconductor is dominated by band-band recombination p . And the bandprocesses, the quasineutrality condition gives n n T + n , with band recombination rate transforms into: R =

Ez =

(Tn )(Tn

It is important to mention that under quasineutrality approximation we only work with one of the continuity equations for electrons or holes. From equations (18) and (20),

+r ,

p0

n

(20)

1 1 (qn + 5/2)) T0 e ,

113

n0 xx

+

p0 xx ,

n1 2 xx Hn n0 ( xx

+

(

(

+

(

n0 xz Hn n0 ( xx

+

2 p0 xz Hp ) , p0 2 xx )

n0 p0 xx /n 0 ) + 1 ( xx / p0 ) , n0 p0 ( xx + xx )

(qn

(

p1 2 xx Hp p0 xx )

n0 xx / T0 ) n0 ( xx

+ +

n0 xz Hn

p0 2 ( xx / p0 ) , p0 xx )

p0 Dzz 1)

(

p0 n0 xz Hp)(Dzz n0 p0 2 xx + xx )

p0 Dzz 2)

(

p0 n0 xz Hp )( zz n0 p0 2 xx + xx )

n0 xz Hn

n0 (Dxz Hn +

( (

n0 xx

+

p0 1 Dxz Hp ) , p0 xx )

n0 p0 xz Hn + 2 Dxz Hp) , n0 p0 ( xx + xx )

International Journal of Thermal Sciences 137 (2019) 110–120

S. Molina-Valdovinos

It is known that the nonequilibrium electron concentration generates a new contribution in the magnetoresistance [16]. In the particular case of null recombination processes, the nonequilibrium temperature and electron concentration are algebraically related by equation (23). Then the electron temperature generates a new contribution to the magnetoresistance. To see this let us calculate the average value of the current density over the semiconductor cross section,

Jx =

1 2b

b

j dz b x

=

1 2b

b b

(jnx + jpx ) dz =

(Ex , Hy ) Ex ,

Qnz |z =±b = ±

1,2 represents the contribution of inelastic scattering of electrons at the boundaries; 1,2 = 0 corresponds to the absence of the surface mechanisms, and 1,2 means a good thermal conductivity through the surface. Applying the boundary conditions (34), we obtain the nonequilibrium electron temperature,

Tn (z ) = Hy E x [ 1 sinh(z / l ) +

(27)

2 1 Hy

0 [1 4 Hy

2bEx

2

n (z ) +

( n (b )

n ( b))

( Tn (b)

Tn ( b))

2bEx 5 Hy

+

3

Tn (z ) , (28)

n (z ) =

where Tn (z ) is the average electron temperature, n (z ) is the average electron concentration, and 0 represents the linear conductivity due to the electrons and hole carriers, 1 represents the classical coefficient of magnetoresistance. The terms 2 and 3 represent the contribution to the density of current due to change in nonequilibrium electron concentration and nonequilibrium temperature. 4 and 5 are a contribution due to the spatial variation of redistribution of nonequilibrium temperature of electrons. Now, the problem is reduced to solving Eq. (21) and finding the nonequilibrium electron temperature. In the following sections, we solve it for different cases and boundary conditions.

2 3 EEx 2

l

2

Tn (z ) =

n0 2 xx Ex

a0

,

1e

z/ l

+

2e

z /l

+ EEx2,

(29)

(30)

l 2 e 2 /(qn

(31)

jnz |z =±b = 0, From equations (24) and (31), we get

A=

1

Hy E x ,

(32)

2

a0

Tn , z

[1

2

4

= (a 0/ l )(

1

= a1 [(

2

= a1 (

3

= [2

4

= (a 0 /l )(

1

1

+

+

4

2)cosh(2b /l

2 )sinh(b / l

1 2 sinh(b /l

(35)

sinh(z / l )],

sinh(z / l )] and

1

cosh(z / l )

4

3

z

B,

(36)

are:

) + ((a 0 /l )2 +

1 2 )sinh(2b / l

(37)

),

(38)

) + 2(a0 / l )sinh(b/ l )]/ 0,

(39)

)/ 0,

) + (a 0 /l )(

2 )sinh(b/ l

1

2

1, 2 , 3

2)cosh(b/ l

1

3 2

sinh(z /l )

cosh(z /l )

3

0

2l

Hy Ex

(E x , Hy ) =

The electron heat flux (25) takes the following form,

Qnz = a1 Hy E x

cosh(z / l )

)]

1

+

2 )cosh(b/ l

(40)

)]/ 0,

(41)

)/ 0,

sinh(b /l ) b

+ EEx2 1

3l

sinh(b /l ) b

,

(42)

Fig. 1 shows the nonequilibrium electron temperature Tn (z ) versus thickness (z) in absence of volumetric and surface recombination processes. We consider a semiconductor sample with numerical values m p = 0.2m 0 , given by T0/kB = 200 K , sample with b = 0.02 cm, m n 6 6 m 0 = 9.11 × 10 31kg, p0 = 2.48 × 10 s, n0 = 2.48 × 10 s, c = 3.0 × 108 ms−1, Nt = 0 , n 0 p0 = 1.45 × 1014 cm−3, E x = 0.5 Vcm−1, Hy = 20000 Oe, qn = qp = 1/2 , kB = 1.38066 × 10 23J −1 K [3,26,27]. The graphs show the relationship between the thickness b and the thermal diffusion length l . The figures represent the cases of a) b/ l = 1, b) b/ l = 2.2, c) b/ l = 3.3, d) b/ l = 5, e) b/ l = 10 , and f) b/ l = 25. The dot-dashed purple, green, blue, red, brown, cyan and black lines represent the cases i) 1 = 2 = 0 , ii) 1 = 2 = 0.0001, iii) 1 = 1, 2 = 0 , iv) 1 = 0, 2 = 1, v) 1 = 2 = 1, vi) 1 = 2 = 100 and vii) 1 = 2 respectively. By substitution of equations (23), (35) and (36) into (28), we can compute the average value of the current density Jx = (Ex , Hy ) E x , where the conductivity is given by

= + 5/2) T0 . From equations (23) and (30) we here E = have four unknown constants A, B, 1, and 2 . If we consider the condition of open-circuit, and leave aside the surface recombination processes, the electron current densities at the two semiconductor boundaries are [14], n0 2 xx l / a 0

3 1

Hy E x

Tn (z ) =

with l = a0 / n n (Tn ) is the thermal diffusion length. The analytic solution for Eq. (29) is given by:

Tn (z ) =

2 cosh(z / l

The constant B can be determined by considering that n (z ) = 0 , which means there is no generation of charge carriers along the z-axis. From (23) we have B = 3 Tn (z )/ 2 . In this case, the average temperature is given by the expression:

Consider that in our system the electrons are being heated by the interaction of the external electric field with the electrons (Joule heating) and lost energy due to electron-phonon interaction. Under weak recombination R = 0 , we do not have energy due to the recombination processes. If we take into account these mechanisms, the energy balance equation transforms into:

Tn (z ) z2

3

where the coefficients

3. Results and discussion

2

EEx2 [1

The electron temperature is generated by two contributions. The first contribution arises when we apply a magnetic field perpendicular to the electric field and it is proportional to Hy Ex . The magnetic field separates the electrons with high energy and low energy and piles up over the surfaces of the sample. The second contribution is linked to the Joule heating, EEx2 . From Eq. (23), we obtain the nonequilibrium electron concentration,

here

(E x , Hy ) =

(34)

Tn (± b),

1,2

(33)

n0 1 here a1 = ( 4 2 5 1)/ 2 = xz (qn + 5/2) T0 e . If we consider the absorption of the carrier energy at the boundaries of the semiconductor are z = ± b. The boundary conditions can be written in the form [14,15],

0

a2 2 l sinh(b / l ) b

+

0

+

0 a2

1

2 1 Hy

1 +

a3 E 4 sinh(b / l ) b

3 l sinh(b / l )

b

a3 1 sinh(b / l ) 2 Hy b

1 4 3

Hy2

Ex Hy

EEx2,

(43)

where a2 = ( 3 2 2 2 3)/ 2 , a3 = ( 5 2 describes the 4 3)/ 2 , classical effect of the magnetoresistance, a3 1 sinh(b/ l ) Hy2 / b represents a new contribution that depends on the thickness of the sample and the boundary conditions and 1 4 Hy2 / 3 is a new contribution due to the 2 1 Hy

114

International Journal of Thermal Sciences 137 (2019) 110–120

S. Molina-Valdovinos

Fig. 1. Variation of nonequilibrium electron temperature along the z-axis. The graphs show the relationship between the thickness b and the thermal diffusion length l : a) b/ l = 1, b) b/ l = 2.2 , c) b/ l = 3.3 , d) b/ l = 5, e) b/ l = 10 , and f) b/ l = 25. The dot-dashed purple, green, blue, red, brown, cyan and black lines represent the cases i) 1 = 2 = 0 , ii) 1 = 2 = 0.0001, iii) 1 = 1, 2 = 0 , iv) 1 = 0, 2 = 1, v) 1 = 2 = 1, vi) 1 = 2 = 100 and vii) 1 = 2 respectively.

diffusion of charge carriers along the z-axis. The term proportional to EEx2 represents the effect of the Joule heat on the electrical conductivity. The term proportional to E x Hy appears due to the fact that the absorption of heat on the surfaces of the sample are different 1 2 (this contribution is similar to the nonreciprocity effect studied in Refs. [14,15]). In the same way, if 1 > 2 , this contribution increases the electrical conductivity. While if 1 < 2 , the electrical conductivity diminishes (with fixed electric and magnetic fields). If 1 = 2 then the coefficients 2 = 4 = 0 are equal to zero and in equation (43) the term proportional to E x Hy disappears. Magnetoresistance effect in low magnetic fields were experimentally studied in p-Ge [28], n-Si [29], GaAs [30], n-InSb [31]. Nonreciprocity effect was experimental studied in p-Ge [28]. Fig. 2 shows the electrical conductivity versus electric field. a) n 0 = p0 = 1.45 × 1014 cm−3, m n = m p = 0.26m 0 , 2 = 0, 1 = 1, Hy = 20000 Oe and b) n 0 = 1.45 × 1014 cm−3, p0 = 1.45 × 106 cm−3, m n = 0.26m 0 , m p = 0.6m 0 , 1 = 1, 2 = 0 , Hy = 20000 Oe. The dot-dashed purple, green, blue, red, brown, cyan and black lines represent the cases i) b/ l = 1, ii) b/ l = 2.2, iii) b/ l = 3.3, iv) b/ l = 5, v) b/ l = 10 , and vi) b/ l = 25. If n 0 = p0 then a2 < 0 and the electrical conductivity is sublineal but when n 0 p0 then a2 > 0 and the electrical conductivity n 0 = p0 = 1.45 × 1014 cm−3, change to superlinear. In c) m n = m p = 0.2m 0 , 1 = 1, 2 = 0 , Hy = 20000 Oe, b/ l = 5 and d) n 0 = 1.45 × 1014 cm−3, p0 = 1.45 × 106 cm−3, m n = 0.26m 0 , m p = 0.6m 0 , 1 = 1, 2 = 0 , Hy = 20000 Oe, b/ l = 5. The purple, green, blue and red lines sow the cases i) 1 = 2 = 0 , ii) 1 = 2 = 0.0001, iii)

= 2 = 1, iv) 1 = 2 , and v) l respectively. In the case of an adiabatic system 1 = 2 = 0 , it fixes the coefficients 2 = 3 = 4 = 0 , which correspond to the case when no heat flows through the surfaces (z = ± b) of the sample. With this consideration, the boundary conditions transform into Qn |z = ± b = 0 . The nonequilibrium electron temperature is: 1

Tn (z ) =

1 Hy Ex

sinh(z / l ) + EEx2,

(44)

The variation of the electron concentration is

n (z ) =

3 1 Hy Ex 2

sinh(z / l ) +

1 3

Hy E x z

2

3 2

EEx2,

(45)

with 1

=

a1 l , a 0 cosh(b/ l )

(46)

These results show that it is not necessary to have generation or recombination processes over the surfaces of the sample to have heating or cooling on the surfaces. We can have heating or cooling as a consequence of the redistribution of carriers along the z-axis and the energy acquired by the charge carriers due to the electric and magnetic fields. For an adiabatic system 1 = 2 = 0 in absence of volumetric and surface recombination processes. The dot-dashed purple line in Fig. 1a–f shows the relationship between the thickness b and the thermal diffusion length l . At z = 0 the nonequilibrium electron concentration is n (0) = 2 3 EEx2/ 2 , and the nonequilibrium electron 115

International Journal of Thermal Sciences 137 (2019) 110–120

S. Molina-Valdovinos

Fig. 2. Electrical conductivity versus electric field. a) n 0 = p0 and b) n 0 p0 .The dot-dashed purple, green, blue, red, brown, cyan and black lines represent the cases i) b/ l = 1, ii) b/ l = 2.2 , iii) b/ l = 3.3 , iv) b/ l = 5, v) b/ l = 10 , and vi) b/ l = 25 respectively. c) n 0 = p0 and d) n 0 p0 . The black, red, green, and blue lines represent , and iv) l respectively. the cases i) 1 = 2 = 0 , ii) 1 = 2 = 0.0001, iii) 1 = 2

temperature is Tn (0) = EEx2 . The electron temperature at z = ± b takes the maximum and minimum value respectively. The difference of temperature between the surfaces of the sample is 0 the difference of temperature is Tn = 2 1 Hy E x sinh(b / l ) , when b zero, i.e., the temperature is homogeneous in all the sample, Tn = T0 (equilibrium temperature). In the case of the sample thickness b less than the thermal diffusion length, the nonequilibrium electron temperature is linear (see Fig. 1a, b/ l = 1, dot-dashed purple line). If we 1, then Tn (z ) a1 Hy Ex z /l a0 + EEx2 , the electron take the limit b/ l temperature is linear tends to horizontal line (dashed brown line). It means that the Joule heat is responsible for uniformly heating all the 1 the electron temperature is consample. In the opposite limit b/ l stant along all the sample until distances of the order of l 1 near to sample surfaces (see Fig. 1c–d, b/ l = 10,25, dot-dashed purple line). In the surface z = b the electrons with high energy tend to accumulate and heat the surface. In the opposite surface z = b the electrons with low energy tend to accumulate and cooling the surface. The electron heat flux along the z-axis takes the form:

Qnz = a1 1

cosh(z /l ) Hy Ex , cosh(b/ l )

(E x , Hy ) =

0

2 1 Hy

1

a3 a1 tanh(b/ l ) 2 Hy a 0 (b / l )

1 4

Hy2 +

2 0 a2 EEx ,

3

(48) Note that 1 describes the typical magnetoresistance for bipolar semiconductors, and 1 4 describes a new contribution to the magne3

toresistance due to the redistribution of the nonequilibrium carriers. In p and if qn = qp = 0 ( ( ), the case of an intrinsic semiconductor n it is independent of ε). The magnetoresistance coefficients 1 and 1 4 / 3 do not disappear, taking values: 1

= e 2 / c 2m n

n0 m p p0

=

(49)

1 4 / 3,

In this case, the magnetoresistance coefficients

1

and

1 4 3

cancel

each other out. The coefficient associated to the Joule heating 3 = 0 is canceled. The contribution to the magnetoresiatance due to the current density is Jx =

0

1

a3 a1 tanh(b / l ) 2 Hy a 0 (b / l )

E x . Note that in the classical

theory of magnetoresistance, this does not happen. Here we can see that a a tanh(b / l ) the contribution to the magnetoresistance given by 3 1a (b / l ) Hy2 de0 pends on the thickness 2b of the sample. In the case of tiny samples, i.e., 0 ) the contribution to the when the size of the sample goes to zero (b a a tanh(b / l ) a3 a1 2 Hy . In the opposite magnetoresistance is preserved 3 1a (b / l ) Hy2 a0 0 case if the sample has a large transverse section b then the cona a tanh(b / l ) 0. tribution to magnetoresistance is null 3 1a (b / l ) Hy2 0 On the other hand, when the frequencies of the electrons and holes are equal, i.e. n0 = p0 (qn = qp) , and the masses of the electrons and n1 p1 n0 p0 n0 p0 = xx = xx = xz holes are equal, m n = m p , it follows that xx , xx , xz , n1 p1 n0 p0 n0 p0 Dzz = Dzz , Dzz = Dzz , Dxz = Dxz . The coefficients of the magnetoresistance 1 and 1 4 do not cancel each other and take the form of

(47)

1, the electron heath flux is canceled in all the In the limit b/ l 1 the electron heath flux sample, Qn (z ) 0 . In the opposite limit b/ l is constant (Qnz a1 Hy Ex ) along all the sample until distances of the order of l 1 near to sample surfaces. At z = ± b the electron heat flux is canceled. This happens due to the fact that the electron diffusion and the electron heat flux compensate the heating generated due to the deflection of the charge carriers generated by the magnetic field along the z-axis. The electrical conductivity over the semiconductor cross section changes by,

3

n0 n0 n0 n0 n1 n0 = xx / xx , and 1 4 = xz Dxz / xx Dzz . From Eq. (17) it can be seen 3 that the electric field along the z-axis is 1

116

International Journal of Thermal Sciences 137 (2019) 110–120

S. Molina-Valdovinos

Ez =

2

n0 zz n0 xx

n0 a E H cosh(z / l ) Tn (z ) 1 x y = zzn0 , z a 0 cosh(b/ l ) 2 xx

The thermoelectric field is generated by the redistribution of charge carriers and by the electric field Ex2 , contrary to the thermoelectric field found in equation (50), which was dependent on E x Hy . Finally, we analyze the case when we have an energy relaxation mechanism at the surfaces of the sample 1 2 and the electric field does not heat. As a consequence, the Joule power is approximately zero, n0 2 1[14,15]. From (25) the 0 . This condition is valid when EEx2 xx Ex energy balance equation is transformed to become

(50)

Here we have a thermoelectric field generated by the redistribution of charge carriers. This result is contrary to the case when we do not consider the effect of temperature of the charge carriers where the electric field is canceled under the same conditions [16]. Another interesting case happens when a surface of the sample has good thermal conductivity ( 1 ) and the other surface has a normal thermal conductivity, ( 2 is finite). This means that the surface with good thermal conductivity dissipates the heat over the surface quickly. The nonequilibrium electron temperature and electron concentration have the same form as equations (35) and (36), but the coefficients 1, 2 , 3 and 4 are transformed to become

= (a 0/ l )cosh(2b/ l ) +

sinh(2b/ l ),

1

= a1 cosh(b/ l )/ 0,

(52)

2

= a1 sinh(b / l )/ 0,

(53)

3

= [2 2sinh(b /l ) + (a 0 /l )cosh(b / l )]/ 0,

(54)

4

= (a 0 /l )sinh(b / l )/ 0,

(55)

Tn (z ) =

EEx2

1

n (z ) =

1 cosh(b / l )

1

Hy Ex z +

3

+

2

2 EEx2,

0

1 4

2 1 Hy

1

Hy2 +

3

0 a2 1

(57)

tanh(b / l ) (b / l )

1

EEx2

0 and if b

3

2

n0 2 zz Ex

2a 0

l sinh(z / l ) , cosh(b/ l )

2

cosh(z /l )

z ,

(62)

2 1 Hy

1

0

a2 2 sinh(b / l ) (b / l )

+

a3 1 sinh(b / l ) 2 Hy b

a3 E 4 sinh(b / l ) b

1 4 3

Hy2

E x Hy ,

(63)

=

each

other

and

n0 n0 n0 n0 xz Dxz / xx Dzz .

take

the

form

of

1

=

3

n1 n0 xx / xx ,

and

From Eq. (17) it can be seen that the electric

0 (l If we have no electron-phonon interaction, then 0 ). In this particular case the energy balance equation transform to become,

1 4 3

2

is constant, ( 0 a2 EEx2 ).

Tn (z ) = z2

here a0 = ( by

Tn (z ) =

n0 2 xx Ex

a0 6 2

n0 2 xx Ex

a0

,

(64)

5 3)/ 2

=

n0 xx (qn

+ 5/2) T0 e . The solution is given

z2 + Cz + D, 2

2

(65)

The same solution can be obtained from Eq. (35), if we take the limit 0 , then the nonequilibrium electron temperature is, when z / l , b/ l

n0 n0 n0 n0 = xz Dxz / xx Dzz . From Eq. (17) it can be seen that the electric 3 field along the z-axis is

Tn (z ) = z

3 2

3.1. Null relaxation frequency

3 1 4

n0 zz n0 xx

3

sinh(z / l )

field along the z-axis is zero, Ez = 0 .

When the frequencies of the electrons and holes are equal, i.e. n0 = p0 (qn = qp) , and the masses of the electrons and holes are equal, n1 p1 n0 p0 n0 p0 n0 p0 = xx = xx = xz = Dzz m n = m p , it follows that xx , xx , xz , Dzz , n1 p1 n0 p0 Dzz = Dzz = Dxz , Dxz . The coefficients of the magnetoresistance 1 and n1 n0 1 4 do not cancel each other and take the form of 1 = xx / xx , and

Ez =

1

2 (b / l )

0

1 4

Hy2 as discussed previously. The contribution to the electrical conductivity due to b 0 then Joule heating has a size dependence, if 0 a2

3 2 sinh(b / l )

cancel

(58) 1 and

2

3

tanh(b/ l ) EEx2, (b / l )

The magnetoresistance has two contributions

(61)

cosh(z / l )],

redistribution of charge carriers along the z-axis, but as the surface dissipates the heat quickly the temperature goes to zero. When the frequencies of the electrons and holes are equal, i.e. n0 = p0 (qn = qp) , and the masses of the electrons and holes are equal, m n = m p . This n1 p1 n1 p1 n0 p0 n0 p0 n0 p0 = xx = Dzz = xx = xz = Dzz implies that xx , xx , xz , Dzz , Dzz , n0 p0 Dxz = Dxz . The coefficients of the magnetoresistance 1 and 1 4 do not

Unlike the electron temperature, the electron concentration is generated by Hy Ex and the Joule heat Ex2 . The electrical conductivity takes the form

(E x , Hy ) =

2

The analysis of the magnetoresistance is the same discussed previously. In the case of 1 = 2 the coefficient 2 = 4 = 0 . If we have an adiabatic system, we understand that 1 = 2 = 0 . It follows from this that 2 = 3 = 4 = 0 and 1 = a1 l / a0 cosh(b / l ) . If both surfaces have good thermal conductivity, it is understood , thus the coefficients 1 = 2 = 4 = 0 and the nonethat 1,2 quilibrium electron temperature is zero, Tn (z ) = 0 , but the nonequilinrium electron concentration is linear, n (z ) = 1 Hy E x z . There is

).

cosh(z /l ) tanh(b/ l ) + cosh(b/ l ) (b / l )

3

3 1

Hy Ex

(E x , Hy ) =

The nonequilibrium electron concentration takes the form

n (z ) =

(60)

The average value of the conductivity is given by

The nonequilibrium temperature is not affected by the magnetic field. In Fig. 1a–f, the brown, cyan and black lines show the cases v) . The nonequilibrium 1 = 2 = 1, vi) 1 = 2 = 100 and vii) 1 = 2 electron temperature is zero at z = ± b. This means that both surfaces dissipate energy quickly with the surrounding medias and thermalize to the equilibrium temperature T0 . The maximum temperature value is

(

Tn (z ) = 0,

where the coefficients 1, 2 are given by equations (38) and (39). The coefficient 2 is less than 1and the nonequilibrium temperature is antisymmetric. From Eq. (23), we obtain the nonequilibrium electron concentration,

(56)

given at z = 0 , Tn (z = 0) = EEx2 1

2

l

Tn (z ) = Hy Ex [ 1 sinh(z /l )

In the inverse case, if 2 and 1 is finite, the coefficients 1 and a1sinh(b/ l )/ 0 and 2 are the same as presented above and 2 = a 0 sinh(b/ l )/l 0 . 4 = , then the If both surfaces have good thermal conductivity 1,2 coefficients 1 = 2 = 4 = 0 and 3 = 1/cosh(b / l ) and the nonequilibrium electron temperature is transformed to become

cosh(z / l ) , cosh(b/ l )

Tn (z ) z2

The nonequilibrium electron temperature changes to become:

(51)

0

2

2

Tn (z ) =

(59) 117

n0 2 xx Ex

a0

z2 + 2

1 Hy E x z

2 Hy E x ,

(66)

International Journal of Thermal Sciences 137 (2019) 110–120

S. Molina-Valdovinos

to infinity, Tn (z )

.

3.2. Current voltage characteristic (CVC) without Hy In the case of Hy = 0 , the Joule heat produced in the sample combined with the absorption of heat through the surfaces (z = ± b) can generate nonequilibrium charge carriers. This mechanism generates nonlinearity in the current-voltage characteristic (CVC) [32,33]. There are other mechanisms that generated nonlinearity in CVC in bipolar semiconductors like as impact ionization [32,34], carrier lifetime changes [35], intervalley redistribution of carriers [36]. Another mechanism that generates nonlinearity in CVC is due to metal-semiconductor contact, for example, Mott's law [37], Schottky barrier [38,39], carrier injection in p-n contact [32,40], etc. The importance of hot electrons come with the practical applications to semiconductor devices [22,41], for example in solar cells [42], electrical and thermal properties of chip-level high-power GaN-based light emitting diodes [43], dual-gated bilayer graphene hot electrons bolometers [44]. From Eqs. (35) and (36), we can obtain the nonequilibrium electron temperature,

Fig. 3. Nonequilibrium electron temperature ( Tn (z ) ) versus z in absence of volumetric and surface recombination processes. The purple, red and blue lines represent the cases a) b) and c) 1 = 2 = 1, 1 = 1, 2 = 0, 1 = 0, 2 = 1respectively.

The parabolic temperature is due to the Joule heating and the linear contribution is due to the heat generated by Hy Ex . By substitution of Eq. (66) into Eq. (23), we obtain this nonequilibrium concentration, n0 2 3 xx Ex

n (z ) =

2a 0

b2 + 3

z2

2

3

Here above the coefficients 1 = a1 ( 2

1

= a1 b (

+

2)/(a 0 (

1

+

2 )/(a 0 ( 1

1

2)

Hy E x z ,

2

1and

2 ) + 2b

+

3 1

1

1

+ 2b

Tn (z ) = [1

(67)

+

are transformed into

2

(68)

1 2 ),

(69)

(E x , Hy ) = +

0 2 0

a2 +

n0 b2 a2 xx

6a0

0

1

n0 a3 xx

a1

a3 1 Hy2

1 4 3

3

3 2

[ 3cosh(z /l ) +

sinh(b/ l )/(b /l )

Jx =

Hy2

sinh(z / l )] EEx2,

(71)

4 sinh(z / l

)

2] EEx2,

(72)

0

1 + a2 1

3sinh(b / l

(b / l )

)

EEx2 Ex ,

(73)

The excess of nonequilibrium carriers generated by the heating of sinh(b / l ) electrons generates a new contribution ( 3 (b / l ) EEx2 ) to the CVC. This

Hy Ex

Ex2,

4

In the equation above, the coefficients 3 and 4 are given by (40) and (41). Fig. 4 shows the nonequilibrium electron temperature versus thickness z. The parameters used are T0/kB = 200 K , sample with m p = 0.26m 0 , n0 = 2.48 × 10 9 s, p0 = 2.48 × 10 9 s, b = 0.02cm, m n n 0 p0 = 1.45 × 1014 cm−3, E x = 0.8Vcm−1, qn = qp = 1/2 . The dotdashed purple, brown, cyan and dashed black lines represent the cases e) f) g) a) 1 = 2 = 0, 1 = 2 = 100 and 1 = 2 = 1, respectively. The green line represent the case b) 1 = 2 1 = 2 = 0.0001, it implied that the coefficient 4 = 0 and the electron temperature is maximum in the center of the sample and tend to cooling at the surfaces z = ± b. The blue and red lines represent the cases c) 1 = 1, 2 = 0 , and d) 1 = 0, 2 = 1respectively. The nonlinear density of current along the x-axis takes the form,

Fig. 3 shows the electron temperature versus thickness z in absence of volumetric and surface recombination processes. We consider a ntype semiconductor sample with numerical values given by T0/kB = 200 K , sample with b = 0.02cm, m n = 0.2m 0 , n0 = 2.48 × 10 9 s, n 0 = 1.45 × 1014 cm−3, E x = 1Vcm−1, Hy = 20000 Oe, Nt = 0 , qn = qp = 1/2 . The red and blue lines represent the cases b) 1 = 1, 2 = 0 and d) 1 = 0, 2 = 1respectively. The brown, green and black lines represent the cases a) 1 = 2 = 0.0001, c) 1 = 2 = 1, and e) 1 = 2 = 100 respectively. In this case, we reach a temperature difference Tn = 7.16 K . The electrical conductivity is given by 2 1 Hy

cosh(z / l )

and nonequilibrium electron concentration

n (z ) =

2),

3

(70)

In the second and third term, we can see the effect of the redistribution of the electron concentration and temperature in the magnetoresistance effect. If the coefficients are given as 1 = 2 then 2 = 0 and 1 = a1/(a 0 + b 1) . We can see a size dependence in the coefficient 0 the coefficient takes the value 1 = a1/ a 0 and when 1when b b the coefficient value is 1 = 0 . If both surfaces have good thermal conductivity 1,2 , then the coefficients 1 = 2 = 0 . In the electrical conductivity, the contribution to the magnetoresistance disappears 2 1 Hy as well as the contribution proportional to Hy Ex . 0 , the coefficients Finally in the case of an adiabatic system 1,2 are 2 = 0 and 1 = a1/ a 0 . This implies that the nonequilibrium temn0 E 2 2 z

xx x + 1 Hy E x z . But, this physically is not perature is Tn (z ) = a0 2 possible, we have an adiabatic system (closed system) and we do not have electron-phonon interaction (we do not have loss of energy), which means that the system absorbs energy from the fields and is continuously heating. In this case, the nonequilibrium temperature goes

Fig. 4. Nonequilibrium electron temperature versus thickness z. The purple, green, blue, red, brown, cyan and black lines represent the cases a) 1 = 2 = 0 , b) 1 = 2 = 0.0001, c) 1 = 1, 2 = 0 , d) 1 = 0, 2 = 1, e) 1 = 2 = 1, f) respectively. 1 = 2 = 100 and g) 1 = 2 118

International Journal of Thermal Sciences 137 (2019) 110–120

S. Molina-Valdovinos

new contribution comes from the dependence of the width of the sample and the relaxation mechanism over the surfaces of the system. Experimental measurements of CVC where studied in n-Si [45] samples and p-Ge [46] at 77°K. The effect of energy relaxation of carriers at the surfaces was studied in Refs. [45,47]. In the case of an adiabatic system, 1 = 2 = 0 . This implies that 3 = 4 = 0 . In an adiabatic system, the electron system is thermally insulated at the surfaces and there is not a mechanism for energy relaxation of carriers at the surfaces, i.e., the mechanisms of surface recombination are absent and the electron heat flux is null over the surfaces. The nonequilibrium temperature for electrons is constant n0 E 2

along the z-axis, Tn (z ) = EEx2 = n xx (Tx ) . See the dot-dashed purple line 0 n0 0 in Fig. 4. On the other hand the nonequilibrium electron concentration n0 E 2 2 3 xx x

is n (z ) =

Jx =

0 [1

. The nonlinear density of current takes the form

n 0 n0 (T0) 2 a2 EEx2] Ex

+

(74)

If a2 < 0 then the CVC is sublinear. In the case of a2 > 0 , the CVC is superlinear. Finally, if a2 = 0 then the CVC is linear (Ohm Law). A simple case to see the behavior of the electrical conductivity is when the frequencies of the electrons and holes are equal, i.e. n0 = p0 (qn = qp) , and the masses of the electrons and holes are equal, m n = m p . This n1 p1 n1 p1 n0 p0 n0 p0 n0 p0 = xx = Dzz = xx = xz = Dzz , xx , xz , Dzz , Dzz , implies that xx n0 p0 Dxz = Dxz . In this case the coefficient a2 is

a2 =

(qn + 2) 0 T0

n0 xx

(75)

then if qn < 2then a2 > 0 , and the CVC is superlinear. In the case of qn > 2 then a2 < 0 , and the CVC is sublinear. Finally, if qn = 2 then a2 = 0 , and the CVC is linear (Ohm Law). If the surfaces (z = ± b) have good thermal conductivity 1,2 , cosh(z / l ) cosh(b / l )

then Tn (z ) = EEx2 1

Fig. 5. Electrical conductivity versus electric field. a) n 0 = p0 and b) n 0 p0 . The black, red, green, and blue lines represent the cases i) 1 = 2 = 0 , ii) , and iv) l respectively. 1 = 2 = 0.0001, iii) 1 = 2

. In Fig. 4, the brown, cyan and da-

shed black lines represent the cases e) 1 = 2 = 1, f) 1 = 2 = 100 and g) 1 = 2 respectively. The electron temperature is maximum in the center of the sample and in the surfaces (z = ± b) dissipate the heat over the surfaces with the surrounding medias quickly and thermalize to the equilibrium temperature T0 . In this case the coefficients 3, 4 change by 4 = 0 and 3 = 1/cosh(b / l ) . The electrical conductivity takes the form

Jx =

0

tanh(b / l ) EEx2 Ex , (b / l )

1 + a2 1

The nonlinear density of current is transformed to become

Jx =

then

tanh(b / l ) (b / l ) + a2 EEx2]

(76)

0 and density of current is transformed to

Ex . become Jx = 0 [1 Now, under the condition 1,2 and if we do not consider electron-phonon interaction (l ) the electrical conductivity becomes Jx = 0 E x (Ohm law). If we have strong electron-phonon interaction 0 ) electrical conductivity is transformed to become (l

Jx =

0

tion

(l )

Jx =

0

n0 b 2 a2 xx Ex2 3a0

1+

the

(

1 (b / l )

1 + a2 1

)

EEx2

conductivity

is

transformed

n (z ) =

n0 2 xx Ex

a0 n0 2 3 xx Ex

2a 0

2

z2 , 2 z2

into

Ex .

(77)

b2 , 3

n0 2 xx b

6a 0

Ex2 E x ,

(79)

In this work we studied the case of heating or cooling of a semiconductor sample in presence of external electric and magnetic fields. We find that under weak recombination there is a relation between nonequilibrium electron concentration and nonequilibrium electron temperature. The nonequilibrium electron temperature and electron concentration depend on the z-coordinate and the heating mechanisms considered in the model. The nonequilibrium electron temperature and electron concentration have a term proportional to Hy Ex and another term proportional to Ex2 . For example, if we take into account the Joule heating and do not consider the heating due to electron-phonon interaction, the nonequilibrium temperature and electron concentration are quadratic functions of the z-coordinate. If we do not consider heating due to electron-phonon interaction and Joule heating, the nonequilibrium temperature and electron concentration are linear functions of the z-coordinate. If we consider an adiabatic system and take into account electron-phonon interaction and Joule heating, the

If the surface heat conductivity is not zero ( 1,2 0 ) and we have no 0 (l electron-phonon interaction, then 0 ). The nonequilibrium electron temperature and the nonequilibrium electron concentration are,

Tn (z ) =

a2

4. Conclusion

E x . If we have large electron-phonon interac-

electrical

1

0 ) the contribution to the when the size of the sample goes to zero (b density of current is transformed into Ohm's law, Jx = 0 E x . Fig. 5 shows the electrical conductivity versus electric field. a) n 0 = p0 = 1.45 × 1014 cm−3, m n = m p = 0.26m 0 and n0 = b) 1.45 × 1014 cm−3, p0 = 1.45 × 106 cm−3, m n = 0.26m 0 , m p = 0.6m 0 . Fig. 5a–b shows the cases i) 1 = 2 = 0 , ii) 1 = 2 = 0.0001, iii) , and iv) l that correspond to equations (73), (74), 1 = 2 (76) and (79) respectively.

0 ), the contribution when the size of the sample goes to zero, i.e., (b tanh(b / l ) 1 and the density of current is transformed into Ohm's law, (b / l ) Jx = 0 E x . In the opposite case if the sample has a large transverse section b

0

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S. Molina-Valdovinos

nonequilibrium temperature is an antisymmetric function with respect to the z-axis. It is shown that the nonequilibrium temperature and charge carriers modify the density of current from linear to nonlinear. The first change to the current density comes with the appearance of magnetoresistance effect. The redistributions of carriers originate two new contributions to the magnetoresistance, one of the contributions is linked to the diffusion of charge carriers along the z-axis, the other one is related to the thickness 2b of the sample and the heat absorption over the surfaces. It was shown that the density of current exhibits linear dependence on the magnetic field Hy Ex ; this coefficient originates due to the absorption of heat over the surfaces of the sample z = ± b. This contribution can increase or decrease the conductivity, if (i) the electric or magnetic field is inverted, or (ii) the heat absorption over one surface (z = + b ) increases or decreases with respect to the absorption of the other surface (z = b ). When the absorption of heat over the surfaces are equal, this contribution disappears. When the magnetic field is null Hy = 0 , the current-voltage characteristic (CVC) deviates from the linear response due to the Joule heating. We find that the CVC has two contributions associated to the nonlinearity, the first contribution a2 EEx2 is due to the Joule heating, the tanh(l b) second contribution a2 l b EEx2 is due to the geometry and boundary conditions. It is worthwhile to stress that the model of the weak recombination above is by no means a general one. It can be refined by considering the volumetric recombination, the interface scattering, and surface recombination processes. This processes affect the heat flux energy and the continuity equation and hence the nonequilibrium distribution of charged carriers and the nonequilibrium distribution of energy.

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