Nonlinear Transport and Optical Response

A nonlinear material response is the part of a current or polarization that is higher than first order in the applied field. It can produce rectification, harmonic generation and a transverse current even when the linear Hall response vanishes. A measured harmonic identifies a frequency channel; identifying its microscopic origin also requires the sample symmetry, a controlled weak-drive regime and a model of relaxation and measurement. This page develops that distinction, concentrating on second-order electric response about a stationary material state.

Helpful background. Kubo Formula in Materials supplies physical currents and contact terms; Symmetry of Bloch States supplies crystal transformations. The Berry-dipole calculation uses Berry Curvature in Band Dynamics and Boltzmann Transport. Nonlinear Quantum Optics explains polarization and propagation, while Transport Measurements supplies contact and background tests.

Response order and frequency mixing

Use SI units and the Fourier convention

$$\begin{aligned} E_a(t)&=\int\frac{d\omega}{2\pi}E_a(\omega)e^{-i\omega t},\\ E_a(-\omega)&=E_a(\omega)^*. \end{aligned}$$

Write $j=j^{(1)}+j^{(2)}+\cdots$ for the change from the reference-state current. Define the second-order conductivity without a factorial:

$$j_a^{(2)}(\nu)={}\sum_{bc}\int\frac{d\omega}{2\pi}\, \sigma_{abc}^{(2)}(\nu;\omega,\nu-\omega) \times E_b(\omega)E_c(\nu-\omega).$$

Thus the output frequency is the sum of the input frequencies. Equivalently, two input-frequency integrals use the factor $2\pi\delta(\nu-\omega_1-\omega_2)$. A coefficient defined as the unscaled second functional derivative of the current requires an additional $1/2!$ in the expansion. Both conventions are possible; their numerical coefficients differ.

The commuting classical input fields allow the observable kernel to be chosen with intrinsic permutation symmetry:

$$\sigma_{abc}^{(2)}(\nu;\omega_1,\omega_2) =\sigma_{acb}^{(2)}(\nu;\omega_2,\omega_1).$$

This exchanges an input index together with its frequency. It does not exchange the measured-current index $a$ with an input index. Reality additionally requires

$$\sigma_{abc}^{(2)}(-\nu;-\omega_1,-\omega_2) =\sigma_{abc}^{(2)}(\nu;\omega_1,\omega_2)^*.$$

Causality says that an input later than the measured current cannot affect it. Stationarity makes the kernel depend on time differences and gives frequency addition. A finite pulse broadens the input spectrum; it does not by itself violate frequency addition. A reference state evolving under a strong pump requires a more general response description. Full permutation of all tensor indices, sometimes used far from optical resonances, must not be imposed without its extra assumptions.

For bulk current density, $[\sigma^{(2)}]=\mathrm{A/V^2}$. For a two-dimensional sheet current, $[\sigma^{(2)}]=\mathrm{A\,m/V^2}$. A chosen thickness converts between the two; it is part of the reported coefficient.

Rectification and the second harmonic

Let $\mathbf E(t)=\operatorname{Re}(\boldsymbol{\mathcal E}e^{-i\omega t})$, where $\boldsymbol{\mathcal E}$ is a peak phasor. Write

$$\mathbf j^{(2)}(t)=\mathbf j_{\rm dc} +\operatorname{Re}(\boldsymbol{\mathcal J}_{2\omega}e^{-2i\omega t}).$$

The preceding normalization gives

$$\begin{aligned} j_{{\rm dc},a}&=\frac12\operatorname{Re}\sum_{bc} \sigma_{abc}^{(2)}(0;\omega,-\omega) \mathcal E_b\mathcal E_c^*,\\ \mathcal J_{a,2\omega}&=\frac12\sum_{bc} \sigma_{abc}^{(2)}(2\omega;\omega,\omega) \mathcal E_b\mathcal E_c. \end{aligned}$$

For the scalar instantaneous law $j=\beta E^2$ and $E=E_0\cos\omega t$, both the dc current and the second-harmonic peak are $\beta E_0^2/2$. Using rms rather than peak fields changes the stated prefactors. With two incident frequencies, sum and difference channels coexist; one observed harmonic can contain more than one mixing process.

For a bounded bulk polarization contribution, $\mathbf j=\partial_t\mathbf P$ implies, at nonzero output frequency and with the same susceptibility normalization,

$$\sigma^{(2)}(\nu;\omega_1,\omega_2) =-i\nu\epsilon_0\chi^{(2)}(\nu;\omega_1,\omega_2).$$

A static optical-rectification polarization is therefore different from a steady dc conduction current. Absorptive photocurrent may involve a singular zero-output-frequency limit; substituting $\nu=0$ into a regular bound-polarization susceptibility does not exclude it. Propagation, phase matching and the field inside a sample are separate from the local coefficient; see Nonlinear Quantum Optics and Optical Conductivity.

Symmetry of the actual sample state

For a spatial symmetry represented by an orthogonal matrix $R$, a local electric-dipole charge-response tensor of a symmetry-invariant state obeys

$$\sigma_{abc}^{(2)} =\sum_{a'b'c'}R_{aa'}R_{bb'}R_{cc'} \sigma_{a'b'c'}^{(2)}.$$

Inversion changes the sign of three polar indices, so it forces this bulk second-order tensor to vanish in an inversion-invariant state. To see the conflict directly, inversion reverses the electric field and current, but a quadratic term $E_bE_c$ keeps its sign. The same invariant sample cannot give both answers unless that term vanishes. A surface, interface, applied bias or ordered domain can have less symmetry than the parent crystal. Spatial-dispersion, magnetic-dipole and electric-quadrupole responses also have different transformation rules. Second-harmonic light from a centrosymmetric specimen does not, by itself, establish inversion breaking throughout its bulk.

Consider a two-dimensional state with mirror $M_x:(x,y)\mapsto(-x,y)$. A component survives only when its three indices contain an even number of $x$ entries. The permitted components are

$$\sigma_{xxy}^{(2)},\quad\sigma_{xyx}^{(2)},\quad \sigma_{yxx}^{(2)},\quad\sigma_{yyy}^{(2)}.$$

In particular, $j_y^{(2)}\propto E_x^2$ is allowed, while $j_x^{(2)}\propto E_x^2$ is forbidden. Further symmetries can remove more components. Time reversal is antiunitary and is not another polar-space matrix in this rule. It does not forbid every second-order transverse response. The linear Onsager relation cannot simply be reused as a symmetry between three nonlinear indices.

Here nonlinear Hall means nonlinear in the electric drive. A curved Hall-voltage-versus-magnetic-field trace can already occur in a response linear in electric field; see Hall Effect. A transverse contact voltage can also result from ordinary anisotropy. A Hall contribution to a dc constitutive law is perpendicular to the electric field for every field direction, not merely transverse at one chosen contact angle.

A different allowed effect in an isotropic chiral medium is the phenomenological magnetochiral term

$$\mathbf E=\rho_0\mathbf j +\gamma(\mathbf B\cdot\mathbf j)\mathbf j.$$

It is longitudinal, quadratic in current and odd in magnetic field; $\gamma$ changes sign between enantiomers. This restricted constitutive example, discussed experimentally by Rikken, Fölling and Wyder (2001), is distinct from a zero-field Berry-dipole Hall current. It is not a universal formula for nonreciprocal transport.

A gauge-consistent material calculation

The nonlinear current must be obtained from the same electromagnetic coupling as the Hamiltonian. In the fixed physical orbital embedding and Peierls approximation of Kubo Formula in Materials, use signed charge $q$ and

$$\begin{aligned} h_{\mathbf A}(\mathbf k)&=h(\mathbf k-q\mathbf A/\hbar),\\ \widehat j_a[\mathbf A]&=-\frac1V\frac{\partial H[\mathbf A]}{\partial A_a}. \end{aligned}$$

For one field direction, suppressing orbital indices and writing primes for wavevector derivatives, the current operator contains

$$\widehat j[A]={}\frac{q}{\hbar V}\sum_{\mathbf k}c^\dagger h'c -\frac{q^2A}{\hbar^2 V}\sum_{\mathbf k}c^\dagger h''c +\frac{q^3A^2}{2\hbar^3 V}\sum_{\mathbf k}c^\dagger h'''c+\cdots.$$

At second order in $A$, one needs the second-order change of state multiplying the first term, the first-order change multiplying the second term, and the explicit third term. Equivalently, the Hamiltonian must supply vertices through cubic order. A triple correlator of the zero-field paramagnetic current alone is generally incomplete. Since $E(\omega)=i\omega A(\omega)$, omitted cancellations can become spurious low-frequency divergences.

Length-gauge and velocity-gauge approaches can both be consistent. Aversa and Sipe (1995) treat the separation of physical and unphysical low-frequency behavior; Parker et al. (2019) develop a systematic velocity-gauge construction. In a wavevector-dependent orbital representation, the coupling needs the corresponding basis connection and projected position elements. Taking ordinary derivatives of an arbitrary truncated band matrix is not a substitute. A phenomenological linewidth cannot repair missing vertices or an inconsistent projection. Population relaxation and optical-coherence decay also need not have the same rate.

A useful check is that a static uniform pure gauge produces no equilibrium bulk current when the state and Hamiltonian are transformed together. A ring threaded by physical flux or a system with occupations held out of equilibrium is a different boundary-value problem. Check the appropriate limit, not a formal cancellation divorced from the state.

Shift and injection photocurrents

Absorption can transport charge by changing a wave packet’s center or by creating an asymmetric population of moving carriers. These are the shift and injection mechanisms in the independent-particle treatment of Sipe and Shkrebtii (2000). The broad term photogalvanic current describes a light-generated dc current; it does not name a third mutually exclusive mechanism.

Displacement during an optical transition

For isolated valence and conduction bands, use the inherited Berry connection $\mathcal A_n^a=i\langle u_n|\partial_{k_a}u_n\rangle$. Let $M_{cv}(\mathbf k)$ be the interband transition amplitude for a fixed incident polarization, proportional to $\sum_b\mathcal E_b r_{cv}^b$, where $r_{cv}^b=i\langle u_c|\partial_{k_b}u_v\rangle$. Away from zeros of that amplitude, the displacement is

$$R_{cv}^a=\mathcal A_c^a-\mathcal A_v^a -\partial_{k_a}\arg M_{cv}.$$

Its sign follows directly from a packet center $x_n^a=\mathcal A_n^a-\partial_{k_a}\arg a_n$: a transition with $a_c\propto M_{cv}a_v$ changes the center by the displayed $R_{cv}^a$. Under $u_n\mapsto e^{i\theta_n}u_n$, the Berry connection changes by $-\partial\theta_n$ and the transition amplitude by $e^{i(\theta_v-\theta_c)}$; these changes cancel in $R_{cv}$. At a true amplitude zero, use the nonsingular rate-weighted expression rather than differentiating an undefined phase. In a smooth local band frame, it contains the combination

$$|M_{cv}|^2R_{cv}^a={}|M_{cv}|^2(\mathcal A_c^a-\mathcal A_v^a) -\operatorname{Im}(M_{cv}^*\partial_{k_a}M_{cv}).$$

Band-gauge patches handle frame singularities separately; they cannot make a vanishing transition amplitude nonzero.

In a weak, independent-transition picture the shift current density has the transparent form

$$j_{{\rm shift},a} =q\sum_{cv}\int_{\rm BZ}\frac{d^d k}{(2\pi)^d} W_{cv}(\mathbf k)R_{cv}^a(\mathbf k).$$

Here $W_{cv}$ is the net excitation rate per momentum state, including occupations and the reverse process when relevant. Each transferred electron has $q=-e$, so positive electron displacement gives negative electric current. Spin or other degeneracy is counted by the resolved bands. A Golden Rule calculation with peak phasors uses the positive-frequency perturbation amplitude $-q\boldsymbol{\mathcal E}\cdot\mathbf r/2$; dropping this half changes the rate by four. This picture fixes interpretation and conventions; it is not a universal interacting formula.

Population injection and its steady limit

An optical transition can instead inject unequal velocities, producing a current-generation rate $\mathcal R_a$. A minimal relaxation model is

$$\frac{dj_{{\rm inj},a}}{dt} =\mathcal R_a-\frac{j_{{\rm inj},a}}{\tau_{\rm inj}}.$$

For a constant rate switched on at $t=0$ with zero initial current,

$$j_{{\rm inj},a}(t) =\mathcal R_a\tau_{\rm inj}(1-e^{-t/\tau_{\rm inj}}).$$

The long-time current is proportional to a relaxation time. In the clean limit at fixed time it is $\mathcal R_at$, a growing population contribution. A calculated clean injection rate therefore does not specify an infinite-time stationary conductivity. Saturation, recombination, extraction and heating can all alter the illumination regime; they must be stated before comparing coefficients.

The Berry-curvature-dipole limit

There is a controlled low-frequency mechanism in an inversion-broken, time-reversal-invariant metal even without real interband absorption. Work at zero magnetic field with isolated bands, a weak spatially uniform field, fixed equilibrium occupations $f_{0n}$ and a constant relaxation time $\tau$. With $q=-e$,

$$\begin{aligned} \hbar\dot{\mathbf k}&=q\mathbf E,\\ \mathbf v_{\rm an}&=-\frac q\hbar\mathbf E\times\boldsymbol\Omega_n. \end{aligned}$$

Time reversal makes the equilibrium Berry-curvature integral vanish. The field changes the occupation by a first-harmonic phasor

$$f_{1n}^{\omega} =-\frac{q\tau}{\hbar(1-i\omega\tau)} \boldsymbol{\mathcal E}\cdot\nabla_{\mathbf k}f_{0n}.$$

Multiplying this occupation change by the anomalous velocity gives a second-order current. Define the Berry-curvature dipole

$$\begin{aligned} D_{bd}&=\sum_n\int_{\rm BZ}\frac{d^d k}{(2\pi)^d} f_{0n}\,\partial_{k_b}\Omega_{n,d}\\ &=-\sum_n\int_{\rm BZ}\frac{d^d k}{(2\pi)^d} (\partial_{k_b}f_{0n})\Omega_{n,d}. \end{aligned}$$

The second equality is periodic integration by parts. It exhibits the Fermi-surface character at low temperature. An entirely filled smooth band gives no such dipole. The first-order occupation is odd in momentum; multiplying the odd Berry curvature can therefore give an even integrand. In this same time-reversal-invariant constant-$\tau$ model, the second-order occupation is even and its group-velocity current vanishes by parity.

For a two-dimensional sheet, set $D_b=D_{bz}$. The second-harmonic sheet current is

$$\boldsymbol{\mathcal J}_{2\omega} =\frac{q^3\tau}{2\hbar^2(1-i\omega\tau)} (\mathbf D\cdot\boldsymbol{\mathcal E}) (\hat{\mathbf z}\times\boldsymbol{\mathcal E}).$$

For real linearly polarized $\boldsymbol{\mathcal E}$, its dc companion is

$$\mathbf j_{\rm dc} =\frac{q^3\tau}{2\hbar^2(1+\omega^2\tau^2)} (\mathbf D\cdot\boldsymbol{\mathcal E}) (\hat{\mathbf z}\times\boldsymbol{\mathcal E}).$$

These signs follow from the preceding signed-charge equations. For $D_x>0$ and electron driving along $x$, both the dc $y$ current and the real part of the second-harmonic $y$ phasor are negative. The dipole has units of length in two dimensions, giving the sheet-conductivity units stated above. The mirror $M_x$ permits $D_x$ but forces $D_y=0$: curvature is axial, while its momentum derivative supplies a polar index.

Sodemann and Fu (2015) establish this mechanism. Its controlled use requires a field-induced momentum displacement small on the scale over which the distribution and band geometry vary, frequencies below relevant interband transitions, and a justified collision approximation. It is not quantized. Disorder-mediated contributions can have the same relaxation-time scaling, as Nandy and Sodemann (2019) show. A fitted nonlinear Hall coefficient or a power of the mobility does not uniquely measure $\mathbf D$.

From a coefficient to a measured harmonic

A calculated current at prescribed internal electric field is not yet an open-circuit voltage. For example, if the transverse linear conductivity is diagonal and finite, the second-order open-circuit constraint gives

$$\mathcal E_{y,2\omega} =-\frac{\mathcal J^{\rm source}_{y,2\omega}} {\sigma^{(1)}_{yy}(2\omega)}.$$

This assumes the generated transverse field is already second order, the longitudinal drive is fixed, and other circuit currents are negligible. A coupled tensor, finite capacitance or optical boundary problem needs its full boundary conditions. The voltage sign additionally requires the contact ordering and path.

Joule heating is an especially useful alternative explanation to calculate explicitly. A current $I_0\cos\omega t$ heats at dc and $2\omega$. Temperature-dependent resistance then mixes those components with the fundamental current to create a $3\omega$ voltage. A temperature gradient and contact thermopower can instead generate a $2\omega$ thermoelectric voltage; a spatially uniform temperature rise alone is not a Seebeck field. The quantitative thermal example below produces a cubic signal without an intrinsic electronic cubic coefficient.

Amplitude, phase and symmetry should agree with one mechanism over the same weak-drive range. Changing thermal coupling can test a heating interpretation, while contact permutations, thickness and polarization can distinguish bulk response from interfaces and circuit mixing. Frequency dependence alone is inconclusive because intrinsic and thermal responses both have time scales.

Hysteresis, switching or a drive-induced phase change means the reference state itself has changed; one stationary Taylor expansion may no longer apply. Thermoelectric Response develops the thermal cross coefficients, and Driven Many-Body Systems discusses absorption and energy exchange with a bath. A claim of unusually large or quantized response still needs the material-specific mechanism and its uncertainties.

Exercises

1. Two drives in one quadratic response

In specified field and current units, let $j=3E^2$ and $E(t)=2\cos\omega t+0.8\cos3\omega t$. Find the dc current and peak amplitudes at $2\omega$, $4\omega$ and $6\omega$. Which output contains both sum and difference processes?

Solution

Expanding with $2\cos a\cos b=\cos(a-b)+\cos(a+b)$ gives

$$j(t)={}6.96+10.8\cos2\omega t +4.8\cos4\omega t+0.96\cos6\omega t.$$

The $2\omega$ amplitude contains $6$ from doubling the fundamental and $4.8$ from the difference $3\omega-\omega$. Its magnitude cannot be assigned to the fundamental alone. All listed oscillating amplitudes are peak amplitudes, not rms values.

2. A mirror-allowed transverse current

For the mirror $M_x$, recover the allowed tensor components and the permitted dipole direction. In units with $q=-1$, $\hbar=\tau=1$, take $D_x=0.07$, $\omega=0.5$ and $\boldsymbol{\mathcal E}=0.1\hat{\mathbf x}$. Find $j_{{\rm dc},y}$ and $\mathcal J_{y,2\omega}$. If the transverse linear sheet conductivity is $2$ in compatible units, real and frequency independent, find the open-circuit $2\omega$ transverse field.

Solution

An even number of $x$ indices permits $xxy$, $xyx$, $yxx$ and $yyy$. Because $\Omega_z$ changes sign under this mirror, $\partial_{k_x}\Omega_z$ is even and $\partial_{k_y}\Omega_z$ is odd: $D_x$ may survive and $D_y$ vanishes. Substitution gives

$$\begin{aligned} j_{{\rm dc},y}&=-0.00028,\\ \mathcal J_{y,2\omega}&=-0.00028-0.00014i,\\ \mathcal E_{y,2\omega}&=0.00014+0.00007i. \end{aligned}$$

The transverse field opposes the source current under the stated open-circuit constraint. Its contact voltage still depends on which contact is subtracted from which. Neither the mirror test nor this calculation proves that a measured signal has only a Berry-dipole origin.

3. Gauge phases and injection time

Verify that $R_{cv}^a$ is invariant under independent conduction- and valence-band phase choices. Then solve the injection equation for a constant rate and compare the clean limit at fixed observation time with the steady limit at fixed relaxation time. Why must the illumination protocol be part of a quoted photocurrent?

Solution

The Berry-connection difference changes by $-\partial_{k_a}\theta_c+\partial_{k_a}\theta_v$. The negative derivative of the transition phase changes by $-\partial_{k_a}(\theta_v-\theta_c)$, canceling it. This is the band-gauge transformation introduced in Berry Curvature; physical displacement cannot depend on the choice of eigenvector phases.

An integrating factor gives $j(t)=\mathcal R\tau(1-e^{-t/\tau})$. At fixed $\tau$, $t\gg\tau$ gives $j=\mathcal R\tau$. At fixed finite $t$, $\tau\to\infty$ gives $j=\mathcal Rt$. There is no finite universal clean stationary current in this rate model. A shift contribution $q\int W R$ measures displacement per net transition, whereas injection also accumulates a velocity imbalance. Comparing either with experiment requires the actual rates, occupations, relaxation and extraction during the observation window.

4. A thermal third harmonic

Let a resistor’s temperature rise $\theta$ obey

$$\begin{aligned} C_{\rm th}\dot\theta+K_{\rm th}\theta&=R_0I^2,\\ I(t)&=I_0\cos\omega t,\\ R(T)&=R_0+R_T\theta. \end{aligned}$$

Assume $|R_T\theta|\ll R_0$ and retain voltage through order $I_0^3$. Use $R_0=100\,\Omega$, $R_T=0.2\,\Omega/\mathrm K$, $K_{\rm th}=10^{-4}\,\mathrm{W/K}$, $C_{\rm th}=10^{-5}\,\mathrm{J/K}$, $I_0=1\,\mathrm{mA}$ and angular frequency $\omega=5\,\mathrm{s^{-1}}$. Find the stationary temperature response and third-harmonic voltage phasor. Does cubic scaling demonstrate an intrinsic electronic third-order response?

Solution

Write $\theta=\theta_{\rm dc}+\operatorname{Re}(\Theta_{2\omega}e^{-2i\omega t})$. With $\tau_{\rm th}=C_{\rm th}/K_{\rm th}$,

$$\begin{aligned} \theta_{\rm dc}&=\frac{R_0I_0^2}{2K_{\rm th}}=0.5\,\mathrm K,\\ \Theta_{2\omega}&=\frac{\theta_{\rm dc}}{1-2i\omega\tau_{\rm th}} =(0.25+0.25i)\,\mathrm K. \end{aligned}$$

Multiplying $R(T)$ by $I(t)$ yields

$$\begin{aligned} \mathcal V_{3\omega}&=\frac{R_TI_0\Theta_{2\omega}}2 =(25+25i)\,\mu\mathrm V,\\ \mathcal V_{\omega}&=R_0I_0+R_TI_0(\theta_{\rm dc}+\Theta_{2\omega}/2)\\ &=(0.100125+0.000025i)\,\mathrm V. \end{aligned}$$

The third harmonic scales as $I_0^3$ through temperature-dependent linear resistance. Its thermal phase and roll-off follow from $\tau_{\rm th}=0.1\,\mathrm s$. Feeding the small temperature correction back into the heating power would first change the voltage at order $I_0^5$, beyond the retained approximation. Cubic scaling alone therefore does not identify an intrinsic electronic nonlinearity.

References

• C. Aversa and J. E. Sipe, “Nonlinear optical susceptibilities of semiconductors: Results with a length-gauge analysis,” Physical Review B 52, 14636–14645 (1995), doi:10.1103/PhysRevB.52.14636.
• S. Nandy and I. Sodemann, “Symmetry and quantum kinetics of the nonlinear Hall effect,” Physical Review B 100, 195117 (2019), doi:10.1103/PhysRevB.100.195117, author preprint.
• D. E. Parker, T. Morimoto, J. Orenstein and J. E. Moore, “Diagrammatic approach to nonlinear optical response with application to Weyl semimetals,” Physical Review B 99, 045121 (2019), doi:10.1103/PhysRevB.99.045121, author preprint.
• G. L. J. A. Rikken, J. Fölling and P. Wyder, “Electrical magnetochiral anisotropy,” Physical Review Letters 87, 236602 (2001), doi:10.1103/PhysRevLett.87.236602.
• J. E. Sipe and A. I. Shkrebtii, “Second-order optical response in semiconductors,” Physical Review B 61, 5337–5352 (2000), doi:10.1103/PhysRevB.61.5337.
• I. Sodemann and L. Fu, “Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials,” Physical Review Letters 115, 216806 (2015), doi:10.1103/PhysRevLett.115.216806, author preprint.